Failure Criteria in Fibre-Reinforced-Polymer Composites

Failure Criteria in Fibre Reinforced Polymer Composites: The World-Wide Failure Exercise

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Failure Criteria in Fibre Reinforced Polymer Composites: The World-Wide Failure Exercise This volume contains 34 previously published papers from the journal Special Issue Composites Science and Technology and four new papers

Edited by

M.J. Hinton QinetiQ Fort Helstead, Kent, UK

A.S. Kaddour QinetiQ Farnborough, Hampshire, UK

P.D. Soden UMIST, Manchester, UK

2004

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First edition 2004

Library of Congress Cataloging in Publication Data A catalog record is available from the Library of Congress. British Library Cataloguing in Publication Data Failure Criteria in Fibre Reinforced Polymer Composites: The World-Wide Failure Exercise 1. Polymeric composites – Fatigue 2. Fibrous composites – Fatigue 3. Laminated materials – Fatigue I. Hinton, M. J. II. Kaddour, A. S. III. Soden, P. D. 620.1'92'0287 ISBN: 0-08-044475-X

䊊 The paper used in this publication meets the requirements of ANSI/NISO Z39.48–1992 (Permanence of Paper). Printed in The Netherlands.

Preface Fibre Reinforced Polymer composite materials (FRP’s) are now widely utilised in many applications including aircraft, yachts, motor vehicles, chemical and process plant, sporting goods and a wide range of military equipment. They are an extremely broad and versatile class of material, encompassing a wide range of fibre and matrix combinations that provide a multiplicity of component design and manufacturing options. Their high strength coupled with light weight leads to their use wherever structural efficiency is at a premium. FRP’s are inherently more complex than metals. By their nature, they are heterogeneous in construction (asymmetric arrays of many thousands of fibres, each with diameter of the order 10 microns, in a polymeric matrix) and they are anisotropic (the strength parallel to the fibres being typically two orders of magnitude greater than that in the transverse directions). Thus, it is perhaps not unsurprising to find that the challenge in predicting the strength of an FRP laminate with accuracy is significantly larger than that in predicting the strength of a conventional metal. In moving from the metals world to the FRP world, a structural designer is faced with many more variables and the need for an additional set of design methods. It is, perhaps, self evident that such methods must be accurate and valid in order to extract the maximum structural performance in terms of strength, deformation and stiffness. The consequences of using methods that have not been benchmarked against satisfactory data are potentially unsafe designs or over design, resulting in unnecessary cost and weight. In most of the early applications of FRP’s (typically military, in the 1960s) this challenge was circumvented by a ‘make and test’ approach which was entirely justified at the time, given the relative novelty of the materials, the absence of proven analytical tools and the relatively poor computation capabilities. Whilst much development work has been conducted since then (and continues to this day) the degree of maturity of the current tools for predicting the strength and deformation of an FRP material, in the general case, has been a somewhat open question. Over the last 12 years the editors of the book have organized and coordinated an international activity, known as the World Wide Failure Exercise, to improve the foundation on which design theories are based, namely the prediction of deformation and failure strength of laminated composite structures. Within the Exercise the leading failure theories for composite laminates have been compared with one another and with experimental data. As the Exercise progressed, the results were published in three special issues, of the international journal, ‘Composites Science and Technology’. The contributors of theoretical papers included many internationally renowned scientists, designers and engineers from six countries and experimental work was gathered from different groups in UK, USA and Germany. This book captures the totality of the Exercise, by bringing together all of the results, encompassing 19 failure theories, and assessing them together in a single volume. The overall strategy and procedures are outlined where each contributor describes their own theory and employs it to solve 14 challenging problems (test cases) set by the organisers. Full details are provided of the test cases, whereby the prediction of failure envelopes and stress strain curves for a range of practical laminates made of unidirectional carbon or glass fibre reinforced epoxy layers were required. The theoretical predictions are compared with one another and with experimental results for each test case. The accuracy of the predictions and the

vi performance of the theories are assessed and recommendations are made on the uses of the theories in engineering design. The unique study reported in this book is intended to serve as a fundamental reference work for university students, research institutions, software houses and design engineers working in the field of prediction of failure strength and deformation in composite structures. All of the necessary information is provided for the methodology to be readily employed for validating and benchmarking new theories, as they emerge. The editors are indebted to all who contributed experimental work to this Exercise (including S.R. Swanson (USA), R.M. Aoki and A.F. Johnson (Germany)) and particularly to the following who contributed the theories which form the critical mass of this book: S. W. Tsai with K. S. Liu and A. Kuraishi (USA) A. Puck with H. Schürmann (Germany) P. Zinoviev with S. V. Grigoriev, O. V. Labedeva and L. R. Tairova (Russia) T. A. Bogetti with C. P. R. Hoppel, V. M. Harik, J. F. Newill and B. P. Burns (USA) G. C. Eckold (UK) N. L. McCartney (UK) C. T. Sun with J. X. Tao (USA) C. C.Chamis with P. K.Gotsis and L. Minnetyan (USA) A. Rotem (Israel) E. C. Edge (UK) L. J. Hart-Smith (USA) W. E. Wolfe with T. S. Butalia (USA) S. J. Mayes with A. C. Hansen (USA) Z. M. Huang (China) R. G. Cuntze and A. Freund (Germany). This has been a truly collaborative effort, being totally reliant on the great perseverance, intellect and patience of the participants in committing their ideas to paper over a prolonged period of time. The editors would also like to acknowledge the support received from the UK MoD Corporate Research Programme and through the sustained programme of collaborative work at UMIST and at QinetiQ. Finally, we wish to acknowledge the encouragement by Bryan Harris, the Editor-in-Chief of the journal, ‘Composite Science and Technology’, at the time this study commenced and by his successor, Karl Schulte, both of whom have been instrumental in the publication process. It was a phone call from Bryan (plus a good lunch in London) that persuaded the editors to take on this Exercise in the first instance. Some 12 years later, we are pleased to announce the completion of the task! Editors:

M. J. Hinton

P. D. Soden

A. S. Kaddour

About the editors Professor Mike Hinton Mike Hinton is a QinetiQ Senior Fellow with more than 30 years of experience in programmes aimed at gaining a fundamental understanding of composite materials and translation into practical hardware. His work includes theoretical and experimental investigations into the response of polymer composite structures when subjected to a wide variety of loading conditions encompassing multi-axial, impact, high strain rate and thermal events. He has developed a number of components for volume production where composites have been utilised to give innovative solutions. He has collaborated extensively with UMIST on many projects and he is a visiting professor in the Department of Mechanical Aerospace and Manufacturing Engineering. Dr A S Kaddour Dr Kaddour is a senior scientist and a structural engineer with broad academic and industrial experience in design of lightweight structures, spanning more than 17 years. He possesses an in-depth understanding of the mechanics of composite materials and composite structures. His work includes modelling failure and improvement of validated computer codes for composites together with developing multi-axial rigs for testing composites tubes under a wide spectrum of loading. He obtained his Ph.D. in 1992 from the University of Manchester Institute of Science and Technology (UMIST) and joined QinetiQ in 1999. He worked in a number of projects sponsored by the UK MoD, DTI and aircraft, space and offshore industries. He has published and co-authored a number of articles in scientific journals. Mr P D Soden Peter Soden was a senior lecturer in the Department of Mechanical Engineering, at the University of Manchester Institute of Science and Technology (UMIST) until he retired in 1999. He is currently a visiting senior research fellow and part-time consultant at the same institution, which is about to be combined with another leading university to become The University of Manchester. For the last 30 years his research has been mainly into the behaviour of load bearing fibre reinforced polymer composite structures, including pressure vessels for use in the chemical and process industry, pipes and sandwich panels for use in the offshore oil industry and filament wound cylinders for a variety of applications. He has been joint supervisor of 30 research contracts and author of 90 papers and has acted as a consultant in the analysis of failure and design of a wide variety of components and structures.

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Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

About the editors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

Section 1

The World-Wide Failure Exercise: Its Origin, Concept and Content

1

Chapter 1.1

The world-wide failure exercise: Its origin, concept and content M.J. Hinton, A.S. Kaddour and P.D. Soden . . . . . . . . . . . . . . . . . . . . .

2

Section 2

Chapter 2.1

Chapter 2.2

Test Cases, Lamina Data and Experimental Results Under Biaxial Loads Lamina properties, lay-up configurations and loading conditions for a range of fibre reinforced composite laminates P.D. Soden, M.J. Hinton and A.S. Kaddour Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates: Failure exercise benchmark data P.D. Soden, M.J. Hinton and A.S. Kaddour . . . . . . . . . . . . . . . . . . . . .

Section 3

Description of the Individual Failure Theories by their Originators

Chapter 3.1

Prediction of composite laminate fracture: Micromechanics and progressive fracture P.K. Gotsis, C.C. Chamis and L. Minnetyan . . . . . . . . . . . . . . . . . . . . Failure criteria for use in the design environment G.C. Eckold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stress-based Grant–Sanders method for predicting failure of composite laminates E.C. Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Predicting transverse crack formation in cross-ply laminates L.N. McCartney . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Predictions of the original and truncated maximum-strain failure models for certain fibrous composite laminates L.J. Hart-Smith. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Predictions of a generalized maximum-shear-stress failure criterion for certain fibrous composite laminates L.J. Hart-Smith. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Failure analysis of FRP laminates by means of physically based phenomenological models A. Puck and H. Schürmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 3.2 Chapter 3.3

Chapter 3.4 Chapter 3.5

Chapter 3.6

Chapter 3.7

29

30

52 97

98 121

140 157

179

219

264

x Chapter 3.8 Chapter 3.9

Chapter 3.10 Chapter 3.11

Chapter 3.12 Chapter 3.13

Chapter 3.14

Chapter 3.15 Chapter 3.16

Chapter 3.17

Section 4

Prediction of laminate failure with the Rotem failure criterion A. Rotem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prediction of failure envelopes and stress/strain behavior of composite laminates C.T. Sun and J.X. Tao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A progressive quadratic failure criterion for a laminate K-S. Liu and S. W. Tsai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A strain-energy based failure criterion for non-linear analysis of composite laminates subjected to biaxial loading W.E. Wolfe and T.S. Butalia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The strength of multilayered composites under a plane-stress state P.A. Zinoviev, S.V. Grigoriev, O. V. Lebedeva and L. R. Tairova . . . . . . . Predicting the nonlinear response and progressive failure of composite laminates T.A. Bogetti, C.P.R. Hoppel, V.M. Harik, J.F. Newill and B.P. Burns . . . . The predictive capability of failure mode concept-based strength criteria for multidirectional laminates R.G. Cuntze and A. Freund. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Composite laminate failure analysis using multicontinuum theory J.S. Mayes and A.C. Hansen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A bridging model prediction of the ultimate strength of composite laminates subjected to biaxial loads Z.M. Huang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expanding the capabilities of the Ten-Percent Rule for predicting the strength of fibre–polymer composites L.J. Hart-Smith. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Comparative Study of Failure Theories and Predictions for Fibre Polymer Composite Laminates: Part (A)

298

316 334

353 379

402

429 490

518

597

643

Chapter 4.1

A comparative study of failure theories and predictions for fibre polymer composite laminates: Part (A) A.S. Kaddour, M.J. Hinton and P.D. Soden . . . . . . . . . . . . . . . . . . . . . 644

Section 5

Comparison Between the Individual Theoretical Predictions and Experimental Results

Chapter 5.1

Chapter 5.2 Chapter 5.3

702

Application of progressive fracture analysis for predicting failure envelopes and stress–strain behaviors of composite laminates: A comparison with experimental results P.K. Gotsis, C.C. Chamis and L. Minnetyan . . . . . . . . . . . . . . . . . . . . 703 Failure criteria for use in the design environment G.C. Eckold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726 A comparison of theory and experiment for the stress-based Grant–Sanders method E.C. Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739

xi Chapter 5.4

Chapter 5.5 Chapter 5.6

Chapter 5.7 Chapter 5.8

Chapter 5.9 Chapter 5.10

Chapter 5.11

Chapter 5.12

Chapter 5.13

Chapter 5.14

Chapter 5.15

Section 6

Chapter 6.1

Comparison between theories and test data concerning the strength of various fibre-polymer composites L.J. Hart-Smith. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prediction of ply crack formation and failure in laminates L.N. McCartney . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Failure analysis of FRP laminates by means of physically based phenomenological models A. Puck and H. Schürmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Rotem failure criterion: Theory and practice A. Rotem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The prediction of failure envelopes and stress/strain behavior of composite laminates: Comparison with experimental results C.T. Sun, J. Tao and A.S. Kaddour. . . . . . . . . . . . . . . . . . . . . . . . . . . A progressive quadratic failure criterion, part B A. Kuraishi, S.W. Tsai, K.S. Liu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . A strain-energy-based non-linear failure criterion: Comparison of numerical predictions and experimental observations for symmetric composite laminates T.S. Butalia and W.E. Wolfe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A coupled analysis of experimental and theoretical results on the deformation and failure of composite laminates under a state of plane stress P.A. Zinoviev, O.V. Lebedeva and L.P. Tairova . . . . . . . . . . . . . . . . . . . Predicting the nonlinear response and failure of composite laminates: Correlation with experimental results T.A. Bogetti, C.P.R. Hoppel, V.M. Harik, J.F. Newill and B.P. Burns . . . . The predictive capability of failure mode concept-based strength criteria for multi-directional laminates – Part B R.G. Cuntze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A comparison of multicontinuum theory based failure simulation with experimental results J.S. Mayes and A.C. Hansen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlation of the bridging model predictions of the biaxial failure strengths of fibrous laminates with experiments Z-M. Huang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Predictive Capabilities of Nineteen Failure Theories and Design Methodologies for Polymer Composite Laminates. Part B: Comparison With Experiments

770 810

832 877

890 903

922

943

961

976

1026

1045

1072

Predictive capabilities of nineteen failure theories and design methodologies for polymer composite laminates. Part B: Comparison with experiments A.S. Kaddour, M.J. Hinton and P.D. Soden . . . . . . . . . . . . . . . . . . . . . 1073

xii Section 7 Chapter 7.1

Recommendations for Designers and Researchers Resulting from the World-Wide Failure Exercise

1222

Recommendations for designers and researchers resulting from the world-wide failure exercise P.D. Soden, A.S. Kaddour and M.J. Hinton . . . . . . . . . . . . . . . . . . . . . 1223

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1253

Section 1 The World-Wide Failure Exercise: Its Origin, Concept And Content

2

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 QinetiQ Ltd. Published by Elsevier Ltd. All rights reserved

CHAPTER 1.1

The world-wide failure exercise: Its origin, concept and content M. J. Hintona, A. S. Kaddourb and P. D. Sodenc a b

QinetiQ, Fort Halstead, UK QinetiQ, Farnborough, UK c UMIST, Manchester, UK

Abstract This paper sets the scene for an investigation into the status of polymer composite failure theories. It provides a historical context for the origin of the study and it contains a detailed account of the methodology employed by the organisers in striving to attain an impartial, independent and broad ranging assessment of the leading failure theories that forms the subject of this book. The study has become known within the composites community as the ‘World-Wide Failure Exercise’ (WWFE) and in some circles, the ‘Failure Olympics’. The paper includes a comprehensive description of the process by which the theories were selected, a brief description of the theories themselves, the data and instructions supplied to the contributors and the chronology of the key milestones.

1. Background to the WWFE – origin and rationale The origin of the WWFE can be traced to an ‘experts meeting’ held at St Albans (UK) in 1991, on the subject of ‘Failure of Polymeric Composites and Structures: Mechanisms and Criteria for the Prediction of Performance’, Ref. [1]. The meeting was organised by the UK Science and Engineering Research Council (now the Engineering and Physical Sciences Research Council) together with the UK Institution of Mechanical Engineers (I Mech E), with the specific aim of establishing the level of confidence within the research and design communities in the current methods for failure prediction of fibre reinforced polymer composites (FRP’s). Experts from many countries attended and the meeting took the form of a series of formal presentations interspersed with informal discussion groups. Two key findings emerged: (a) There is a lack of faith in the failure criteria in current use – Attendees concluded that, even at the lamina or laminate level, there was a lack of evidence to show whether any of the criteria could provide accurate and meaningful predictions of failure, over anything other than a very limited range of circumstances.

3 (b) There is no universal definition of what constitutes ‘failure’ of a composite – In the broadest sense, a designer would define ‘failure’ as the point at which the structure or component ceases to fulfil its function. The definition of ‘failure’ must, therefore, be application specific – a pipe designer might consider weeping of fluid through the pipe wall as constituting failure, whereas a bridge designer might consider (say) a 10% loss of stiffness as failure. Attendees concluded that the connection between events at the lamina level and the many definitions of structural failure required by designers, remains to be established. To many people, these conclusions might be unexpected. After all, there is a large body of research into composite materials to draw upon, spanning at least 50 years, and there are numerous examples (aircraft, boat hulls etc) where composite materials are being used widely, and successfully, in primary load bearing structures. One might logically conclude that design procedures (including strength prediction) for FRP structures are fully mature. Closer examination reveals the fact that current commercial design practices place little or no reliance on the ability to predict the ultimate strength of the structure with any great accuracy. Failure theories are often used in the initial calculations to ‘size’ a component (i.e. to establish the approximate dimensions, such as panel thickness, width etc). Beyond that point, experimental tests on coupons or structural elements (such as the notched hot/wet compressive strength tests in aerospace) are used to determine the global design allowables. These are typically set at levels which are less than 30% of the ultimate load carrying capability, thereby providing a wide safety margin to accommodate loss in performance due to fatigue, operating environments, impact and any other unaccounted features. The coupon/ structural element testing approach is widespread in the aerospace industry, where large databases have been established at great expense. Small to medium sized companies tend to follow a broadly similar path, though on a much reduced scale. A ‘make and test’ approach combined with generous safety factors is commonplace, though in niche markets (such as sporting goods) confidence built up over several years is leading to reduced margins (as evidenced by the progression of lighter tennis rackets and skis over the last 10 years). From an industrial perspective, therefore, there has been little pressure in the recent past to investigate or promote the need for improved failure theories, the view being that they are more of an academic curiosity than a practical design aid. This view is beginning to change, in light of the need to reduce the time and cost of bringing new components to the market place, and the desire to exploit the performance of FRP’s in ever more complex applications. There is a growing realisation that ‘make and test’ is simply too slow and too costly. Improved design methods are now being called for which depend upon analytical modelling techniques. Central to this, is the need to establish the level of confidence which is applicable to current failure theories. The 1991 St Albans meeting was memorable for the amount of heat and excitement that was generated over the question of the validity of established failure criteria. During informal intervals when people gathered to continue the disputes, it began to seem to one or two of the attendees that were involved in publishing – Paul Hogg, (then) Editor of the Elsevier Advanced Composites Bulletin, and Bryan Harris, (then) Editor of the Elsevier Composites Science and Technology – abetted and encouraged by Clive Phillips and John Hart-Smith, that here was an excellent subject for a Special Issue of Composites Science and Technology with a possibility of some lively correspondence.

4 Encouraged by Peter Lanagan, then responsible for Elsevier’s Composites journals, Bryan set about trying to find a Guest Editor. He solved the problem by approaching Mike Hinton and Peter Soden (i.e. two of the authors of this paper) who were both active in the field of failure prediction, and had a long history of working together in their research. They agreed to produce a publication that would provide an authoritative source for designers and researchers by making it possible, for the first time, to obtain unbiased comparisons of the merits and short-comings of current failure criteria. Later, in 1994, they were joined in this endeavour by a colleague, Sam Kaddour. It has to be said that Hinton, Soden and Kaddour (hereafter referred to as the ‘organisers’) did not envisage the shear scale and volume of the challenge that was ahead and that would occupy a significant portion of their efforts over the next twelve years!

2. Clarifying the aims of the WWFE A critical first step was to define the objectives of the study. After some discussion the organisers targeted three broad aims: (a) Establishing the current level of maturity of theories for predicting the failure response of fibre reinforced plastic (FRP) laminates. (b) Closing the knowledge gap between theoreticians and design practitioners in this field. (c) Stimulating the composites community into providing design engineers with more robust and accurate failure prediction methods and the confidence to use them. Whilst (a) was the central focus of the study, the organisers felt that (b) and (c) were equally important issues that needed to be addressed. The 1991 St Albans meeting had highlighted a clear division between the arguably academic pursuit of polymer composite failure theories by many theoreticians and the practical needs of designers of composite structures. The organisers’ intent was therefore to exploit the WWFE as a catalyst for healthy debate between theoreticians and designers, with the expectation that this would lead to progress in the field.

3. Constructing a definitive test of a failure theory – key principles At an early stage in the planning process, the organisers decided to capture the key requirements for what would constitute a definitive test of a polymer composites failure theory and for a rigorous comparison of the best failure theories available. Not surprisingly, most of the principles are derived from standard scientific practice for testing any new hypothesis. They are: (a) The organisers of the exercise must maintain true independence from those participants making the predictions. (i.e. there must be no insider dealing). (b) Where predictions based on a given failure theory are required, the best approach is to request that the originator of the theory carries out the calculations. The room for misinterpretation by an intermediary is then removed. (c) A true comparison of theories should use a common set of test cases that are clearly and unambiguously defined. Equally, the parameters to be predicted should be clearly defined, so that direct comparison is facilitated.

5 (d) In order to determine the bounds on the validity of a given theory, it is important to test it over a wide range of conditions and to choose test problems that highlight the differences and similarities between the theories. Careful thought is needed, therefore, to identify the laminate and loading conditions that will test a theory to the full and thereby identify any discriminating features. (e) The test cases should be chosen by the organisers and not by those participants making the predictions. In this way, the test cases are unlikely to favour any one theory. (f) There should be matching, high quality, experimental data available for each of the test cases to be analysed theoretically. Each theory can then be benchmarked directly against experimental observations. The debate can then be moved from whether theories (a) and (b) agree with each other, to which theory matches reality most closely and over what range. (g) The theoretical predictions should be made ‘blind’ in the first instance (i.e. with no knowledge of the equivalent experimental results). The participants should not be given sight of the experimental results for the test cases until all of the papers containing their predictions are complete and in the hands of the organisers. This avoids any suspicion of ‘tuning’ the predictions. (h) Whilst it is accepted that certain models can be ‘tuned’ to improve their accuracy, it is important to be able to discriminate between a ‘blind’ and a ‘tuned’ prediction. Clearly, in most design situations, the reliance will be placed on ‘blind’ predictions. Principles (a) to (h) were adopted for the study, the remaining task being to develop a suitable framework for implementation. In a later section of this chapter, we will return to this key requirements list to check the level of compliance achieved within the framework that was finally adopted.

4. Lessons from previous studies in the literature Given the magnitude of papers in the literature on failure theories for polymer composites, the organisers sought to pick up useful ideas or even a complete methodology for carrying out the WWFE, to meet the principles outlined in section 3. The literature contains a steady stream of work on failure of composite materials, starting in earnest in the 1950’s. Papers containing a major theoretical content far outweigh those with a major experimental emphasis and the literature is particularly sparse in experimental data on the response of FRP materials when subjected to biaxial and triaxial stresses. This is especially surprising, given that most structural components are subjected to multi-axial loads (and hence multi-axial states of stress), and one would expect there to be sufficient, good quality, experimental data available to validate the many failure theories that purport to model such conditions. The literature contains numerous attempts to conduct comparisons between competing theories and between theory and experiment (see for instance Refs [3–6]), and they generally take one of two forms: (a) An author produces a computer programme containing a personal interpretation of a range of failure theories, devised by others and gleaned from the literature. In certain cases, the author has introduced a genuinely new theory of his/her own or a modification to someone else’s theory. The programme is then used to generate failure predictions,

6 usually over a restricted range of materials, layup configurations and loading conditions taken from the literature. Conclusions are then drawn as to the validity or otherwise of the competing theories. (b) An author conducts experiments to measure the strength of a series of laminates. By necessity, the experiments cover a restricted range of materials, layup configurations and loading conditions. The author then takes the approach outlined in (a) to generate theoretical predictions for comparison with the experimental data. Frequently, in the light of the available experimental data, the author introduces modifications (‘tuning’) to one or more of the theories thus improving the fit between theory and experiment. Conclusions are then drawn as to the validity or otherwise of the competing theories. This previous work has provided valuable insights into the differing theoretical approaches that underpin many of the current failure theories. These include the variety in mode of failure that laminates exhibit, and the many experimental difficulties which tend to preclude the observation of true materials failure rather than an artefact of the testing methodology. However, when tested against the key principles for a definitive test of a failure theory, outlined in the preceding section (3), the organisers concluded that no previous studies have come close to satisfying the conditions. The very fact that the 1991 St Albans meeting was so dismissive of the current failure theories as practical design tools, is further evidence for this conclusion.

5. Raising the profile – initial publicity In order to bring the exercise to the attention of the community, a calling notice was issued through Composites Science and Technology, Ref. [2], in 1991 and via the Elsevier stand at the ICCM 9 in Madrid, in 1993. The organisers also presented an outline at the second I Mech E Symposium on Composite Design in London, in 1993. Responses were invited from potential participants in the exercise, and from those experimentalists who would be willing to contribute experimental results (i.e. from a series of detailed tests where comprehensive data was available on laminae or laminates that had been loaded to failure under specific load ratios).

6. Developing the plan for carrying out the WWFE 6.1. Overview of the original two – stage process The principal elements of the plan, as envisaged at the start of the process, involved the following steps: (a) Assemble a comprehensive description of the current, foremost, failure theories for fibre reinforced plastic laminates. (b) Compare their predictive capabilities directly with each other. (c) Compare their predictive capabilities directly with experimental data. At the outset, careful consideration was given as to how the above three steps could be carried out to preserve objectivity and to stand scrutiny from the composites community. Failure

7 prediction for FRP’s is a controversial subject area for a number of reasons. Firstly, there are several, fundamentally different, technical approaches being pursued in academia and in industry, giving rise to some highly polarised opinions. Secondly, there are large investments at stake, as many companies (particularly aerospace and defence) have well established design and qualification procedures, and they would probably be reluctant to expose any limitations in predictive capability to their competitors or their customers. This is particularly true for the vendors of commercial software packages used for designing FRP structures. Clearly there is a significant market for such software packages, and one could understand the reluctance of individual vendors from participating in a ‘failure exercise’ that might identify significant shortfalls in the performance of their package. With this as a backdrop, the organisers began the process of setting up the terms and conditions of the exercise. Considerable thought was given to the practicalities of getting a cooperative ‘failure prediction round robin’, off the ground. In the first instance, visits were made to some of the leading experts, to canvass their views on how a strictly unbiased technical appraisal might be attained and to seek their agreement to participate. Views were also sought from individuals with experience of other ‘round robins’ for ISO test standardisation and validation of specific Finite Element methods. After much debate, the organisers adopted a strategy that contained two sequential stages, designated Part A and Part B. PART A was planned to contain: • A detailed specification of a series of test cases and a format for presenting the predictions, chosen by the organisers to stretch the failure theories to the full. • A series of invited papers from originators of the foremost failure theories available at the present time (or their chosen collaborators). The papers would provide a comprehensive description of the theory employed, and would contain predictions of strength and stress/ strain curves for each test case, using the data provided. • A direct comparison, by the organisers, of the methodologies employed in each theory and of the predictions made for each test case. PART B was planned to contain: • Detailed experimental results for each of the test cases and a full description of how the experiments were conducted. • A second paper from each participant containing a graphical comparison of their original prediction with the experimental results for each test case, comments on the performance of their chosen theory, and the opportunity to show refinements in the light of the experimental data. • An overall comparison of the theories represented in the exercise against the experimental results, presented in such a form that their degree of validity (or otherwise) could be judged by the community. • Shortfalls in the theories and in the experiments, highlighted by the ‘exercise’. The two-phase approach was adopted because it allowed both a ‘blind test’ and a further opportunity for participants to offer refinements to the theories, thereby advancing the science.

8 The timing was chosen such that no participant was in receipt of the experimental data until all of the ‘blind prediction’ papers (i.e. the Part A submissions) had been completed and were with the organisers in their final revised form. Part A would be published before Part B. Further details of this two stage process are presented, in flow diagram format, in Fig. 1.

Fig. 1. Original two stage plan (i.e. Parts A & B) for the WWFE.

9 6.2. The need for an additional stage – Part C When the original two-stage process was first conceived, the expectation of the organisers was that the WWFE would be completed within a maximum duration of 4 years (i.e. 2 years to complete Part A and a further 2 years to complete Part B). This expectation was based on an initial survey of the participants to ascertain their views on a practical timescale for each to produce their Part A and Part B contributions. A lesson learned the hard way by the organisers is that timescales are impossible to enforce in an exercise which is reliant on the goodwill of the participants. In the event, it took more than 10 years to complete the original two-stage plan, two years to set the framework in place, 4 years on Part A and 4 years on Part B. As a result of the lengthy duration of the WWFE, it was natural that additional potential participants, with new and emerging theories, would come forward during the process. An extension of the original plan was devised by the organisers to include the new methodologies into the WWFE. A third stage, Part C, was introduced with the timing fixed to preserve the probity (i.e. ‘blind’ predictions) that is key to the WWFE. The manner by which the new submissions were dealt with mirrors the two stage process that was utilised for the original participants: • New participants were asked to produce a paper in an identical manner to the ‘Part A’ contributions of the original group (i.e. describe their theory and produce ‘blind’ predictions of the test cases, given the identical information). These papers will be referred to hereafter as ‘Part C/A’. • The organisers adopted the same review process for the new and original groups (typically the papers required a number of iterations before acceptance). • The organisers provided the new contributor with the ‘Part B’ instruction pack, only after his/her ‘Part C/A’ paper had reached final acceptance for publication. Thus the new contributor had no possibility of revising his/her ‘Part C/A’ paper after seeing the experimental data. Hence the ‘Part C/A’ papers still retained the principle of being ‘blind’ predictions. • The participants submitted their second contributions, referred to hereafter as ‘Part C/B’. Their second contributions contained a graphical comparison of their original prediction with the experimental results for each test case, comments on the performance of their chosen theory, and the opportunity to show refinements in the light of the experimental data. • The ‘Part C/B’ papers were subjected to the same reviewing process adopted for the original Part B papers. Again several iterations were required before the papers were finally accepted for publication. The revised, three stage, plan for the WWFE is depicted in flow diagram format, in Fig. 2. It should be noted that Part C was run in parallel with Part B such that the new participants did not get sight of any experimental results before they had finalised and submitted their blind predictions. The papers, from which much of this book has been derived, appeared originally in three separate special editions of Composites Science and Technology (Part A, Ref. [7], Part B, Ref. [8], Part C, Ref. [9]) and throughout the book there are many references to those papers. However it should be noted that all 42 papers that made up the output from the exercise have been collected together and can be found here in this one volume. The book also provides an

10

Fig. 2. Revised three part (i.e. A, B & C) plan for the WWFE.

opportunity to rearrange the papers into a more logical sequence. Whilst groups of theories were compared and assessed in separate papers in the special editions they are all assessed together in the book. The original two stage process has been retained by amalgamating the original Part A and Part C/A papers to form a new Part A (Chapter 3). Similarly, the original

11

Fig. 3. Amalgamating the original (Part A and Part B) and the additional (Part C) papers in this book.

Part B and Part C/B papers have been grouped together to form a new Part B (Chapter 5). These changes are depicted in Fig. 3.

7. The process for selecting the failure theories and their advocates for participation in the WWFE A major feature in getting the WWFE off the ground was to attract a critical mass of participants with well established, world class, reputations in the development of failure theories for FRP materials. This was achieved through personal visits made by the organisers (as outlined in 6.1) and was crucial in giving the WWFE credibility in the eyes of the community. This was supplemented with the issue of calling notices and invitations (as outlined in 5) to those who could not be visited personally. The principal objective was to attract input in predicting FRP material failure. These included: Interactive Failure Theories: These are essentially derivatives from theories developed in the 1950’s to describe yielding of metals (Hill), updated to account for material anisotropy and

12 the sign dependent (i.e. tensile or compressive) strength characteristics observed in FRP’s. They are widely used in computer programmes, as the theories can be reduced to simple equations which are easy to embody into efficient algorithms. Physically Based Failure Criteria: The common feature in this class of criteria is that specific modes of failure are defined, each mode being described by a unique equation within the theory. Typically, the criteria will delineate between fibre failure and matrix failure, and some provide further delineation, such as between brittle and ductile matrix failure modes. Thus, in addition to predictions of strength, strain to failure etc, these criteria also provide predictions of mode of failure. Damage Mechanics: A fundamental feature of composite structures is that, in most cases, ‘failure’ is not a unique event, but a gradual sequence of microcracking and delamination leading up to structural collapse. In many applications, structures are operated on a regular basis and in perfect safety whilst containing some damage. The ‘damage mechanics’ community is developing analytical tools to describe the initiation and evolution of damage, such that the complete failure process can be represented. Industrial and Standard Design Code Approaches: Numerous local, national and international standards/guidelines, already exist for the design of certain classes of composite structure. Foremost amongst these are the in-house procedures developed by commercial aerospace companies, national standards for pressure vessels and piping (such as the UK BS 4994, Ref. [10]), and design guides (such as the US Mil Handbook 17, Ref. [11]). Typically these contain clear procedures and design rules for developing a viable structural solution. Commercial Software Vendors/Developers: Design aids for fibre composite structures have been on the market for more that 15 years and there is a wide variety of packages available. These stretch from small packages which embody laminated plate theory, through to large finite element packages which have the capability to simulate the structural response of an entire aircraft under highly transient loading conditions. Typically, the codes contain one or more failure theories which the vendor has selected from the literature. There are many ways to implement a failure theory into a code and this can influence the predictions made. Thus, there is no guarantee that, for instance, a theory used within an FE idealisation and the same theory employed in an analytical model by the originator of that theory, will produce equivalent predictions. For this reason, the organisers approached a representative selection of software vendors (ABAQUS, MSC/NASTRAN, ANSYS, DYNA3D, PDA/PATRAN, ESDU, PAMFISS/PAMCRASH) to participate in the exercise. Table 1 summarises the names of the participants to whom invitation letters were sent, together with details of the theoretical approach they were considered to represent. More than forty groups in the area of fibre composite failure theories, including leading academics and developers of software/numerical codes, were invited to participate in the exercise. Some declined to participate from the outset and others agreed but then withdrew later. Table 2 summarises those invited participants who either declined or later withdrew. Among those who declined was Prof Z Hashin of Tel Aviv University, whose work (e.g. Refs [12–14]) is well known by the composites community. Hashin’s reasoning for declining to participate, is worth noting (quoting from his letter to the organisers):

13 Table 1 A summary of approaches and failure theories originally planned in the exercise. Contributor

Approach represented

Chamis C C (USA) Hart-Smith J (USA)

Micromechanics Generalised Tresca, Max Strain and ‘10% Rule’ theories Design codes Industry ANSYS ESDU International PDA/PATRAN DYNA3D PAMFISS/PAMCRASH PAMSISS/PAMCRASH MSC/NASTRAN ABAQUS Damage models and cracking of composites

Eckold G C (UK) Edge E C (UK) Ellis D (UK) Grayley M E (UK) Hall R L (USA) Hallquist J (USA) Johnson A (Germany) Haug E (France) Stanton E (UK) Hibbitt D (USA) Allen D H, Talreja R, Nairn J, Reifsnider K (USA), McCartney L (UK) Puck A,Schürmann H, Cuntze R G, Aoki R (Germany) Hashin Z and Rotem (Israel) Rosen B W (USA) Zinoviev P (Russia), Skudra A M (Latvia), Chang F-K, Chou T W, Sun C T, Swanson S R (USA), Marom G (Israel), de Wilde P (Belgium), Surrel Y (France), Hansen J S (Canada) Tsai S W (USA) Uemura M (Japan) Sandhu R S and Wolfe W E (USA)

Puck’s theory Rotem and Hashin’s theories MIL-HNDBK-17 General application and methods

Interactive failure theory Statistical prediction of failure Strain energy failure theory

“my only work in this subject relates to failure criteria of unidirectional fibre composites, not to laminates. I do not believe that even the most complete information about failure of single plies is sufficient to predict the failure of a laminate, consisting of such plies. A laminate is a structure which undergoes a complex damage process (mostly of cracking) until it finally fails. The analysis of such a process is a prequisite for failure analysis. While significant advances have been made in this direction we have not yet arrived at the practical goal of failure prediction. I must say to you that I personally do not know how to predict the failure of a laminate (and furthermore, that I do not believe that anybody else does)”. Hashin’s strongly held view served to highlight an extreme position on the validity and bounds of applicability (or otherwise) of failure theories for composite materials and provided a further impetus for completing the WWFE.

14 8. Summary of the theories and participants that are represented in the WWFE The invitation and consultation process resulted in firm commitments from 15 groups to participate in the exercise. The theories represented by these groups are listed below in an alphabetical order. It is worth re-iterating here that four of these groups (Bogetti et al., Cuntze, Mayes and Hansen and Huang) joined the exercise via Part C. Bogetti [15–16] and his colleagues at ARL employed the Maximum Strain theory in a full 3D form, taking into consideration the non-linear shear stress–strain response of the lamina. Chamis [17–18] and his group at NASA have developed two computer programs (a) Integrated Composite Analyser (ICAN) and (b) Composite Durability Structural Analyser (CODSTRAN) which has been integrated with a finite element package that allows analysis of complex structures. Results are presented from each program. Both of Chamis’ programs are based on micromechanics relationships that predict the lamina properties from those of the constituent fibre and matrix. Cuntze’s theory [19–20] gains some of its strong features from that of Puck. It considers five failure mechanisms (two fibre failure and three inter-fibre failure) and its predictions of the three failure envelopes for the unidirectional laminae bear some resemblance to Puck’s. Unlike Puck’s theory, interaction between the failure mechanisms is assumed, due to probabilistic effects. Eckold [21–22] demonstrated a simplified approach which is employed in the British design code for glass fibre reinforced tanks and pressure vessels (BS4994), Ref. [10]. The code was intended primarily for designing plant for the chemical and process industry and applies stringent limits on strain to avoid resin cracking and resin/fibre debonding. Edge [23–24] presented an approach that has been developed by one of the major aerospace companies. The method makes allowance for nonlinearity, assuming that failed plies unload gradually (in contrast to the abrupt ‘ply discount’ technique adopted by others). A number of lamina failure criteria are utilised (maximum stress, interactive failure between transverse tension and shear, and interaction between longitudinal compressive stress and shear stress), the choice being dependent on the biaxial stress state encountered during the analysis. Hart-Smith [25–28] provided three contributions to the current exercise. The first is a Generalized Tresca model [25], which considers shearing of the fibres as one of the dominant failure modes. The second is a 2-D interpretation of the widely used Maximum Strain and Truncated Maximum Strain failure theory [26] with a simplifying assumption to suppress the occurrence of initial failure. The third is a generalisation of Hart-Smith’s wellknown Ten Percent Rule [27] which has been used quite widely in the aerospace industry for many years. Unlike most of the other participants, at the outset of the study Hart-Smith had yet to embody his theories into a computer program. He solved each problem by hand. Huang’s theory [29–30] is micro-mechanics based and it combines a number of ideas (bridging model, anisotropic plasticity and generalised Maximum Stress theories) into a single theoretical framework. The theory provides a basis for obtaining the lamina and laminate moduli and strengths from the constituent fibre and matrix properties.

15 Mayes and Hansen [31–32] have developed a finite element, multi-continuum, micromechanics based theory, reminiscent of the overall approach adopted by Chamis, that takes the lamina nonlinear shear behaviour into account. The theory employs a form of interactive failure criterion at the fibre and matrix constituent level and it uses a finite element method to calculate values of stress at the laminate level. McCartney [33–34] employed an analysis based on damage mechanics. He used a fracture energy based approach to predict the formation and multiplication of transverse tensile cracks as a function of applied strain. The progressive reduction in laminate stiffness due to the presence of the transverse cracks is then predicted. Puck [35–36] and his colleagues in Germany have been working for almost 30 years to develop a mechanistic theory that distinguishes between various modes of failure. The theory employs two independent fracture criteria (for fibre fracture and inter-fibre fracture) and makes allowance for a continuous and gradual loss of stiffness after initiation of cracking. Puck generously acknowledged Hashin’s contribution to his type of approach. Rotem’s contribution [37–38] is based on a theory developed in collaboration with Hashin [21]. It is one of the earliest to concentrate on two separate failure criteria, one for the fibres and the other for the resin. Sun and Tao [39–40] presented results from two separate analytical routes, one linear and the other nonlinear. The linear analysis, which employs a derivation of the Hashin-Rotem lamina failure criteria, Ref. [21], was applied to generate the failure envelopes and stress–strain curves. The detailed nonlinear analysis, which is based on finite element analysis and allows for elastic/plastic material properties and progressively increasing matrix crack density, was used in the test cases involving prediction of stress/strain curves. Tsai has been associated with the development of failure theories for polymer composites, for more than 30 years. He is a strong advocate of the quadratic failure criterion with linear terms (the Tsai-Wu failure criterion) because (i) it is easy to use, (ii) it is a single valued function, (iii) it is based on a mathematically rigorous framework and (iv) it gives a basis for an extension to cater for three dimensional failure cases. The theory, which he employed in the failure exercise, Refs [41–42], has been used widely and has been disseminated to the community in the form of a spreadsheet based computer program. Wolfe and Butalia, Refs [43–44], employed a strain energy based failure theory which originates from the work of Sandhu and colleagues at Wright Patterson Air Force Base during the early 1970’s. The widely used Maximum Stress theory is represented here by Zinoviev and his colleagues from Russia, Refs [45–46]. Their theory has been highly developed and takes into consideration such factors as the unloading behaviour of cracked laminates, and geometric nonlinearity due to changes in ply angle with increasing deformation. 9. Summary of theoretical methods that are not represented in the WWFE Soni, Ref. [47], presented the results of a survey that had been conducted by the US AIAA Composite Structures Subcommittee to identify the most used failure theories. The results

16 showed that 90% of the designers who replied to the questionnaires, used either the Maximum Stress theory, the Maximum Strain theory or one of the quadratic polynomials theories. It is gratifying to note that all of these major approaches are represented in the WWFE. The organisers believe that a ‘critical mass’ of the most significant, currently available, models for analysing failure in composites has been adequately represented in the exercise. However no exercise like this can claim to have covered all of the theories, as there will always be certain groups who cannot participate for one reason or another (such as those cited in section 6.1). In addition, there have been new developments over the period which are not represented in the WWFE (eg due to there being insufficient time, or an absence of a willing champion for the method, or for reasons of timing that might have compromised the probity of the study). Those not represented include: Commercial vendors of finite element design software: A representative group was invited to participate (see Table 1), but all declined for one reason or another (see Table 2). This was particularly disappointing (though, perhaps, not totally surprising (see section 5.4)). It means that the validity of the FRP failure theories implemented within major commercial finite element codes remains untested.

Table 2 A list of invited individuals who did not contribute. Name

Remarks

Allen D H Chang F-K Chou T W Cuntze R G* de Wilde P Ellis D Grayley M E Hall R L Hallquist J Hansen J S Hashin Z Hibbitt D Johnson A Marom G Reifsnider K Rosen B W Sandhu R S Stanton E Swanson S R Talreja R

Declined invitation to join to the exercise Declined Declined Declined in favour of Prof. Puck Declined because of different expertise Accepted but later withdrew Accepted but later withdrew Declined Declined Accepted but later withdrew Declined Declined Accepted but later withdrew due to pressure of work Declined because of different expertise Declined Very late invitation. Declined Declined but nominated Prof Wolfe Declined Accepted but later withdrew Accepted but later withdrew

* Cuntze later decided to join the exercise in Part C.

17 Theories which do not use lamina based input data: A prime example is the Dissipated Energy Density (DED) approach, which has been developed by Badaliance et al., Refs [48–50], at the Naval Research Laboratory (NRL, USA), and is a significant departure from the existing failure methodologies. The organisers held discussions with the NRL group, with a view to carrying out the experimental work needed (which is unique to the DED method) in order to characterise the materials used in the test cases. The additional experiments have now been carried out by the organisers and by the NRL team. Sufficient data has been obtained so that NRL might apply the DED methodology to a sub-set of the test cases. The expectation is that NRL will carry out the analysis and that the results will be published at a later date (ie separately from the WWFE). Theories based on probabilistic methods: Methods for predicting the failure strength of brittle materials (e.g. ceramics) recognise the need to represent material strength as a statistical distribution, rather than as a single valued function. Given that many of the FRP mechanisms could be classed as ‘brittle’ (e.g. transverse tensile fracture of a lamina), one would expect there to be advocates for probabilistic based FRP failure theories, such as [51]. No such advocate emerged from the publicity/selection stage, though it should be noted that one small portion of Cuntze’s contribution does utilise some probabilistic features. Continuum Damage Mechanics (CDM) models: Champions of this methodology were included in the original call for participation in the WWFE (including the initiator of the concept, Talreja [60]), but none agreed to participate. A contribution from Li et al. at UMIST [52], one of the experts in the CDM methodology, was deliberately discounted from the WWFE on grounds of probity, because of his close association with the organisers. A separate publication is being considered. The micro-mechanics based strain invariant criterion: This method, recently called Strain Invariant Failure Theory and abbreviated as ‘SIFT’, is being developed by Gosse [53] and is part of a growing interaction between Gosse and Hart-Smith within the Boeing company. The organisers have met with Gosse and Hart-Smith on several occasions to gain greater insight into the theory and to determine the nature of the material properties data required for the theory to function. At the time of writing, the predictive capability of this method has not been assessed as, to the best of knowledge of the authors, insufficient information has been published about the materials characterisation requirements of the theory. The expectation is that a further effort might lead to Gosse attempting the WWFE test cases at a later date. Hopefully the examples, given above, serve to show that the organisers have made every effort to engage with the broadest base of theoreticians and to encourage their involvement in the WWFE. Whilst it has not always been possible to get full participation within the framework of the WWFE itself, the expectation is that some of these groups (or their representatives and collaborators) will choose to attempt the test cases defined in the WWFE at a later date, and then produce papers indicating how well their theories have performed. Such new papers will not have the same authority as those generated under the strict rules of the exercise, because the predictions will not be made ‘blind’ (new authors will have full access to ‘Parts A, B and C’ by then). Nevertheless they can only add to the collective understanding of this very complex area.

18 10. Selection of the ‘test cases’ for the WWFE and assembly of datapack 1 The usefulness of the exercise stands or falls on the selection of the test cases (as can be seen in sections 3(c), (d), (e) and (f)). Careful consideration was given to this step in order to ensure that the test problems would provide a thorough examination of the theories (Part A and Part C/A), with high quality experimental data available so that the theories could be benchmarked against reality (Part B and Part C/B)). Fourteen cases were identified, providing a test of the capabilities of the theories to predict stress/strain response, strength and modes of failure for a range of materials, layup geometries and load ratios. In order to avoid any ambiguity in interpretation between the participants, the organisers produced ‘datapack 1’ that defined the test cases, in great detail. It was important to ensure that each participant started from identical descriptions of the test problems, and the materials properties to be used in the calculations. Similarly, to ensure that the predictions of each participant could be compared directly, the organisers provided detailed specifications of the parameters to be predicted, down to the level of the axes and scales for each graph. A draft version of ‘datapack 1’ was issued to the participants for comment and acceptance, to ensure that each participant had been given the opportunity to request any additional data or any additional clarification prior to applying their theory to the test problems. This step was also aimed at heading off any subsequent complaints from participants about lack of input data, should their ‘blind’ predictions be greatly in error from the experimental results. The comments were incorporated into a final, ‘frozen’ version of ‘datapack 1’, which was then ready to issue. The definition of the test cases is the central pillar of the WWFE. Recognising that other theoreticians may wish to use these cases to benchmark models against in the future, the exact information provided to the participants in the exercise is provided in Chapter 2 and Ref. [54]. 11. Initiating Part A (and Part C/A) The organisers initiated Part A (and Part C/A) by issuing a letter and an accompanying ‘datapack 1’ to participants. For completeness, the content of the letter is reproduced and full details of ‘datapack 1’ are provided in Chapter 2.1 of this book. 12. Initiating Part B (and Part C/B) Release of information to the participants in Part B (and Part C/B) only occurred once all of the Part A (and Part C/A) submissions had been received in an acceptable form for publication (ie after satisfying the referees). This step ensured that there was no prospect of any participant being able to modify his/her ‘blind’ prediction, once the experimental data became available. Once Part A (and Part C/A) was concluded and ready to publish, the organisers initiated Part B (and Part C/B) by issuing a letter and an accompanying, datapack (‘datapack 2’) to each participant. The letter provided instructions on the content required in the Part B (and Part C/B) submission. For completeness, the content of the letter is reproduced in the Appendix to this chapter. The datapack was personalised to contain graphs showing the

19 participants own individual theoretical predictions for each test case with experimental results superimposed upon them. 13. The refereeing process Every paper generated within the WWFE was refereed by the organisers and colleagues. This was felt to be the only way to ensure that the participants had complied with the instructions laid down by the organisers, to ensure that submissions from each participant had been treated equally, and to ensure that there would be no premature release or leakage of information into the community. For the Part A (and C/A) submissions, the refereeing process concentrated on ensuring that the descriptions of the theories were comprehensive, that any assumptions made in the predictions were clearly identified, and that the predictions provided were self consistent and complied with the instructions. For the Part B (and C/B) submissions, the refereeing process concentrated on ensuring that the participants had provided full commentary on the performance of their theory against each of the test cases, that any proposed modifications to the theory were clearly described and justified, and that some attempt had been made to comment on the utility of the theory from a designer’s perspective. Future organisers of ‘Round-Robins’ should note that the refereeing process turned into a massive task. • There were examples where detailed checking of the theoretical equations by the organisers identified a number of errors or shortfalls (often in the participant’s computer programme), resulting in the participant needing to revise his/her model and recalculate the blind predictions for every test case. • There were several examples where the participant realised that their theory just wouldn’t work for certain of the test cases and this led to a revision of the theory and a greatly revised paper. • There were examples where the theory was under continual development by the participant, such that just as one draft submission was being refereed, a new submission arrived in the post. Some papers required many iterations (the record being 11) before they reached their final form. The editorial iterations took more than 24 months, in certain cases, and the slowest paper became the rate determining step in ascertaining the final publication date. It became something of a challenge for the organisers to retain the support of both the early finishers, who wished to see their papers published, and the later finishers, some of whom found the refereeing to be a trial. Fortunately all of the participants showed great commitment to maintaining probity within the WWFE. 14. Evaluating the current level of maturity of theories for predicting the failure response of fibre reinforced plastic (FRP) laminates There has been considerable interest in the WWFE from within the composites community and one question asked repeatedly of the organisers has been ‘whose theory is the best?’ This

20 Table 3 An updated list of names and addresses of the organisers, participants (and their groups). Name

Address

Prof M J Hinton*

QinetiQ, Fort Halstead, Sevenoaks, Kent, TN14 7BP, UK. E-mail: [email protected] QinetiQ, Farnborough, Hampshire, GU14 0LX, UK.E-mail: [email protected] UMIST, Manchester, M60 1QD, UK. E-mail: [email protected] NASA, Lewis Research Centre, 21000 Brookpark Road, Cleveland, Ohio, 44135–3191, USA.E-mail: [email protected] or [email protected] AEAT, Harwell, Oxfordshire, OX11 OR, UK.E-mail: [email protected] British Aerospace, Military Aircraft Division, Warton Aerodrome, Preston, PR4 1AX. UK. E-mail: [email protected] The Boeing company, Phantom Works, Huntington Beach, California, USA. E-mail: [email protected] Division of Materials Metrology, National Physical Laboratory, Teddington, Middlesex, UK. TW11 0LW. UK.E-mail: [email protected] Technische Hochchule Darmstadt, Fachgebiet Kostruktiver, Leichtbau und Bauweisen, Petrsen strasse 30, D-64287 Darmstadt, Germany. E-mail: [email protected] Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Technion City, Haifa 32000, Israel.E-mail: [email protected] Purdue University School of Aeronautics & Astronautics, Grissom Hall 325, West Lafayette Indiana 47907, USA.E-mail: [email protected] Aeronautics and Astronautics Department, Stanford University, California 94305, 4035, USA. E-mail: [email protected] Department of Civil Engineering, The Ohio State University, 470 Hitchcock Hall, 2070 Neil Avenue, Columbus, Ohio 43210, USA. E-mail: [email protected], [email protected] Institute of Composite Technologies, Orevo, Dmitrovskii raion, Moskovkaya obl., 141814 Russia. E-mail: [email protected] or [email protected] MAN Technologie AG, D-86153 Augsburg, Germany.E-mail: [email protected] U.S. Army Research Laboratory; AMSRL-WM-MB Aberdeen Proving Ground, MD 21005–5066. E-mail: [email protected], [email protected], [email protected] Division of Mechanical Engineering, Alfred University, Alfred, NY, 14802, USA, Email: [email protected] Department of Mechanical Engineering, University of Wyoming, Laramie, WY, 82071, USA. E-mail: [email protected] Department of Engineering Mechanics, Tongji University, 1239 Siping Road, Shanghai 200092, P. R. China.E-mail: [email protected]

A S Kaddour* P D Soden* C C Chamis (with P K Gotsis and L Minnetyan) G C Eckold E C Edge L J Hart-Smith N L McCartney

A Puck (with H Schürmann)

A Rotem

C T Sun (with J X Tao)

S W Tsai (with K-S Liu and A Kuraishi) W E Wolfe (with T S Butalia)

P A Zinoviev (with S V Grigoriev, O V Labedeva and L R Tairova) R Cuntze (with A Freund)** T Bogetti, C Hoppel, V Harik, J Newill, B Burns** S J Mayes and A C Hansen**

Z-M Huang**

* Organisers. ** Participant in Part C.

21 question over simplifies the richness of the information that has emerged in terms of gaining a better understanding of the strengths and weaknesses of the current models, how they work and how improvements to the current models might be achieved. However, given the principal aim of the WWFE (see section 2) it was incumbent on the organisers to carry out a careful comparison of the 19 theories featured within the study and, in response to the repeated requests, we have also attempted to assess and summarise the overall performance of the theories. The organisers have approached this in three stages: Comparing the predictive capabilities of the theories directly with each other: The Part A (and Part C/A) call resulted in 15 papers being produced, covering 19 different theoretical methods for predicting failure in an FRP material. The papers contained a description of each theory and predictions, in graphical form, of the test cases. The papers appear as Chapter 3 of this book. Starting from this information, the organisers set out to summarise the key features in each theory which included: – – – – –

the types of failure models employed; how many different modes of failure the theory could discriminate between; whether linear or non-linear analysis was carried out; the degree of reliance on software and numerical methods; allowance for thermal stresses (or not).

Taking each test case in turn, the predictions (i.e. initial and final failure envelopes or representative stress–strain curves) for all theories were superimposed so that the similarities and differences could be highlighted. In addition, bar charts were constructed to demonstrate the levels of agreement between the predicted initial and final failure stresses and strains. Finally, the organisers provided a commentary on the possible sources of discrepancies between the predictions of the various failure theories. The detailed analysis is presented as Chapter 4 of this book. Comparing the predictive capabilities of the theories directly with experimental data: The Part B (and Part C/B) call resulted in 16 papers being produced, covering the 19 different theoretical methods presented in Part A (and Part C/A) for predicting failure in an FRP material. The papers contained a commentary on the degree of correlation between the individual predictions and the experimental data, and a description of any refinements in theory introduced to resolve shortfalls. The papers appear as Chapter 5 of this book. Starting from this information, the organisers carried out a systematic and detailed comparison, to identify the strengths and weaknesses of each theory, together with a ranking of the overall effectiveness of each theory. Careful reading of this analysis and study of the figures provided will inform the reader when selecting an appropriate failure theory for use in a given design situation. It will also provide a qualitative assessment of the likely accuracy and reliability of the prediction in a given circumstance. The detailed comparison is presented as Chapter 6 of the book. Providing recommendations for use by designers: Based on the analysis presented in Chapter 6, five of the most promising theoretical approaches were selected and explored in greater depth as a basis from which to construct the

22 best approach for making safe and economic predictions in a wide range of engineering design applications. The detailed analysis is presented as Chapter 7 of the book.

15. Maintaining the profile – publicising the results Maintaining the interest, publicity and involvement of the international community throughout the duration of the exercise has been a major driver for the organisers. This has been addressed by providing the composites community with regular updates on the progress of the exercise in the form of papers, talks and seminars given by the organisers in a number of national and international conferences and gatherings, see for instance [56–59]. In all cases, considerable discussion was stimulated as a result.

16. Concluding remarks Theories for the prediction of failure in polymer composite materials have been in existence for 50 years or more, yet this book arose from a clear lack of faith in the credibility of the science in this domain and a need for an objective review of the current scene. This introductory paper has attempted to set the WWFE in this wider context whilst also endeavouring to capture the thought processes of the organisers in arriving at the final format for the exercise. The organisers believe that the WWFE has provided a unique forum to examine the robustness of the leading theories for predicting failure in laminated composites, in an objective and dispassionate manner. We also believe that the lessons learned and recorded herein may prove useful to those who may wish to organise a benchmarking exercise of similar type in the future. The organisers are aware that the procedure adopted for identifying and selecting participants in this exercise was not foolproof, but was pragmatic. Encouragingly, the failure theories used by more than 90% of current designers are represented in the exercise. However, there are clearly many other leading members of academia and industry, who could have contributed to the exercise, and there may be certain approaches that have been overlooked. This book has been written with the clear intent that all of the necessary information is contained here, such that future groups can employ the same test cases as a benchmark. We encourage those groups to publish any promising alternative approaches, not represented to date. At the outset of the WWFE the organisers decided to capture the key requirements for what would constitute a definitive test of a polymer composites failure theory and for a rigorous comparison of the best failure theories available (see section 3). Having established a means of conducting the exercise, it is useful to revisit the key requirements to check the degree of compliance of the implementation plan for the WWFE. Table 4 contains a mapping of the requirements and the mitigating actions. As can be seen, the organisers believe that the process adopted for the WWFE meets the requirements for objectivity and independence, in full. The collected papers in this book are a distillation of the work of many of the world’s leading experts in this field. As such it forms a unique and powerful insight into the subject.

23 It is not written for the faint hearted but for the deep specialist who will be prepared to invest time and effort in studying the nuances within each contribution and in the results of the wide range of theories represented here. Read on.

Table 4 Comparing the plan adopted for the WWFE against the principles required for a definitive study. Index No.

Principle

Mitigation

3(a)

The organisers of the exercise must maintain true independence.

3(b)

The originator of the theory carries out the calculations.

3(c)

Use a common set of test cases for each theory.

3(d)

Test the theory over a wide range of conditions and choose test problems that highlight the differences and similarities between the theories. The test cases should be chosen by the organisers. Matching, high quality, experimental data must be available for each of the test cases. The theoretical predictions should be made ‘blind’.

– The organisers disbarred themselves from contributing a theory of their own. – All participants received exactly the same instructions, in parallel. – Results from all participants were released simultaneously, either via international journal publication or conference paper (ie no one participant had access to the results of another prior to finalising their submission). – The organisers expressly selected participants such that wherever possible they were the originators (or their co-workers) of the theory. The organisers: – compiled and supplied a common set of test problems – supplied a common input data set – supplied clear instructions on what to predict and how to present the information. The organisers selected test problems to cover a wide range of materials, layups, stress ratios, and parameters required in the predictions.

3(e) 3(f)

3(g)

3(h)

Within the WWFE it must be possible to discriminate between a ‘blind’ and a ‘tuned’ prediction.

All test cases were chosen by the organisers (and only the organisers). The test cases were selected from the most suitable high quality experimental data available to the organisers, after they had requested ‘donations of such via letters in journals and notices at conferences. The organisers devised a two-part process whereby the participants described their theory and made their predictions before access was given to the experimental results (ie ‘blind’). In the second part of the process, after the participants had made their predictions and had seen the experimental results, they were given the opportunity to revise their theory and predictions. Clear distinction was maintained in the WWFE between the ‘blind’ and the revised (readers may judge if these were ‘tuned’) predictions.

24 References 1 ‘Failure of polymeric composite structures: Mechanisms and Criteria for Prediction of Performance’, SERC/I.Mech.E Annual Expert Meeting, 23–25 September 1991, Sopwell House, St.Albans, UK, (A report complied by C E Neal-Sturgess). 2 P D Soden and M J Hinton, Failure criteria for composite laminates: Comparison of failure theories with experimental results: A call for experimental data, Compos Sci Techno, V45, 87–88, 1992, see also ‘Failure criteria for composite laminates’, Compos Sci Techno, V45, pp 283–284, 1992. 3 J H Greenwood, ‘German work on GRP design’, Composites, pp 175–184, July 1977. 4 S R Swanson and A P Christoforou (1987), ‘Progressive failure in carbon/epoxy laminates under biaxial stress’, J Eng Mat Tech, Trans ASME, V109, pp 12–16. 5 M J Owen and D I Rice (1982), ‘Biaxial strength behaviour of glass-reinforced polyester resins’, Composite materials: testing and design, ASTM STP 787, (Edited by I M Daniel), pp 124–144. 6 P Tang (1988), ‘A multiaxial failure criterion for composites’, in Computational probabilistic methods, AMD/ASME, V93, published by ASME, New York, USA, pp 87–96. 7 M J Hinton, P D Soden and A S Kaddour, ‘Failure Criteria In Fibre Reinforced Polymer Composites’, a special issue of Composites Science and Technology, V58: No 7, published by Elsevier Science Ltd, Oxford, UK, 1998. 8 M J Hinton, P D Soden and A S Kaddour, ‘Failure Criteria In Fibre Reinforced Polymer Composites: Comparison with experiments’, a special issue of Composites Science and Technology, V62: No 11/12, published by Elsevier Science Ltd, Oxford, UK, 2002. 9 M J Hinton, A S Kaddour and P D Soden, ‘Failure Criteria In Fibre Reinforced Polymer Composites: Additional Theories, Comparison and Recommendations’, V64, No 3–4, 2004, Published by Elsevier Science Ltd, Oxford, UK, 2004. 10 BS4994: Specification for design and construction of vessels and tanks in reinforced plastics, BSI, London, 1987. 11 MIL-HNDBK-17, Department of Defence, Washington D.C., 20025, USA, 1996. 12 Z Hashin, ‘Failure criteria for unidirectional fiber composites’, J Appl Mech, V47, pp 329–334, 1980. 13 Z Hashin, ‘Finite thermoelastic fracture criterion with application to laminate cracking analysis’, J Mechanics and Physics of Solids, V44: No 7, pp 1129–1145, 1996. 14 A Rotem A and Z Hashin, “Failure modes of angle ply laminates”, J Comp Mat, V 9, pp 191–206, 1975. 15 T A Bogetti, C P R Hoppel, V M Harik, J F Newill and B P Burns, ‘Predicting the nonlinear response and progressive failure of composite laminates’, Compos Sci Techno, V64, pp 329–342, 2004, see also Chapter 3.13 in this book. 16 T A Bogetti, C P R Hoppel, V M Harik, J F Newill and B P Burns, ‘Predicting the nonlinear response and failure of composite laminates: correlation with experimental results’, Compos Sci Techno, V64, pp 477–485, 2004, see also Chapter 5.12 in this book. 17 P K Gotsis, C C Chamis and L Minnetyan, ‘Prediction of composite laminate fracture: micromechanics and progressive fracture’, Compos Sci Techno, V58, pp 1137–1150, 1998, see also Chapter 3.1 in this book. 18 P K Gotsis, C C Chamis and L Minnetyan, ‘Application of progressive fracture analysis for predicting failure envelopes and stress–strain behaviors of composite laminates: A comparison with experimental results’, Compos Sci Techno, V62, pp 1545–1560, 2002, see also Chapter 5.1 in this book. 19 R G Cuntze and A Freund, ‘The predictive capability of Failure Mode Concept – based strength criteria for multidirectional laminates’, Compos Sci Techno, V64, pp 343–377, 2004, see also Chapter 3.14 in this book. 20 R G Cuntze, ‘The predictive capability of failure mode concept – based strength criteria for multidirectional laminates: Part B’, Compos Sci Techno, V64, pp 487–516, 2004, see also Chapter 5.13 in this book.

25 21 G C Eckold, ‘Failure criteria for use in the design environment’, Compos Sci Techno, V58, pp 1095–1106, 1998, see also Chapter 3.2 in this book. 22 G C Eckold, ‘Failure criteria for use in the design environment – Part B’, Compos Sci Techno, V62, pp 1561–1570, 2002, see also Chapter 5.2 in this book. 23 E C Edge, ‘Stress based Grant-Sanders method for predicting failure of composite laminates’, Compos Sci Techno, V58, pp 1043–1044, 1998, see also Chapter 3.3 in this book. 24 E C Edge, ‘Theory v. experiment comparison for stress based Grant-Sanders method’, Compos Sci Techno, V62, pp 1571–1590, 2002, see also Chapter 5.3 in this book. 25 L J Hart-Smith, ‘Predictions of the original and truncated maximum strain failure models for certain fibrous composite laminates’, Compos Sci Techno, V58, pp 1151–1178, 1998, see also Chapter 3.5 in this book. 26 L J Hart-Smith, ‘Predictions of a generalised maximum-shear-stress failure criterion for certain fibrous composite laminates’, Compos Sci Techno, V58, pp 1179–1208, 1998, see also Chapter 3.6 in this book. 27 L J Hart-Smith, ‘Expanding The Capabilities Of The Ten-Percent Rule For Predicting The Strength Of Fibre-Polymer Composites’, Compos Sci Techno, V62, pp 1515–1544, 2002, see also Chapter 3.17 in this book. 28 L J Hart-Smith, ‘Comparison between theories and test data concerning the strength of various fibre-polymer composites’, Compos Sci Techno, V62, pp 1591–1618, 2002, see also Chapter 5.4 in this book. 29 Zheng ming Huang, ‘Correlation of the bridging model predictions of the biaxial failure strengths of fibrous laminates with experiments’, Compos Sci Techno, V64, pp 529–548, 2004, see also Chapter 5.15 in this book. 30 Zheng ming Huang, ‘A bridging model prediction of the tensile strength of composite laminates subjected to biaxial loads’, Compos Sci Techno, V64, pp 395–448, 2004, see also Chapter 3.16 in this book. 31 S J Mayes and A C Hansen, ‘Composite Laminate Failure Analysis Using Multicontinuum Theory’, Compos Sci Techno, V64, pp 379–394, 2004, see also Chapter 3.15 in this book. 32 S J Mayes and A C Hansen, ‘A comparison of multicontinuum theory based failure simulation with experimental results’, Compos Sci Techno, V64, pp 517–527, 2004, see also Chapter 5.14 in this book. 33 L N McCartney, ‘Predicting transverse crack formation in cross-ply laminate’, Compos Sci Techno, V58, pp 1069–1082, 1998, see also Chapter 3.4 in this book. 34 McCartney L N, ‘Predicting Ply Crack Formation And Failure In Laminates’, Compos Sci Techno, V62, pp 1619–1632, 2002, see also Chapter 5.5 in this book. 35 A Puck and H Schürmann, ‘Failure analysis of FRP laminates by means of physically based phenomenological models’, Compos Sci Techno, V58, pp 1045–1068, 1998, see also Chapter 3.7 in this book. 36 A Puck and H Schürmann, ‘Failure analysis of FRP laminates by means of physically based phenomenological models- Part B’, Compos Sci Techno, V62, pp 11633–1672, 2002, see also Chapter 5.6 in this book. 37 A Rotem, ‘Prediction of laminate failure with Rotem failure criterion’, Compos Sci Techno, V58, pp 1083–1094, 1998, see also Chapter 3.8 in this book. 38 A Rotem, ‘The Rotem failure criterion theory and practice’ Compos Sci Techno, V62, pp 1663–1672, 2002, see also Chapter 5.7 in this book. 39 C T Sun and J X Tao, ‘Prediction of failure envelopes and stress strain behaviours of composite laminates’, Compos Sci Techno, V58, pp 1125–1136, 1998, see also Chapter 3.9 in this book. 40 C T Sun, J Tao and A S Kaddour, ‘Prediction of failure envelopes and stress–strain behaviour of composite laminates: Comparison with experimental results’, Compos Sci Techno, V62, pp 1672–1682, 2002, see also Chapter 5.8 in this book. 41 K-S Liu and S W Tsai, ‘A progressive quadratic failure criterion of a laminate’, Compos Sci Techno, V58, pp 1023–1032, 1998, see also Chapter 3.10 in this book.

26 42 A Kuraishi, S W Tsai and K Liu, ‘A progressive quadratic failure criterion, Part B’, Compos Sci Techno, V62, pp 1682–1696, 2002, see also Chapter 5.9 in this book. 43 W E Wolfe and T S Butalia, ‘A strain energy based failure criterion for nonlinear analysis of composite laminates subjected to biaxial loading’, Compos Sci Techno, V58, pp 1107–1124, 1998, see also Chapter 3.11 in this book. 44 T S Butalia and W E Wolfe, ‘Strain energy based non-linear failure criterion: comparison of numerical predictions and experimental observations for symmetric composite laminates’, Compos Sci Techno, V62, pp 1695–1710, 2002, see also Chapter 5.10 in this book. 45 P Zinoviev, S V Grigoriev, O V Lebedeva and L R Tairova, ‘Strength of multilayered composites under plane stress state’, Compos Sci Techno, V58, pp 1209–1224, 1998, see also Chapter 3.12 in this book. 46 P Zinoviev, Olga V Lebedeva and Ludmilla P Tairova, ‘Coupled analysis of experimental and theoretical results on the deformation and failure of laminated composites under a plane state of stress’, Compos Sci Techno, V62, pp 11711–1724, 2002, see also Chapter 5.11 in this book. 47 S R Soni, ‘A new look at commonly used failure theories in composite laminates’, in 24th Int Conf Structures and Structures Dynamics and Materials, May 1983, Lake Tohoe, AIAA/ASME, ASCE/ AHS, pp 171–179. 48 P W Mast, J G Michopoulos, R W Thomas, R Badaliance and I Wolock, ‘Characterisation of straininduced damage in composites based on the dissipated energy density: Part I- Basic scheme and formulation’, Int J of Theor. & Applied Fract. Mech., V22, pp 71–96, 1995. 49 P W Mast, J G Michopoulos, R W Thomas, R Badaliance and I Wolock, ‘Characterisation of straininduced damage in composites based on the dissipated energy density: Part II- Composite specimens and naval structures’, Int J of Theor. & Applied Fract. Mech., V22, pp 97–114, 1995. 50 P W Mast, J G Michopoulos, R W Thomas, R Badaliance and I Wolock, ‘Characterisation of straininduced damage in composites based on the dissipated energy density: Part III- General material constitutive relation’, Int J of Theor. & Applied Fract. Mech, V22, pp 115–125, 1995. 51 M Uemura and H Fakunaga, ‘Probabilistic burst strength of filament wound cylinder under internal pressure’, J Compos Mater, V15, pp 462–480, 1981. 52 S Li, S R Reid and P D Soden, ‘A continuum damage model for transverse matrix cracking in laminated fibre reinforced composites’, Phil Trans Roy Soc Lond A, V356, pp 2379–2412, 1998. 53 J Gosse and S Christensen, ‘Strain Invariant Failure Criteria for Polymers in Composite Materials’, Paper AIAA-2001–1184, pp 45–55, 2001. 54 P D Soden, M J Hinton and A S Kaddour, ‘Lamina properties and lay-up configurations and loading conditions of a range fibre reinforced composite laminates’, Compos Sci Techno, V58, pp 1011–1022, 1998, see Chapter 2.1 in this book. 55 P D Soden, M J Hinton and A S Kaddour, ‘Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates: Failure exercise benchmark data’, Compos Sci Techno, V62, pp 1489–1514, 2002, see Chapter 2.2 in this book. 56 M J Hinton, A S Kaddour and P D Soden, ‘Failure criteria for FRP’s- A world study’, (C536–018), 15–16 Sept 1998, IMechE HQ, London. 57 M J Hinton, A S Kaddour and P D Soden, ‘Prediction of world-wide failure theories: Part B: Some experimental comparisons’, 15–16 March 1999, IMechE HQ, London, UK. 58 M J Hinton, P D Soden and A S Kaddour, ‘On the maturity of world-wide failure theories: Comparison with experiments for ± 55° GRP laminates’, presented at ICCM-12, July 1999, Paris, France. 59 M J Hinton, P D Soden and A S Kaddour, ‘Predicting failure in fibre composites: Lessons learned from the World-Wide Failure Exercise’, in ICCM-13, Bejing, China, June 2001. 60 R Talreja, ‘A continuum mechanics characterisation of damage in composite materials’, Phil Trans Roy Soc Lond A, V339, pp 195–226, 1985.

27 Appendix to Chapter 1.1 Instructions to contributors for Part B Enclosed are copies of graphs showing your theoretical predictions of strength and stress– strain curves for selected laminates with experimental results superimposed upon them. Please prepare a short paper on the correlation between the results, for publication in a Special Issue (Part B) of the Composites Science and Technology journal. The Special Issue will contain: – An introduction by the editors containing an explanation of the exercise and the problems analyzed in ‘Part A’ and a description of the experimental results provided for use in Part B (similar to the paper enclosed). – A comparison of the experimental results with other published data. – A number of papers (including yours) by individual contributors comparing their own predictions with the experimental results provided. – Concluding comments by the editors on the overall comparison between the theory and experiments and areas requiring further work. Your paper should contain: – The enclosed graphs comparing your predictions with the experimental results. – Any comments you wish to make on the experimental results and your theoretical results and the correlation between them. Authors must limit their comments to comparing their own theory with experimental results. Your paper could also contain: – A description of further refinement of your model made since ‘Part A’ of the exercise was issued. – Additional graph or graphs to demonstrate refinements or special features of your method of analysis. ( If any of your additional figures are modified versions of the original figures, please use exactly the same scales). – A brief note of any proposed improvements to your method of analysis. – Comments on the nature and effect of your predicted failures and their relevance to the experimental data and to design practice. Contributors whose method of analysis takes into account ply grouping (differences in effective ply thicknesses) could compare the predicted performances of the ± 45° glass/ MY750 epoxy laminate under equal biaxial tensile loading (SR = 1:1), assuming two alternative stacking sequences: (a) The symmetric model + 45°/–45°/–45°/ + 45° and (b) The antisymmetric model + 45°/–45°/ + 45°/–45° (suitably constrained)

28 Contributors with integrated failure analysis and structural analysis packages may analyze the simple tubular specimen geometry and loading case described in the Appendix to these instructions* as additional contribution. Your paper should be sent to us as soon as possible and not later than the deadline date, when that has been agreed by all participants (including yourself). If you do not wish to comment on the results or you are unable to meet the agreed deadline, then the enclosed graphs would be taken as your contribution to Part B.

* See the Appendix to Chapter 2.2.

Section 2 Test Cases, Lamina Data and Experimental Results Under Biaxial Loads

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

30

CHAPTER 2.1

Lamina properties, lay-up configurations and loading conditions for a range of fibre reinforced composite laminates P. D. Sodena, M. J. Hintonb and A. S. Kaddourc a UMIST, Manchester, UK QinetiQ, Fort Halstead, Kent, UK c QinetiQ, Farnborough, Hampshire, UK b

Abstract This paper gives details of the input data and a description of the laminates provided to all participants in an exercise to predict the strength of composite laminates. The input data include the elastic constants and the stress strain curves for four unidirectional laminae and their constituents. Six types of laminates, chosen for the analysis, are described together with the lay-up, layer thicknesses, stacking sequences and the loading conditions. Consideration is given as to why these six laminates were selected and of the challenges imposed by the selected problems. The detailed instructions issued to the contributors are also presented.

1. Introduction In order to confirm the current state-of-the art of predicting failure in composite laminates, an exercise was launched by Hinton, Soden and Kaddour, Ref. [1]. In this exercise, selected workers, including leading academics and developers of software and numerical codes, were invited to submit papers describing their current theory and its intended application, predict the deformation and strength of selected laminates under a variety of biaxial loads and compare their prediction with experimental results provided. The process of selecting and setting a manageable, useful and balanced set of tasks to achieve the aims of the exercise was by no means simple or easy. However, six laminates were selected to represent a wide range of parameters. These parameters include the type of composite material (fibre and matrix), the type of laminate lay-up (unidirectional, angle ply, cross ply, quasi-isotropic etc.) and the loading conditions. The choice of laminates and loading conditions was determined, to some extent, by the availability of experimental data for use at a later stage of the exercise. The availability of data for particular laminates had, in turn, a large influence on the selection of materials for the exercise. The theoretical predictions and details of the theories used by the individual contributors are presented in Refs [2–13, 24–28] and the predictions of various theories are compared in

31 Ref. [14]. In all of these predictions, the authors were asked to employ exactly the same input data to allow rational comparison between the predictions of various theories. The purpose of the present paper is to provide full details of the input data provided to all participants. The input data included the elastic constants, stress strain curves and strengths for four unidirectional laminae and their constituents. The six types of laminates chosen for the analysis, the lay-ups, layer thicknesses, stacking sequences and the loading conditions for each laminate are described.

2. Material properties 2.1. General Many different types of composite materials are available. To make the exercise manageable, consideration was limited to continuous fibre reinforced thermosetting plastics. Taking into consideration the availability of suitably extensive experimental data for laminates, two important and widely used classes of fibres (carbon and Eglass) and one group of resin systems (epoxy resins) were selected for the exercise. A unidirectional (UD) lamina made of continuous fibres in a softer matrix was considered to be the basic building block for the multidirectional laminates. The properties of the laminate depend very much on the properties of the laminae. The behaviour of each lamina is, in turn, governed by its constituents, i.e. the properties of the fibres, the surrounding matrix, the interface and the relative amount of fibres and matrix in the lamina. Tables 1 to 10 present typical data for the properties of four unidirectional laminae, four epoxy resin matrices and four types of Eglass or carbon fibres. The stress strain behaviour of composite laminates is sometimes highly non-linear, particularly in shear. Figures 1 to 7 show typical stress strain curves for the selected laminae under a variety of uniaxial loadings. This data was presented to the participants. For performing theoretical analysis of the mechanical behaviour of multidirectional laminates under various loadings, most theories require the properties of each of the individual layers in the laminates. The properties required include: elastic constants and

Fig. 1. In-plane shear stress strain curve for E-glass/MY750/HY917/DY063 epoxy lamina.

32

Fig. 2. In-plane shear stress strain curve for E-glass/LY556/HT907/DY063 epoxy lamina.

Fig. 3. In-plane shear stress strain curve for AS4/3501–6 epoxy lamina.

Fig. 4. In-plane shear stress strain curve for T300/BSL914C epoxy lamina.

33

Fig. 5. Transverse compressive stress strain curve for E-glass/MY750/HY917/DY063 epoxy lamina.

Fig. 6. Transverse compressive stress strain curve for AS4/3501–6 epoxy lamina.

Fig. 7. Longitudinal tensile stress strain curve for AS4/3501–6 epoxy lamina.

34

Fig. 8. Schematic showing the co-ordinate system used for a unidirectional lamina.

thermal properties, strengths, failure strains, in some cases, full stress–strain curves and occasionally fracture toughness. Some methods of analysis require information on the properties of the constituent fibres and matrix. The three dimensional elastic constants for an orthotropic UD lamina consist of the following independent in-plane and through-thickness properties, see for instance Ref. [15]: E1, E2, E3, G12, G13, G23, ␷12, ␷13, ␷23 where the subscripts 1, 2 and 3 refer to the three mutually perpendicular principal material directions. Figure 8 shows a schematic diagram of a UD lamina with the co-ordinate system used. The rest of the Poisson’s ratios can be obtained by applying the reciprocal Maxwell relations, Ref. [15], which give ␷ij/Ei = ␷ji/Ej. Four of these constants (E1, E2, ␷12 and G12) pertain to the in-plane behaviour of thin laminae and the rest are related to the through-thickness (direction 3) behaviour. It is usually assumed that a unidirectional fibre reinforced lamina can be treated as transversely isotropic. For a transversely isotropic lamina, the independent elastic constants are reduced to five because E2 = E3, G12 = G13, ␷12 = ␷13 and G23 = E2/2(1 + ␷23), see Ref. [15]. For two dimensional (plane stress) analysis only four independent constants are required. Methods of measuring these properties are described in a number of references, e.g. Ref. [15]. Orthotropic composites generally possess nine strengths and nine failure strain values. These are longitudinal tensile and compressive properties X1T, ␧u1T, X1C and ␧u1C, transverse tensile and compressive properties X2T, ␧u2T, X2C and ␧u2C, through-thickness tensile and compressive properties X3T, ␧u3T, X3C and ␧u3C and in-plane and through-thickness shear properties S12, ␥u12, S13, ␥u13, S23 and ␥u23. The assumption of transverse isotropy and plane stress conditions reduces the number of uniaxial strength properties required to five for a UD lamina. The fracture energy of a unidirectional composite is yet another property that some analyses rely upon. Normally, GIC is determined from a double cantilever beam (DBC) test, see for instance Refs [16,17] for description of other methods used. There are a number of methods for measuring strengths and failure strains in tension, compression and in shear, Refs [15,17–20] and the failure properties obtained from different methods can be different. Some properties (e.g. compression and shear strengths, Refs [21,22]) are particularly difficult to determine accurately. Indeed, composite material characterisation is one of the key areas of concern and hence the problem of standardising test

35 methods for determining design allowables is currently being addressed by several agencies in the USA and Europe. The difficulties in measuring mechanical properties sometimes lead to a wide range of values being quoted for the same property of the same material. Where ever possible the material properties given in Tables 1–10 were obtained from the same source as experimental data for the laminates considered and are consistent with properties published elsewhere (see Ref. [23]). Inevitably, some of the materials data given in Tables 1–3 will be inaccurate.

2.2. Properties of the fibres Four types of fibres were selected in the analysis, two types of Eglass fibres and two types of carbon fibres. They were chosen for consistency with data for particular laminates. The fibres are: – – – –

E-Glass fibres, Silenka, 1200tex E-Glass fibres, Gevetex, 21
K43 T300 carbon fibres AS4 carbon fibres

The properties assumed for these fibres are listed in Table 2. Both types of E-glass fibres are isotropic while the carbon fibres are anisotropic. In the latter case, the modulus along the fibre direction is much higher than that in the transverse direction. Determining the properties of the fibres is not always straightforward as the fibres are normally of diameters in the range 5–20 m, and hence they are difficult to handle. In order to obtain the mechanical properties, indirect methods are usually adopted. In these methods, tests are carried out on unidirectional laminae where the fibres are embedded within a matrix. Suitable micromechanics relations are then used to extract the properties of the fibres form the results of UD tests. As a result, some variations of the extracted properties are expected, Ref. [23].

2.3. Properties of the matrices A variety of matrices are available for fibre reinforced composite materials. Epoxy resins have been employed in a number of load bearing application. Four types of epoxy matrices were used in the analysis. These are: (1) (2) (3) (4)

MY750/HY917/DY063 (Ciba-Geigy) LY556/HT907/DY063 (Ciba-Geigy) 3501–6 (Hercules) BSL914C

Typical properties for these four matrix materials are shown in Table 3. The data listed in Table 3 were selected from a wide range of values available in the literature, Ref. [23].

36

Table 1 Mechanical and thermal properties of four unidirectional laminae. Fibre type

AS4

T300

E-glass 21
K43 Gevetex

Silenka E-Glass 1200tex

Matrix Specification Manufacturer Fibre volume fraction Vf % Longitudinal modulus E1 GPa Transverse modulus E2 GPa In-plane shear modulus G12 GPa Major Poisson’s ratio 12 Through thickness Poisson’s ratio 23 Longitudinal tensile strength XT MPa Longitudinal compressive strength XC MPa Transverse tensile strength YT MPa Transverse compressive strength YC MPa In-plane shear strength S12 MPa Longitudinal tensile failure strain 1T % Longitudinal compressive failure strain 1C % Transverse tensile failure strain 2T % Transverse compressive failure strain 2C % In-plane shear failure strain 12u % Strain energy release rate GIC J/m2 Longitudinal thermal coefficient 1 10–6/°C Transverse thermal coefficient 2 10–6/°C Stress free temperature °C Curing

3501–6 epoxy Prepreg Hercules 60 126* 11 6.6* 0.28 0.4 1950** 1480 48 200** 79** 1.38 1.175 0.436 2.0 2 220 –1 26 177

BSL914C epoxy Prepreg DFVLR 60 138 11 5.5* 0.28 0.4 1500 900 27 200 80** 1.087 0.652 0.245 1.818 4 220 –1 26 120

LY556/HT907/DY063 epoxy Filament winding DLR 62 53.48 17.7 5.83* 0.278 0.4 1140 570 35 114 72** 2.132 1.065 0.197 0.644 3.8 165 8.6 26.4 120 2 hr at 120°C 2 hr at 150°C

MY750/HY917/DY063 epoxy Filament winding DRA 60 45.6 16.2 5.83* 0.278 0.4 1280 800 40 145** 73** 2.807 1.754 0.246 1.2 4 165 8.6 26.4 120 2 hr at 90°C 1.5 hr at 130°C 2 hr at 150°C

* Initial modulus. ** Nonlinear behaviour and stress strain curves and data points are provided.

Table 2 Mechanical and thermal properties of four fibres. Fibre type

AS4

T300

E-glass 21
K43 Gevetex

Silenka E-Glass 1200tex

Longitudinal modulus Ef1 GPa Transverse modulus Ef2 GPa In-plane shear modulus Gf12 GPa Major Poisson’s ratio f12 Transverse shear modulus Gf23 Longitudinal tensile strength XfT MPa Longitudinal compressive strength XfC MPa Longitudinal tensile failure strain f1T% Longitudinal compressive failure strain f1C% Longitudinal thermal coefficient f1 10–6/°C Transverse thermal coefficient f2 10–6/°C

225 15 15 0.2 7 3350 2500 1.488 1.111 –0.5 15

230 15 15 0.2 7 2500 2000 1.086 0.869 –0.7 12

80 80 33.33 0.2 33.33 2150 1450 2.687 1.813 4.9 4.9

74 74 30.8 0.2 30.8 2150 1450 2.905 1.959 4.9 4.9

Table 3 Mechanical and thermal properties of four matrices. 3501-6 epoxy

BSL914C epoxy

LY556/HT907/DY063 epoxy

MY750/HY917/DY063 epoxy

Manufacturer Modulus Em Gpa Shear modulus Gm Gpa Poisson’s ratio m Tensile strength YmT MPa Compressive strength YmC MPa Shear strength Sm MPa Tensile failure strain mT% Thermal coefficient m 10–6/°C

Hercules 4.2 1.567 0.34 69 250 50 1.7 45

DFVLR 4.0 1.481 0.35 75 150 70 4 55

Ciba Geigy 3.35 1.24 0.35 80 120 – 5 58

Ciba Geigy 3.35 1.24 0.35 80 120 – 5 58

37

Matrix type

38 Table 4 Data for in-plane shear stress strain curve of AS4/3501-6 lamina. Strain %

Stress MPa

Strain %

Stress MPa

0.000 0.076 0.114 0.152 0.190 0.228 0.266 0.305 0.344 0.383 0.424 0.465 0.507

0.0 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5

0.551 0.587 0.596 0.644 0.746 0.860 0.991 1.142 1.319 1.527 1.772 2.000

35.0 37.0 37.5 40.0 45.0 50.0 55.0 60.0 65.0 70.0 75.0 79.0

2.4. Properties of the unidirectional laminae Table 1 lists typical values of properties of four different unidirectional (UD) laminae used in the exercise. The four UD laminae are: – E-Glass/MY750 epoxy (Silenka E-glass/MY750/HY917/DY063) – E-Glass/LY556 epoxy (Gevetex E-glass/LY556/HT907/DY063)

Table 5 Data for in-plane shear stress strain curve of T300/BSL914C epoxy lamina. Strain %

Stress MPa

Strain %

Stress MPa

0.000 0.182 0.273 0.364 0.455 0.548 0.644 0.747 0.864 1.004

0.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0

1.185 1.431 1.777 2.002 2.272 2.743 2.984 3.447 4.0

55.0 60.0 65.0 67.5 70.0 73.5 75.0 77.5 80.0

39 Table 6 Data for in-plane shear stress strain curve of E-glass/MY750/HY917/DY063 epoxy lamina. Strain %

Stress MPa

Strain %

Stress MPa

0.0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

0.0 5.830 11.660 17.490 23.320 29.150 34.980 37.705 41.298 44.535 47.446 50.056 52.391 54.475 56.331 57.982 59.447 60.748 61.902

1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.2 3.4 3.5 3.7 3.8 3.9 4.0

62.927 63.839 64.653 65.382 66.041 66.639 67.188 67.696 68.173 68.624 69.055 69.472 70.273 71.039 71.409 72.109 72.432 72.728 72.991

– AS4 Carbon/epoxy (AS4/3501–6) – T300 Carbon/epoxy (T300/BSL914C) The constituent fibres and the epoxy resin matrices were as described in sections 2.2 and 2.3 above. The material properties were derived by a variety of methods and the data and its sources are discussed in Ref. [23]. Cooling after curing results in residual stresses and the assumed stress free temperatures are included in Table 1.

3. Details of laminates selected Undoubtedly, there are many interesting and unresolved problems in the area of predicting the stress strain characteristics and fracture of composite laminates. Selecting the laminates and loading conditions to be analysed was not easy. The following factors were taken into consideration: • The problems should cover a wide range of lay-ups. For this reason, six different lay-ups were chosen. These lay-ups are 0° Unidirectional lamina, (90°/ ± 30°/90°), (0°/ ± 45°/90°), ( ± 55°), (0°/90°) and ( ± 45°) multi-directional laminates.

40 Table 7 Data for in-plane shear stress strain curve of E-glass/LY556/HT907/DY063 epoxy lamina. Strain %

Stress MPa

Strain %

Stress MPa

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7

0.0 5.830 11.660 17.490 23.320 29.150 34.980 37.705 41.298 44.535 47.446 50.056 52.391 54.475 56.331 57.982 59.447 60.748

1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.2 3.4 3.5 3.7

61.902 62.927 63.839 64.653 65.382 66.041 66.639 67.188 67.696 68.173 68.624 69.055 69.472 70.273 71.039 71.409 72.109

• A wide range of loading conditions should be analysed. The problems selected include generating the complete failure envelopes under two types of combined stresses, namely combined direct stresses ␴y versus ␴x and combined direct stress and shear stress (␴y versus ␥xy) or (␴x and ␥xy). • The problems should include predicting the stress strain curves under both uniaxial and biaxial loading because changes in laminate stiffness may be critical in some applications. • The laminates analysed should develop different types of damage due to loading in shear, tension and compression, transverse and parallel to the fibre direction. • Both linear and non-linear properties of unidirectional laminae should be considered. • Experimental results should be available which could be used to check the effectiveness of the theoretical predictions. The six types of laminate lay-ups selected were analysed under a variety of loading conditions. The instructions to participants (see Appendix), specified how loads were to be applied and how results were to be presented. Table 11 summarises laminate type, material type and the graphical results requested. Figure 9 shows a diagrammatic representation of a failure envelope. The laminates selected are as follows: (a) 0° Unidirectional lamina of thickness 1 mm. (b) Balanced and symmetric (90°/ ± 30°/90°) laminate. Layer orientation: 90°/ + 30°/–30°/ –30°/ + 30°/90°. The total thickness is 2 mm. The thickness of the ± 30° plies is 81.8% and that of the 90° plies is 17.2% of the total thickness of the laminate.

41 Table 8 Data for transverse compressive stress strain curve of E-glass/MY750 /HY917/ DY063 epoxy lamina. Strain %

Stress MPa

0.000 –0.062 –0.123 –0.185 –0.247 –0.309 –0.371 –0.434 –0.499 –0.566 –0.640 –0.723 –0.822 –0.944 –1.103 –1.200

–0.0 –10.0 –20.0 –30.0 –40.0 –50.0 –60.0 –70.0 –80.0 –90.0 –100.0 –110.0 –120.0 –130.0 –140.0 –145.0

(c) Balanced and symmetric (0°/ ± 45°/90°) quasi-isotropic laminate. Layer orientation: 90°/ + 45°/–45°/0°/0°/–45°/ + 45°/90°. The total thickness of the laminate is 1.1 mm, and all the plies have identical thickness. (d) Balanced and symmetric ( ± 55°) angle ply laminate. The layer orientation is + 55°/–55°/–55°/ + 55°. The total thickness is 1 mm and the thickness of each lamina is 0.25 mm. (e) Balanced and symmetric (0°/90°) cross ply laminate. The layer orientation is 0°/90°/90°/0°. The total thickness is 1.04 mm and each ply has a thickness of 0.26 mm. (f) Balanced and symmetric ( ± 45°) angle ply laminate. The layer orientation is + 45°/–45°/–45°/ + 45°. The total thickness is 1 mm and each ply has a thickness of 0.25 mm. Schematic diagrams showing the loading directions, layer and laminate dimensions and stacking sequence of the laminates are shown in Figs 10 to 15. Note that the angles of the fibres in each layer are measured from x direction as shown in Fig. 8. The selection of the laminates and the loading conditions was based upon the following considerations.

3.1. Selection of 0° unidirectional laminae Many failure theories are formulated on the basis of the behaviour of a single lamina and applied using laminate theory which assumes that a lamina subjected to combined stresses will behave identically whether it is isolated or within a multidirectional laminate. Before preceding to analyse the behaviour of laminates, participants were first asked to predict failure envelopes for unidirectional laminae under biaxial loads. The objective was to demonstrate

42 Table 9 Data for transverse compressive stress strain curve of AS4/3501-6 lamina. Strain %

Stress MPa

0.000 –0.091 –0.182 –0.273 –0.364 –0.455 –0.545 –0.636 –0.728 –0.819 –0.911 –1.003 –1.096 –1.191 –1.288 –1.388 –1.493 –1.604 –1.723 –1.854 –2.000

0.0 –10.0 –20.0 –30.0 –40.0 –50.0 –60.0 –70.0 –80.0 –90.0 –100.0 –110.0 –120.0 –130.0 –140.0 –150.0 –160.0 –170.0 –180.0 –190.0 –200.0

and compare the assumptions and predictions at this basic level which will presumably be reflected in the accuracy of more complex laminate predictions. Figure 10 shows diagrams of 0° unidirectional lamina under various types of combined biaxial loads (␴y and ␴xy) (␴x and ␴xy) and (␴y and ␴x). The unidirectional composite materials selected for the analysis were: E-glass/MY750 subjected to combined ␴y and ␴x. E-glass/LY556 subjected to combined ␴y and ␴xy. T300/914C subjected to combined ␴x and ␴xy.

3.2. Selection of ( ± 30°/90°) laminate This ( ± 30°/90°) E-glass/LY556 laminate, Fig. 11, was selected partly because experimental results are available for the final failure under combined biaxial loading of ␴y with ␴x and ␴x

43 Table 10 Data for longitudinal tensile stress strain curve of AS4/3501-6 lamina. Strain %

Stress MPa

0.0 0.157 0.235 0.312 0.388 0.463 0.537 0.610 0.682 0.754 0.824 0.893 0.962 1.029 1.095 1.160 1.224 1.287 1.380

0.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0 1100.0 1200.0 1300.0 1400.0 1500.0 1600.0 1700.0 1800.0 1950.0

with ␴xy. The construction of the laminate is such that the thickness of the ± 30° plies, being 82.8% of the total thickness, is different from that of the 90°plies, which form 17.2% of the total thickness of the laminate. That laminate is not quasi-isotropic, a number of different modes of failure are expected to be encountered under biaxial loading and the final failure stresses are not expected to be simply due to fibre failure.

3.3. Selection of (0°/ ± 45°/90°) laminate Quasi isotopic laminates are an important class of composites and most familiar to the aerospace industries. One of the common quasi-isotropic laminates is that chosen for study in the exercise which consisted of (0°/ ± 45°/90°) lay-up, made of AS4/3501–6 carbon/epoxy material, Fig. 12. Unlike the ( ± 30°/90°) laminate described above, the (0°/ ± 45°/90°)s quasiisotropic laminates are expected to exhibit the same strengths when loaded in the 0° and 90° directions. Experimental results are available for failure of this type of laminates under combined tension-tension and tension-compression biaxial loading. Therefore, one of the tasks given to the contributors was to generate the biaxial failure stress envelope (␴y versus ␴x).

44

Table 11 Summary of laminate types, material types and plots required form contributors. Laminate type

Material type

Plots required and description of loading conditions

0° unidirectional lamina

E-glass/LY556/HT907/DY063 T300/BSL914C E-glass/MY750/HY917/DY063

1 – ␴y versus ␶xy failure stress envelope 2 – ␴x versus ␶xy failure stress envelope 3 – ␴y versus ␴x failure stress envelope

(90°/ ± 30°/90°) laminate

E-glass/LY556/HT907/DY063

4 – ␴y versus ␴x failure stress envelope 5 – ␴x versus ␶xy failure stress envelope

(0°/ ± 45°/90°) laminate

AS4/3501-6

6 – ␴y versus ␴x failure stress envelope 7 – Stress strain curves under uniaxial tensile loading for (␴y/␴x = 1/0). 8 – Stress strain curves for ␴y/␴x = 2/1

± 55° angle ply laminate

E-glass/MY750/HY917/DY063

9 – ␴y versus ␴x failure stress envelope 10 – Stress strain curves under uniaxial tensile loading for ␴y/␴x = 1/0 11 – Stress strain curves for ␴y/␴x = 2/1

(0°/90°) cross ply laminate

E-glass/MY750/HY917/DY063

12 – Stress strain curve under uniaxial tensile loading for ␴y/␴x = 0/1

± 45° angle ply laminate

E-glass/MY750/HY917/DY063

13 – Stress strain curves for ␴y/␴x = 1/1 14 – Stress strain curves for ␴y/␴x = 1/–1

45

Fig. 9. Schematic of a failure envelope where initial and final failure stages are marked.

In addition, two stress strain curves were requested. The contributors were asked to predict the stress strain curves under uniaxial tensile loading (␴y/␴x = 1/0) and the stress strain curves under biaxial tension (␴y/␴x = 2/1). 3.4. Selection of ( ± 55°) laminate The mechanical response of angle ply laminates should provide useful information on the ability of failure theories to predict various forms of failure induced by the presence of various stress components. Three stress components normally exist in the individual plies and those are tension (or compression) parallel and transverse to the fibre direction and shear stresses. However, the relative magnitude of these stresses depends on fibre orientation and loading conditions. The ± 55° laminate, Fig. 13, was selected on the basis of the following considerations – Its wide spread use in industrial pipework. – The availability of experimental results on the failure under a wide range of biaxial failure stresses including those in the compression-compression quadrant. For the ± 55° laminates as for the quasi-isotropic laminate, one of the tasks given to the contributors was to generate the biaxial failure stress envelope (␴y versus ␴x). In addition, two stress strain curves were requested. The contributors are asked to predict the stress strain curves under biaxial tension (␴y/␴x = 2/1) and the stress strain curves under uniaxial tensile loading (␴y/␴x = 1/0) which is expected to produce non-linear response. 3.5. Selection of 0°/90° laminate Cross ply laminates consisting of layers oriented at 0° and 90° are a classical example used for studying the development of matrix cracking (transverse tension cracking) and its effect

46

Fig. 10. Diagrams of 0° unidirectional lamina under various types of combined biaxial loads of (␴y and ␥␹␷) (␴x and ␥␹␷) and (␴y and ␴x).

on the load carrying capacity of laminated composites. One of the simplest forms of cross ply laminates, Fig. 14, is made of plies oriented at 0°/90°/90°/0°, all of the same thickness. The case chosen for analysis in the exercise is that of 0°/90°/90°/0° laminate under uniaxial tension.

3.6. Selection of ( ± 45°) laminate Experimental results are available on the behaviour of the ± 45° laminate under stress ratios ␴y/␴x = 1/1 and ␴y/␴x = 1/–1 (see Fig. 15) which could be regarded as equivalent to a 0°/90° laminate loaded under biaxial tension and pure shear, respectively. Hence, the theoretical solutions requested should provide an insight into the degree of understanding of 0°/90° laminates under loading cases other than uniaxial tension.

47

Fig. 11. Diagram showing the lay-up and loading configurations of the 90°/ ± 30°/90° laminate. Note the total thickness of the laminate is 2 mm where h1 = 0.172 mm and h2 = 0.414 mm.

Fig. 12. Diagram showing the lay-up and loading configurations of the (0°/ ± 45°/90°)s laminate. Note that the total thickness of the laminate is 1.1 mm and all the layers have the same thickness.

Fig. 13. Diagram showing the lay-up and loading configurations of the ± 55° laminate. Note that the total thickness of the laminate is 1 mm and all the four layers have the same thickness.

48

Fig. 14. Diagram showing the lay-up and loading configurations of the 0°/90°/0° laminate. Note the total thickness of the laminate is 1.04 mm, b = 0.52 and a = 0.26 mm.

Fig. 15. Diagram showing the lay-up and loading configurations of the ± 45° laminate. Note that the total thickness of the laminate is 1 mm and all the layers have the same thickness.

4. Concluding remarks – This paper presented details of the mechanical properties of the fibres, matrices and the unidirectional laminae used in the failure exercise. – The materials selected have been widely used in practical applications and their properties are reasonably well characterised. – Nonlinear stress strain data has been presented as well as elastic constants for linear elastic analysis. – In some cases, the literature gave a wide range of values for the same property and in other cases no data were available. The values chosen were in some cases arbitrary and inaccurate but this should not devalue the first part of this exercise as all the participants used exactly the same data. – The six laminates chosen for analysis by the participants are considered to be representative of a wide range of composite laminates encountered in practical use in a variety of industries. – A wide range of practical biaxial loading conditions were also specified which should produce a variety of modes of failure. In some cases a succession of failures may occur before the laminate can no longer carry load.

49 – In some cases the stress strain behaviour of the laminates is expected to be linear and in some other cases highly non-linear. – The solutions for this wide range of problems should help in stimulating researchers and engineers working in the area of composites to tackle the some of complex and challenging problems associated with design of composite laminates and should highlight some of major problems and gaps in current knowledge and practice.

References 1 M J Hinton, A S Kaddour and P D Soden, ‘The World-Wide Failure Exercise : Its Origin, Concept And Content’, see Chapter 1 in this book. 2 P K Gotsis, C C Chamis and L Minnetyan, ‘Prediction of composite laminate fracture: micromechanics and progressive fracture’, Compos Sci Techno, V58, pp 1137–1150, 1998, see also Chapter 3.1 in this book. 3 G C Eckold, ‘Failure criteria for use in the design environment’, Compos Sci Techno, V58, pp 1095–1106, 1998, see also Chapter 3.2 in this book. 4 E C Edge, ‘Stress based Grant-Sanders method for predicting failure of composite laminates’, Compos Sci Techno, V58, pp 1043–1044, 1998, see also Chapter 3.3 in this book. 5 L N McCartney, ‘Predicting transverse crack formation in cross-ply laminate’, Compos Sci Techno, V58, pp 1069–1082, 1998, see also Chapter 3.4 in this book. 6 L J Hart-Smith, ‘Predictions of the original and truncated maximum strain failure models for certain fibrous composite laminates’, Compos Sci Techno, V58, pp 1151–1178, 1998, see also Chapter 3.5 in this book. 7 L J Hart-Smith, ‘Predictions of a generalised maximum-shear-stress failure criterion for certain fibrous composite laminates’, Compos Sci Techno, V58, pp 1179–1208, 1998, see also Chapter 3.6 in this book. 8 A Puck and H Schürmann, ‘Failure analysis of FRP laminates by means of physically based phenomenological models’, Compos Sci Techno, V58, pp 1045–1068, 1998, see also Chapter 3.7 in this book. 9 A Rotem, ‘Prediction of laminate failure with Rotem failure criterion’, Compos Sci Techno, V58, pp 1083–1094, 1998, see also Chapter 3.8 in this book. 10 C T Sun and J X Tao, ‘Prediction of failure envelopes and stress strain behaviours of composite laminates’, Compos Sci Techno, V58, pp 1125–1136, 1998, see also Chapter 3.9 in this book. 11 K-S Liu and S W Tsai, ‘A progressive quadratic failure criterion of a laminate’, Compos Sci Techno, V58, pp 1023–1032, 1998, see also Chapter 3.10 in this book. 12 W E Wolfe and T S Butalia, ‘A strain energy based failure criterion for nonlinear analysis of composite laminates subjected to biaxial loading’, Compos Sci Techno, V58, pp 1107–1124, 1998, see also Chapter 3.11 in this book. 13 P Zinoviev, S V Grigoriev, O V Labedeva and L R Tairova, ‘Strength of multilayered composites under plane stress state’, Compos Sci Techno, V58, pp 1209–1224, 1998, see also Chapter 3.13 in this book. 14 A S Kaddour, M J Hinton and P D Soden, ‘A comparison of the predictive capabilities of current failure theories for composite laminates: Additional contributions’, Compos Sci Techno, V64, pp 449–476, 2004, 2002, see also the up-date Chapter 4 in this book 15 S W Tsai, ‘Composite design’, 4th edition, Think Composites, Dayton, Ohio, 1988. 16 I M Daniel, G Yaniv and J W Auser (1987), ‘Rate effects on delamination fracture toughness of graphite/epoxy composites’, in 4th Int Conf Composite Structures, V2, I M Marshall Ed., Paisely College of Technology, Scotland, pp 2.258–2.272, Elsevier Applied Science Publishers, London. 17 I M Daniel and O Ishai (1994), ‘Engineering Mechanics of Composite Materials’, Oxford University Press, Oxford, UK.

50 18 Y M Tarnopol’skii and T Y Kincis (1985), ‘Static test methods for composites’, in Handbook of Composites, V3, Failure Mechanics of Composites, Edited by G C Sih and A M Skudra, Elsevier Science Publishers B.V, pp 215–275. 19 S R Swanson, M J Messick and G R Toombes (1985), ‘Comparison of torsion tube and Iosipescu in-plane shear test results for a carbon fibre reinforced epoxy composite’, Composites, V16, pp 220–224. 20 S R Swanson and G R Toombes (1989), ‘Characterisation of prepreg tow carbon/epoxy laminates’, J Engg Mater Techno, Trans ASME, V111, pp 150–153. 21 R Y Kim and A S Castro (1993), ‘Failure of carbon fibre-reinforced epoxy composites under combined loading’, in Proc. ICCM-9, Vol V, pp 15–22, edited by Miravete A, University of Zaragona, Woodhead Publishing Ltd, Spain. 22 C T Sun and A W Jun (1994), ‘Compressive strength of unidirectional fibre composites with matrix nonlinearity’, Compos Sci Techno, V52, pp 577–587. 23 P D Soden, M J Hinton and A S Kaddour, ‘Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates’, see Chapter 2.2 in this book. 24 T A Bogetti, C P R Hoppel, V M Harik, J F Newill and B P Burns, ‘Predicting the nonlinear response and progressive failure of composite laminates’, Compos Sci Techno, V64, pp 329–342, 2004, 2002, see also Chapter 3.15 in this book. 25 R G Cuntze and A Freund, ‘The predictive capability of failure mode concept – based strength criteria for multidirectional laminates’, Compos Sci Techno, V64, pp 343–377, 2004, 2002, see also Chapter 3.14 in this book. 26 S Mayes and A C Hansen, ‘Composite laminate failure analysis using multicontinuum theory’, Compos Sci Techno, V64, pp 379–394, 2004, 2002, see also Chapter 3.15 in this book. 27 Z-m Huang, ‘A bridging model prediction of the tensile strength of composite laminates subjected to biaxial loads’, Compos Sci Techno, V64, pp 395–448, 2004, 2002, see also Chapter 3.16 in this book. 28 L J Hart-Smith, ‘Expanding the capabilities of the ten-percent rule for predicting the strength of fibre-polymer composites’, Compos Sci Techno, V62, pp 1515–1544, 2002, see also Chapter 3.17 in this book.

Appendix to Chapter 2.1 Instructions to contributors The instructions provided to each contributor are listed below in their original form. The attached notes specify the unidirectional lamina properties, layer thicknesses, stacking sequences and loading for each of the laminates we would like you to analyse. The in-plane loads (section stresses) should be applied in the x and y directions defined in the diagrams provided for each laminate. The section stresses ␴x and ␴y are defined in the usual way as the in-plane loads per unit width divided by the total thickness of the laminate. In your calculations assume that the loads are increased monotonically, keeping the ratios of ␴x/␴y, ␥xy/␴x and ␥xy/␴y constant. Record and tabulate the magnitude of the section stresses (and if appropriate the type and location of failure) at which each failure is predicted. Repeat the calculation to cover the range of stress ratios (2 or 4 quadrants) indicated by the graphs provided for each laminate.

51 Plot the results using the scales provided for each laminate. Draw curves through the results to represent the initial (inner) and final (outer) failure envelopes. Indicate any intermediate failure points. Plot the section stress versus strain curves as requested for particular laminates using the scales provided. It would be helpful if you would also send us your tables of results, but these will probably not be included in the paper. Results in the form of data files sent to us by E-mail or on floppy disks would be appreciated. We are asking all contributors to use the same material properties even if you have reservations about the values provided. Your theory may not require all the lamina properties provided (e.g. some software assumes linear elastic properties). In that case please employ your usual assumptions and neglect any information which is not needed. If your theory requires additional (or different) information from that provided, please let us know as soon as possible and we will endeavour to provide that information. If you have default values for any missing parameters (e.g. interaction coefficients), we prefer you to use those. In some cases the theory employed may not be intended to be applied to the whole range of laminates specified here. In that case, you may opt to analyse only some of the laminates but please explain the reasons for not analysing the other laminates in your paper. The paper should describe your failure theory and method of application to laminates in sufficient detail to allow your predictions to be reproduced by others, comment on the nature and effects of the failures predicted and, if appropriate, how your predictions could be used for design. After receiving all the theoretical papers with your permission to publish the results, the experimental results will be superimposed on the theoretical predictions. The superimposed graphs will be sent back to you together with tables of the experimental results for your future use and information on how the experimental results were obtained. The second paper (part B) would present graphs of superimposed results with any comment you may wish to make on the correlation between experiment and theory. You may choose to add a figure (or figures) to demonstrate refinement or particular features of your approach. You could indicate any future development to your theory which would allow you to consider a wider variety of laminates than those you are able to analyse immediately. For those participants who have integrated failure analyses and structural analysis packages, details of the simple specimen geometry that you may opt to analyse as part of the second paper, will be sent as soon as you request them.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 QinetiQ Ltd. Published by Elsevier Ltd. All rights reserved

52

CHAPTER 2.2

Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates: Failure exercise benchmark data P. D. Sodena, M. J. Hintonb and A. S. Kaddourc a UMIST, Manchester, UK QinetiQ, Fort Halstead, Kent, UK c QinetiQ, Farnborough, Hampshire, UK b

Abstract In Part A of the World Wide Failure Exercise, presented in Chapters 3 and 4, all contributors were given exactly the same set of material properties and were asked to predict the strength and deformation of the same set of laminates under a range of specified loading conditions. In Part B of the Exercise, available experimental results are superimposed on the theoretical predictions and returned to the contributors for comment. The test data were for (a) 0° unidirectional laminae under biaxial direct and shear loads (b) (90°/ ± 30°)s, (0°/ ± 45°/90°)s, ( ± 55°)s multi-directional laminates under biaxial loads and (c) stress strain curves for (0°/ ± 45°/90°)s, ( ± 55°)s, ( ± 45°)s and (0°/90°)s laminates under uniaxial and biaxial loads. This paper briefly describes the experimental results issued in Part B of the Exercise and their origin and limitations. Comments are made on the material properties given for the unidirectional fibre reinforced layers and constituents used in Part A of the Exercise and on approximations in the laminate models specified in Part A. Keywords: Failure exercise, biaxial loads, tubes, torsion, internal/external pressure, composite laminates.

1. Introduction The present paper is one of a series of papers related to the world wide failure exercise, which was launched in order to determine the accuracy of current theories for predicting failure in composite laminates. That has been achieved by comparing the predictions of twelve different failure theories with experimental data. This paper presents the experimental data and comments on the experiments. It was issued to all the participants in the failure exercise before they compared their theoretical predictions with the experimental results. The experimental data covered 14 test cases involving (a) 0° unidirectional laminae under biaxial direct and shear loads (b) (90°/ ± 30°)s, (0°/ ± 45°/90°)s, ( ± 55°)s multi-directional

53 laminates under biaxial loads and (c) stress strain curves for (0°/ ± 45°/90°)s, ( ± 55°)s, ( ± 45°)s and (0°/90°)s laminates under uniaxial and biaxial loads. It has been noted that whilst there is a considerable body of experimental data describing the strength and mechanical response of composites, there appears to be a relatively limited number of studies where a wide range of loading conditions have been applied to the same laminate. This is partly due to the difficulties involved in such experiments. Almost all of the experimental results chosen for use in this Exercise were derived from tests on tubular specimens. Testing of tubes avoids problems associated with free edge effects that are encountered with coupon and other specimens and a wide range of biaxial and triaxial stresses can be applied by subjecting tubular specimens to combinations of internal or external pressure, torsion and axial load. The testing of tubular specimens can, however, involve problems including: – The avoidance of failures at end constraints and end fittings and the minimisation of end effects. – Tubular specimens can experience various forms of buckling when they are subjected to circumferential or axial compression or torsion loading. – Tubes may exhibit changes in geometry during loading but these effects are usually ignored when processing experimental results. Tubular specimens can fail in a variety of ways. Unless otherwise stated the specimens considered here failed by fracture. Some results are given for leakage failures in cases where the specimens were subjected to internal pressure. Very few experimental results are available for ‘initial’ failure, or the onset of damage such as resin cracking. In some cases the initial failures have no obvious effect on the mechanical behaviour of the structure but in other cases ‘initial’ failure may be expected to influence the shape of the stress–strain curves and for this reason contributors were asked to predict the shape of stress–strain curves. In most cases all of the experimental results selected for any given set of loading cases were from tests carried out by the same investigators in the same laboratory. References are given to the sources of the experimental data and, where possible, to publications giving more details of the specimens and test procedures for each set of experiments. Each specimen was tested to failure at a fixed ratio (SR)1 of circumferential to axial stresses in the test section and the loads were usually increased continuously until fracture occurred. Unless otherwise stated the applied stresses were calculated based on initial (un-deformed) tube dimensions with no allowance made for change of shape during loading. The laminated tubes were constructed from unidirectional fibre reinforced composite plies. The ply angle  is specified as the angle between the fibre direction and the axis of the tube. Note that  = 90° represents the circumferential direction and y (or ) and x (or z) are the applied stresses in the circumferential and the axial directions, respectively. Unless otherwise stated the fibre volume fraction in the laminated specimens is approximately the same as the unidirectional laminae (UD) provided in Part A. The cure procedure followed in many cases that recommended by the resin manufacturers and was the same as for the UD laminates in Part A.

1

Similar definition of stress ratio is used when the loading involves normal and shear stresses.

54 Whenever possible, the material properties issued in Part A were obtained from publications by the same investigators who tested the laminates presented here. In some cases, the investigators had employed the same materials in experimental studies of different laminates. Sources of discrepancy between the predictions and the experimental results include combinations of the following: (I) (II) (III) (IV)

Errors and approximations in the theory. Errors in the experimental results. Errors in the given material properties. Differences between the model laminates specified for analysis and the construction of the specimens used in the experiments.

The following sections present the experimental data for each of the laminates, comment on the source of the unidirectional laminate data presented in Part A and discuss some of the assumptions in the laminate models.

2. Description of experimental data for the exercise 2.1. Biaxial failure envelope for unidirectional E-glass/epoxy lamina under combined transverse and shear loading (x versus xy) The results were given by Hütter et al., Ref. [1], and were obtained from experiments on circumferentially wound filament wound tubes. The tubes were 60 mm internal diameter, 2 mm thick and were constructed from 62% by volume Vetrotex 21
K43 E-glass fibre (Gevetex) rovings and a Ciba Geigy epoxy resin system LY556/HT907/DY063 mixed weight proportions of 100 : 85 : 4. The tubes were cured at 100°C for 2 hours and post cured at 150°C for 2 hours. The tubes were tested under torsion combined with axial tension or compression and the biaxial data are shown in Fig. 1 and listed in Table 1. Details of the tests are given by Hütter et al., Ref. [1]. Also shown in Fig. 1 are the data supplied to contributors for use in their analysis. This data was taken from the original results presented in an earlier paper by the originators of the biaxial test program, Krauss and Schelling, Ref. [2], and do not agree exactly with the strength obtained from the other test data presented in Fig. 1 which were reported by Hütter et al., Ref. [1].

2.2. Biaxial failure envelope for unidirectional T300/914C carbon/epoxy lamina under combined longitudinal and shear loading (y versus xy) The results of Schelling and Aoki, Refs [3,4], are used in the Exercise. The specimens were in the form of axially wound tubes made from prepreg T300/BSL914C carbon/epoxy. The tubes were tested under combined axial tension or compression and torsion. All the tubes were end reinforced and grips were used to transmit the torque to the tubes.

55

Fig. 1. Biaxial failure stress envelope for 0° unidirectional lamina made glass/epoxy under transverse and shear loading (y versus xy).

Table 1 Data for biaxial failure stresses of 0° unidirectional lamina under combined transverse and shear loading (y versus xy). Hütter et al., Ref. [1]. Axial stress y MPa

Shear stress xy MPa

40.0 26.9 30.7 34.0 18.0 –137.8 –142.0 –132.3 –104.6 –134.6 –99.4 –70.5 –122.0 –44.1 –133.3 0.00

0.0 36.0 32.3 12.8 51.3 0.0 0.0 0.0 46.7 28.9 64.5 96.6 54.6 81.9 20.7 61.2

56

Fig. 2. Biaxial failure stress envelope for 0° unidirectional lamina made of T300/BSL914C under longitudinal and shear loading (x versus xy).

One set of tubes was tested under combined axial tension and torsion. The results, Fig. 2, showed that the shear failure stresses tended to increase when simultaneous moderate axial tension stress was applied and decreased at higher tensile stresses. Tests under combined axial compression and torsion were carried out at two laboratories on tubes of similar materials. The tubes were 32 mm in diameter and 1.9–2.3 mm thick. The results are plotted in Fig. 2 and the data used are listed in Table 2. A wide range of scatter is seen in the shear strength, when no axial load was applied. No explanation was available for the apparently different values between the shear strengths obtained from tension/torsion and compression/torsion data sets when no axial load was applied. The volume fraction of fibres, Vf = 0.56, measured by Deutschen Forschungs und Versuchsanstalt für Luft- und Raumfarht (DFVLR) for the specimens used in the combined tension and shear tests is slightly lower than that given in Part A of the Exercise.

2.3 Biaxial failure envelope for unidirectional lamina under combined longitudinal and transverse loading (y versus x) The results reported by Al-Khalil et al., Ref. [5], were selected for this failure envelope. Most of the results were obtained from testing nearly circumferentially wound tubes under combined internal pressure and axial load. The winding angle, measured from the axial direction of the tubes, was ± 85° rather than 90°. The specimens tested were filament wound glass/epoxy tubes produced by QinetiQ (formerly known as The Defence Evaluation and Research Agency (DERA), Fort Halstead, Kent, UK). The specimens were 100 mm inner

57 Table 2 Data for biaxial failure stresses of 0° unidirectional lamina under combined longitudinal and shear loading (y versus xy). Schelling and Aoki, Refs [3,4]. Hoop stress y MPa

Shear stress xy MPa

1 – Tests at DFVLR, tubes (Vf = 0·56) 1318.9 1481.1 1500.7 1522.8 1376.1 1480.6 1309.8 1417.3 1435.4 983.4 815.6 716.2 756.9 172.2 0.0 0.0 0.0 0.0

0.0 0.0 0.0 20.4 31.0 46.9 56.4 68.4 73.7 122.3 113.0 107.7 129.9 57.8 55.2 63.5 64.6 66.5

2 – Tests at DFVLR on tubes (D = 32 mm, h = 1·9–2·3 mm) –297.8 –469.4 –790.1 –798.7 –854.8 –939.3

104.5 94.6 46.7 37.4 0.0 0.0

3 – Tests at MBB on tubes with D = 32 mm, h = 2·2 mm 0.0 0.0 –126.0 –260.3 –412.4 –559.6 –678.3 –659.9 –735.6 –835.7

86.3 101.3 94.7 85.7 82.7 78.6 49.6 27.2 0.0 0.0

58 diameter, 300 mm long and approximately 0.95 mm (2 covers2) or 1.2 mm (3 covers) thick. The tubes were made of E-glass/epoxy material with a volume fraction of fibres approximately 0.6. The E-glass fibre reinforcement was Silenka 051L, 1200 tex and the epoxy resin system was Ciba–Geigy MY750/HY917/DY063 mixed in weight proportions of 100 : 85 : 2. The curing cycle was 2 hours at 90°C followed by 1.5 hours at 130°C and 2 hours at 150°C. The ends were reinforced with circumferentially wound E-glass fibre reinforced epoxy resin, leaving an un-reinforced test section 60 mm long. A rubber lining was applied to the inside surface of the specimens. The tubes were tested under combined internal pressure and axial compression using the technique described by Soden et al., Ref. [6,7]. During the tests, the pressure was increased at a steady rate (typically 2MPa/min) and the axial load was increased continuously to maintain a constant predetermined ratio of applied hoop to axial stresses within the gauge length until the specimen fractured. The tubes were tested under ten different stress ratios. Eight specimens were tested at a nominal stress ratio SR = 1 : 0 and the stress–strain curves, measured at this stress ratio, were linear up to failure. The test results are shown in Fig. 3 together with the tension and compression strengths in directions parallel and perpendicular to the fibres which were measured by different methods and were issued as basic properties for the unidirectional composite lamina.

Fig. 3. Biaxial failure stress envelope for 0° unidirectional lamina made of Glass/epoxy under longitudinal and transverse loading (y versus x).

2

See section (4) for comments on filament wound tube construction.

59 The hoop  and axial x stresses were calculated from the measured pressure P and axial load F using: PRi h

(1)

PRi F + h 2 Rih

(2)

 = and x =

where Ri is the inner radius of the un-deformed tubes and h is the mean thickness. Equations (1) and (2) ignore any change in diameter. However, in plotting the failure stresses, correction was made for the axial tensile stress which arises at large displacements due the diameter at the centre of the gauge length becoming greater than the diameter at the ends of the tubes, AlKhalil et al., Ref. [5]. The corrected data is listed in Table 3. 2.4. Biaxial Failure Envelope of ( ± 30°/90°) E-glass/epoxy laminates under biaxial stress loading Extensive work was reported by Krauss and Schelling, Ref. [2], Forster and Knappe, Ref. [8] and Hütter et al., Ref. [1], on the failure behaviour of (90°/ ± 30°/90°)s E-Glass/LY556/ HT907/DY063 epoxy tubes under combined pressure and axial load and combined torsion and axial load. Details of the material used are described in section 2.1 above. The tubes were made by wet filament winding and had a volume fraction of fibres 0.62. The tubes were 60 mm inside diameter and 2 mm thick and had a gauge length of 80 mm for tests involving axial compression and 180 mm in tests involving axial tension. They were end reinforced and tested with a liner inside. The wall thickness consisted of two circumferentially wound (90°) layers, one at the inside and the other at the outside of the tube, and hellically wound ± 30° central layers. The thickness of the 90° circumferential layers was 17.2% and the + 30° and –30° layers together made up 82.8% of the total thickness of the tubes. Therefore, the material is not quasi-isotropic and the strength depends on the loading direction. Tables 4 and 5 show the data points for failure under combined direct stresses (y versus x) and for failure under combined torsion and axial load (x versus xy). Figures 4 and 5 show the failure stresses under combined axial and circumferential loading and under combined axial load and torsion, respectively, see and Hütter et al., Ref. [1]. All the specimens tested under internal or external pressure were lined with a flexible adhesive liner. No data was available on initial failure or leaking and all the data are for final rupture. The few test results carried out under external pressure and axial compression were reported to be governed by buckling. 2.5. Biaxial failure envelope of angle ply ± 55° E-glass/epoxy laminates The specimens tested were filament wound glass/epoxy tubes produced by QinetiQ (formerly known as The Defence Evaluation and Research Agency (DERA), Fort Halstead, Kent, UK), see section 2.3 above for material description. A simple ± 55° helical winding pattern with a single winding angle was employed. (The winding angle is measured between the fibre

60 Table 3 Data for biaxial failure stresses of nearly unidirectional ( ± 85°) lamina under combined longitudinal and transverse loading (y versus x). The results were taken from Table 1 of Al-Khalil et al., Ref. [5], unless otherwise stated. Hoop stress y MPa

Axial stress x MPa

0.0 1280.0** –800.0*** 1170.0 1041.0 1134.0 1086.0 1181.0 1029.0 1200.0 1124.0 1249.0 1200.0 1138.0 1115.0 906.0 834.0 696.0 510.0 206.0 0.0

40.0* 0.0 0.0 35.0 29.0 34.0 31.0 36.0 29.0 37.0 33.0 –4.0 –42.0 –59.0 –79.0 –67.0 –90.0 –109.0 –121.0 –136.0 –145.0****

* From axial tension tests on hoop wound tubes. ** From internal pressure tests on hoop wound tubes. *** This result was obtained from axial compression test on small cylindrical specimens. **** This was obtained from tests on hoop wound tubes under axial compression.

direction and the tube axis). The tubes were either 100 mm or 51 mm inner diameter and were of various thicknesses, see Tables 6(a), 6(b) and 6(c). The mean thickness was determined from measurements taken at 40 positions around the surface of each specimen. The fibre volume fraction measured using burn-off tests was approximately 60% for the thin tubes used for determining strength of the tension–tension quadrant of the failure envelope shown in Fig. 6 but higher for thick tubes. The tubular specimens tested under combined internal pressure and axial load were 100 mm inside diameter, typically 1 mm (2 cover) thick and the overall length 300 with 60 mm gauge length. The ends of the tubes were reinforced and fitted with special grips. The test rigs and technique used were described by Soden et al., Refs [6,7]. Some of the tests were carried out with a flexible plastic liner (Table 6(a)) and some without (Table 6(b)). The

61 Table 4 Biaxial failure stresses for ( ± 30°/90°) E-glass/epoxy laminates under combined hoop and axial loading. Hütter et al., Ref. [1]. Axial stress x (MPa)

Hoop stress y (MPa)

Axial stress x (MPa)

Hoop stress y (MPa)

Axial stress x (MPa)

Hoop stress y (MPa)

292.3 592.3 730.8 557.7 720.5 548.7 –266.7 –268.0 –264.6 628.2 605.1 664.1 39.9 –179.2 223.8 110.3

312.0 157.5 385.5 269.3 384.0 405.0 67.5 66.0 36.0 277.5 222.0 102.0 –107.2 –82.4 –78.4 –91.2

–183.3 –348.5 483.7 144.4 61.60 667.3 148.8 616.0 471.0 –164.2 –195.1 –115.5 48.70 633.9 577.5 385.0

–78.4 –40.0 –44.8 299.0 335.4 344.1 315.6 129.6 336.0 195.0 138.0 247.5 301.5 157.5 202.5 351.0

474.8 577.2 535.2 531.3 559.5 605.7 531.3 559.5 –346.5 –354.2 0.0 0.0 0.0 0.0 0.0

312.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 291.0 267.0 289.5 –113.6 312.0

specimens tested with a liner all failed by rupture. Some of the tubes tested exhibited significant damage and deformation before the final fracture (e.g. see section 2.6). Most of the specimens tested without a liner failed by weeping or jetting of the test liquid (oil) through the tube wall (Specimens that failed in this way are identified in Table 6(b)). Full test results for all the thin walled tubes have been documented in detail by Soden et al., Refs [6,7]. Some of the thin walled tubes which were tested under axial compression failed by buckling and these results have been replaced with new results for thicker walled tubes that failed by bursting rather than buckling, see Table 6. For biaxial compression tests which include testing thick tubes, the overall lengths of the 100 mm and 51 mm diameter specimens were 370 and 185 mm, respectively. In most of the tests the ratio of the specimens inside radius : wall thickness (Ri : h) was 5 and in some of the tests the specimens wall thickness was increased to evaluate the influence of shell buckling. The mean values of the volume fraction of fibres was Vf = 0.68. The ends of the specimens had additional reinforcement and inner end plug contours carefully designed to avoid end failures and excessive stress concentrations. No outer grips were needed in the biaxial compression tests. The test techniques have been described by Kaddour and Soden, Ref. [9], and the results which are listed in Table 6(c) and plotted in Fig. 6 have been described in detail by Kaddour, Soden and Hinton, Ref. [10]. Almost all the tubes were tested without using any liner or elastomeric coating. Only one test was carried out using an elastic coating on the outer surface and the behaviour of the tube was similar to the others. All the tubes for which the results are plotted failed by rupture and not by buckling.

62 Table 5 Biaxial failure stresses for ( ± 30°/90°) E-glass/epoxy laminates under combined axial load and torsion. Hütter et al., Ref. [1]. Axial stress x (MPa)

Shear stress xy (MPa)

Axial stress x (MPa)

Shear stress xy (MPa)

Axial stress x (MPa)

Shear stress xy (MPa)

577.2 531.3 282.3 436.3 513.3 462.0 410.7 64.2 128.3 128.3 192.5 256.7 –346.5 –354.2

0.0 0.0 228.9 140.1 100.8 173.4 251.2 199.9 222.1 248.6 274.2 284.5 0.0 0.0

0.0 320.8 320.8 38.5 256.7 102.6 154.0 205.3 308.0 359.3 359.3 410.7 462.0 513.3 535.2

233.2 258.9 261.4 231.5 278.5 198.2 211.0 218.7 283.6 206.7 216.2 159.8 120.5 15.4 0.0

–256.7 0.0 –64.2 –154.0 –308.0 –333.7 –38.5 –102.7 –154.0 –192.5 –282.3 –308.0 531.3 559.5 605.7

194.8 265.7 271.7 240.9 138.4 75.2 238.4 269.1 227.3 223.8 162.3 145.2 0.0 0.0 0.0

Fig. 4. Biaxial failure stress envelope for (90°/ ± 30°/90°) laminate made of Glass/epoxy under combined loading (y versus x).

63

Fig. 5. Biaxial failure stress envelope for (90°/ ± 30°/90°) laminate made of Glass/epoxy under combined loading (x versus xy).

2.6. The stress–strain curve for ± 55° angle ply laminate under uniaxial tension y : x = SR = 1 : 0 and SR = 2 : 1 of the Exercise The stress–strain curves under uniaxial tension y : x = 1 : 0 and y : x = 2 : 1, used in the Exercise, were taken from the work of Al-Khalil, Ref. [11], at UMIST. The laminates used for testing ± 55° E-glass/epoxy tubes were fabricated by wet filament winding. These tubes were of the same materials and were produced by the same manufacturer as those used for the biaxial envelope (Fig. 6) and for the ± 85° tubes, see section 2.3 above. The tubes were 100 mm inner diameter, 310 mm overall length, 60 mm gauge length and typically 1 mm thick and a fibre volume fraction of 0.6. The actual tube construction consists of four plies (2 covers) with fibre directions at 55°/–55°/55°/–55°, relative to tube axis. The tubes were end reinforced with circumferentially wound E-glass/epoxy material and were tested under internal pressure. The test equipment and procedures used by Al-Khalil, Ref. [11], were similar to those described by Soden et al., Refs [6,7]. The hoop stress  (y) was calculated using Equation (1) and thus no allowance was made for bulging or change in dimensions of the tube under pressure. The strains were measured at the mid-section of the outer surface by bonding three or four pairs of electrical resistance strain-gauges in the axial and circumferential directions. There was a considerable (up to ± 20%) variation in the readings from individual strain gauge readings. The stress–strain curves presented are obtained from selected gauges. 2.6.1. Data for SR = 1 : 0 The circumferential tension (SR = 1 : 0) tests were carried out with and without plastic liners using the test rig shown in Fig. 7(a). Loose pistons with low friction seals were fitted into the end of the tube. The pressure end load acting on the pistons was carried by tie bars and not by the specimen. The axial loads due to seal friction and due to bulging of the specimen under pressure were neglected.

64 Table 6(a) Biaxial test results of ± 55° GRP tubes subjected to combined internal pressure and axial loading. (The tubes were 100mm inside diameter and typically 1mm thick and Vf = 0.6, unless otherwise shown), Soden et al., Ref. [6]. Tests carried out without using a liner. Hoop stress y Mpa

Axial stress x MPa

Type of Failure

69.0 106.0 137.0 134.0 197.0 209.0 300.0 271.0 302.0 268.0 491.0 615.0 852.0 775.0 820.0 736.0 605.0 362.0 410.0 321.0 318.0 191.0 64.7 110.7 0.0 0.0

92.0 106.0 116.0 103.0 124.0 122.0 151.0 135.0 151.0 134.0 196.0 205.0 257.0 234.0 234.0 133.0 0.0 0.0 0.0 –17.0 –17.0 –47.0 –129.0* –112.5* 69.0 76.0

[S] [S] [S] [S] [5] [S] [S] [S] [S] [S] [S] [S] [F] [F] [J] [J] [J] [J] [J] [J] [J] [G,S] [J] [J]

[S]: Weeping (Spots of oil). [F]: Specimens fractured without initial failure. [J]: Jetting of oil. [G]: Combination of buckling and interlaminar shear. * 5.1 mm thick specimen, Vf = 0.68.

Table 7 gives the data for hoop stress, hoop strain and axial strain used for plotting the stress–strain curves and the typical stress–strain curves for SR = 1 : 0 chosen for the Exercise are shown in Fig. 8. These were taken from a test on a lined specimen carried out by AlKhalil, Ref. [11]. The failure hoop stress was approximately 595 MPa, the hoop failure strain 8.8% and the axial failure strain –10.9%. The mean failure hoop strain, using 6 strain-gauges

65 Table 6(b) Biaxial test results of ± 55° GRP tubes subjected to combined internal pressure and axial loading. The tubes were 100mm inside diameter and typically 1mm thick and Vf = 0.6, unless otherwise shown, Soden et al., Refs [6,7]. For tests carried out using a liner. Hoop stress y MPa

Axial stress x MPa

0.0 0.0 107.0 198.0 331.0 374.0 525.0 599.0 723.0 736.0 741.0 717.0 750.0 835.0 803.0 914.0 939.0 867.0 921.0 817.0 761.0 676.0 516.0 594.0 638.0 622.0 544.0 492.0 256.0 114.3

74.0 62.0 143.0 198.0 280.0 288.0 332.0 349.0 365.0 368.0 370.0 358.0 375.0 334.0 321.0 305.0 283.0 262.0 263.0 148.0 138.0 67.0 0.0 0.0 0.0 0.0 0.0 –27.0 –65.0 –115.5*

* 5.1 mm thick specimen, Vf = 0.68.

on two specimens, was 8.7 ± 1.2 (%). The mean axial strain, using 6 strain-gauges on two specimens, was –12.1 ± 1.6 (%) (The selected axial strain curve was at the lower strain bound of the results). The failure stress for this specimen is consistent with the results used to plot the biaxial failure envelope for the ± 55° tubes and with other results by Al-Khalil, Ref. [11] and Soden

66 Table 6(c) Details of biaxial test results of ± 55° GRP tubes subjected to combined external pressure and axial compression, Kaddour et al., Ref. [10]. The tubes had a typical fibre volume fraction of 0.68. Tubes were of different diameters and thicknesses. Diameter mm 51 100 100 + 51 51 51 51 100 100 51 51 51 51 51 51 51 100 100 51 51 51 51 51 100 51 100**

Thick. mm 4.8 14.4 14.3 7.13 8.82 10.0 8.77 9.43 9.57 4.78 4.94 4.65 4.76 4.66 4.67 7.13 9.595 9.9 5.87 5.2 4.24 4.78 4.73 9.51 4.64 9.55

* MPa

x* MPa

–680 –709 –715 –798 –836 –807 –888 2.5 2.5 –94 –269 –260 –403 –540 –520 –693 –557 –588 –769 –705 –476 –549 –384 –289 –276 –339

–347 –365 –366 –409 –430 –417 –457 –152 –149 –195 –273 –267 –329 –320 –352 –406 –235 –249 –312 –282 –174 –186 –96 –57 –42 –53

+ This specimen was tested with rubber coating on the outside surface. * These are the stresses at the inside surface. ** In this test, a special end plug was used which allows for the end pressure to be applied over the thickness of the specimen only. Therefore, the results of this test include some unknown friction.

et al., Ref. [6], for the same stress ratio of 1 : 0 on lined specimens. The strength of unlined specimens for use in the Exercise is 386 MPa. This value was taken from the work of Soden et al., Ref. [6], who reported two test values of 362 and 410 MPa for unlined tubes which failed by oil jetting through the wall thickness, and it is consistent with the mean strength of unlined tubes reported by Al-Salehi et al., Ref. [12], on tubes without end reinforcement which was 427 ± 13 MPa. In the latter work, the failure strains of unlined tubes were y = 4.36 ± 0.61 (%) and x = –4.85 ± 0.44 (%), which are also in agreement with those selected for the Exercise and marked in Fig. 8.

67

Fig. 6. Biaxial failure stress envelope for ± 55° angle ply laminate made of Glass/epoxy under combined loading (y versus x).

2.6.2. Data at SR = 2 : 1 The specimens were tested under internal pressure with the ends closed using the rig shown in Fig. 7(b). The axial stress was taken to be half of the hoop stress, i.e SR = 2 : 1. The stress– strain curves chosen for the Exercise are shown in Fig. 9. These were typical curves taken from one test on a lined specimen carried out by Al-Khalil, Ref. [11]. The failure hoop stress, computed using Equation (1), was approximately 668 MPa. The hoop failure strain was 2.5%, which was very typical of other gauge readings. The axial failure strain was 4.2%, although other axial gauges gave strains as low as 3%. Table 8 gives the data for hoop stress, hoop strain and axial strain used for plotting the stress–strain curves. The specimens failed by extensive cracking parallel to and across the fibres, presumably due to transverse tension and fibre tension fractures in the gauge length. Other tests were carried out in Refs [6,11,13] at the same stress ratio of 2 : 1 and gave mean strengths of lined tubes as 684, 692 and 736 MPa, respectively. The mean strengths of unlined tubes reported by Soden et al., Ref. [6], and Kaddour et al., Ref. [13], which failed by oil leaking through the wall thickness, were 280 and 320 MPa, respectively. The strength of unlined and lined specimens shown in Fig. 9 are 280 MPa and 736 MPa. These were taken from Soden et al., Ref. [6], and are the same as the strengths for the failure envelope in Fig. 6. The shape of the stress strain curves in Fig. 9 is quite similar to that of other results for tests carried out at UMIST, although there was some variation in the axial strain readings, Refs

68

Fig. 7. Schematic diagrams of test rigs used in the biaxial tests ± 55° and ± 45° angle ply tubes. (a) For open end burs test, i.e. for SR = 1 : 0. (b) For SR = 2 : 1 and SR = 1 : 1 and others.

69 Table 7 Data for stress–strain curves for ± 55° angle ply tubes made of E-glass/epoxy material tested under y : x = 1 : 0, Al-Khalil, Ref. [11]. Hoop stress y MPa

Hoop strain y%

Axial strain x%

Hoop stress y MPa

Hoop strain y%

Axial strain x%

Hoop stress y MPa

Hoop strain y%

Axial strain x%

0.0 12.3 25.2 37.1 51.4 60.7 75.4 84.2 95.2 106.2 121.6 131.7 144.6 157.2

0.0 0.04 0.12 0.18 0.26 0.31 0.39 0.45 0.52 0.61 0.72 0.82 0.94 1.05

0.0 –0.01 –0.04 –0.07 –0.11 –0.14 –0.19 –0.23 –0.28 –0.34 –0.42 –0.49 –0.59 –0.68

164.8 177.5 191.3 208.0 215.1 231.1 244.7 254.8 269.3 280.8 289.0 300.5 309.9 325.7 338.7

1.14 1.26 1.40 1.57 1.69 1.86 2.04 2.18 2.37 2.55 2.68 2.84 3.01 3.22 3.42

–0.75 –0.85 –0.98 –1.13 –1.25 –1.40 –1.57 –1.71 –1.90 –2.09 –2.23 –2.40 –2.58 –2.80 –3.02

357.0 368.1 382.0 394.3 410.0 423.4 437.3 475.5 506.2 522.4 528.7 557.4 584.1 594.9

3.75 3.92 4.21 4.43 4.68 4.97 5.27 6.11 6.67 7.10 7.52 8.03 8.46 8.78

–3.40 –3.60 –3.95 –4.21 –4.49 –4.87 –5.27 –6.45 –7.27 –8.02 –8.81 –9.74 –10.41 –10.93

Fig. 8. Stress–strain curves for ± 55° angle ply laminate made of Glass/epoxy under uniaxial tensile loading with y : x = 1 : 0.

70

Fig. 9. Stress–strain curves for ± 55° angle ply laminate made of Glass/epoxy under biaxial tensile loading with y : x = 2 : 1.

[6,11,13]. The shape is different from that published by other investigators using different types of glass/epoxy and specimen geometry, see for instance Refs [68,69].

2.7. Biaxial Failure Envelope of quasi-isotropic (90°/ ± 45°/0°)s AS4/3501-6 tubes Biaxial behaviour of quasi-isotropic (90°/ ± 45°/0°)s tubes made of AS4/3501-6 carbon/epoxy material was investigated by Swanson and Nelson, Ref. [14], Swanson and Christoforou, Ref. [15], Swanson and Colvin, Ref. [16], and Colvin and Swanson, Ref. [17]. The tests were carried out by subjecting the tubular specimens to pressure and axial loads. The specimens used for testing AS4/3501-6 pre-preg material were fabricated by hand layup and autoclave cured at 177°C. The wall construction was (90°/ ± 45°/0°)ns where n was 1 for most tests, but was either 2, 3 or 4 for tests used to study the effect of wall thickness. Note that the 90° direction coincides with the circumferential direction of the tubes. In most of the tests involving internal pressure loading, the tubes were 96 mm inside diameter, gauge length 80 mm. The total length of the specimens was 430 mm for standard tests and 330 mm for other tests that did not involve axial tension, Swanson et al., Ref. [18]. The specimens were end reinforced with fibre glass cloth over-wraps, aluminium ring and low modulus epoxy. A thin rubber bladder was used as a liner for tests involving internal pressure.

71 Table 8 Data for stress–strain curves for ± 55° angle ply tubes made of E-glass/epoxy material tested under y : x = 2:1, Al-Khalil, Ref. [11]. Hoop stress y MPa

Hoop strain y%

Axial strain x%

Hoop stress y MPa

Hoop strain y%

Axial strain x%

Hoop stress y MPa

Hoop strain y%

Axial strain x%

0.0 17.00 27.30 42.60 52.60 64.90 78.30 88.20 102.50 114.70 127.40 139.20 151.00 164.10 178.60 189.50 201.10 213.50 225.90 239.50 254.10

0.0 0.04 0.08 0.13 0.16 0.20 0.24 0.28 0.33 0.37 0.42 0.46 0.51 0.56 0.62 0.67 0.71 0.76 0.80 0.86 0.92

0.0 0.02 0.04 0.06 0.07 0.09 0.10 0.11 0.13 0.15 0.16 0.18 0.20 0.22 0.25 0.28 0.30 0.34 0.37 0.40 0.44

265.60 278.00 290.60 298.90 315.10 328.60 333.60 346.40 358.00 348.70 363.10 377.00 390.10 400.20 412.40 426.10 445.60 457.70 474.50 484.60

0.97 1.03 1.08 1.12 1.18 1.22 1.25 1.31 1.37 1.34 1.39 1.44 1.49 1.52 1.57 1.62 1.69 1.74 1.80 1.84

0.48 0.57 0.61 0.75 0.86 0.97 1.03 1.14 1.25 1.24 1.27 1.36 1.44 1.49 1.59 1.68 1.81 1.92 2.05 2.15

499.10 490.60 507.00 520.80 506.70 516.40 537.10 552.50 558.50 582.70 607.10 607.50 623.50 632.70 635.50 644.50 661.30 649.80 637.70 668.10 668.30

1.89 1.86 1.92 1.97 1.92 1.96 2.02 2.07 2.10 2.19 2.28 2.29 2.35 2.39 2.40 2.43 2.50

2.25 2.25 2.33 2.43 2.41 2.46 2.56 2.71 2.81 2.96 3.17 3.28 3.50 3.63 3.76 3.91 4.22

The strains were measured using 3 strain-gauge rosettes, located around the mid-plane on the outer surface of the tubes. The hoop  and axial x stresses were calculated from the measured pressure P and axial load F using  =

PRm h

(3)

F A

(4)

and x =

where Rm is the mean radius, h is the thickness and A is the cross sectional area of the tubes, Swanson and Christoforou, Ref. [15].

72

Fig. 10. Biaxial failure stress envelope for (90°, ± 45°,0°)s laminate made of AS4/3501-6 under combined loading (y versus x).

In addition to the 96 mm inner diameter tubes, tests were also carried out on 38 mm and 51 mm inner diameter tubes, Refs [16–17]. The 38 mm diameter tubes were tested under axial compression only while the 51 mm tubes were used to obtain failure strength data under external pressure with and without axial compression and under internal pressure only with no axial load. The mean axial compressive strength of the 38 mm tubes was 637 MPa while that obtained from the 96 mm tubes was 375 MPa. No explanation was offered for the large difference in the results but Swanson and Colvin, Ref. [16], hinted that buckling of the 96 mm tubes might have resulted in the corresponding low axial strength values. The 51 mm tubes tested under internal pressure had a radius : thickness ratio (R : h) of 25 and those used for external pressure tests had R : h of 13. The final circumferential failure stress under internal pressure (SR = 1 : 0) was 721 MPa (tension) and that under external pressure (SR = –1 : 0) was 317 MPa (compression). Figure 10 shows the experimental failure stresses obtained from the different publications and the data for the axial and circumferential stresses are listed in Table 9. The mode of failure for tests under hoop tension and axial loading was reported to be by fibre fracture, i.e. the final failure was dominated by the fibre strength. In some of the tests, a non-catastrophic failure took place prior to the final failure. The non-catastrophic (or initial)

73 Table 9 Biaxial failure stresses for quasi-isotropic (90°, ± 45°,0°)s AS4/3501-6 laminates under a variety of stress ratios. (These results were collected from different sources). The First Set: Tests on 96 mm tubes, Swanson and Nelson, Ref. [14]. Axial stress x (MPa)

Hoop stress y (MPa)

–305.82 –328.40 –368.43 –393.06 –281.19 –369.45 –264.77 –281.19 –213.46 –248.35 –300.69 –232.96 –87.23 –28.73

0.0 0.0 0.0 0.0 161.97 203.73 243.46 312.73 387.10 399.32 405.43 514.44 777.26 644.83

The Second Set: Tests on tubes (96 mm diameter, 1 mm thick) under internal pressure and axial load, Swanson and Trask, Ref. [22]. Axial stress x (MPa) 428.7 423.5 38.70 220.0 202.0 18.1

Hoop stress y (MPa) 857.4 847.1 721.9 677.1 730.5 677.6

failure was in the form of softening behaviour of the stress–strain curves that followed initially linear behaviour. The softening response, which manifested itself in a change in the slope of the stress–strain curves, was attributed to matrix cracking, Trask, Ref. [9]. Some of the data points of change in the slopes of the stress–strain curves reported by Swanson and Christoforou, Ref. [20], and Trask, Ref. [19], have been extracted and marked on the failure envelope in Fig. 10. Buckling was observed to take place in the tests carried out under external pressure with and without axial compression. Therefore, the test results in the compression–compression quadrant represent structural failure by buckling rather than crushing failure of the material.

74 Table 9 Continued. The Third Set: Tests on tubes (96 mm diameter, 1 mm thick) under internal pressure and axial load, Swanson and Christoforou, Ref. [15] Axial stress x (MPa)

Hoop stress y (MPa)

46.90 172.2 35.80 124.8 54.5 254.0 450.0 442.0 504.0 728.0

667.0 790.0 718.0 696.0 723.0 774.0 849.0 813.0 908.0 969.0

The Fourth Set: Tests on 38.1 mm diameter and approximately 2 mm thick tubes, Colvin and Swanson, Ref. [17]. Axial stress x (MPa)

Hoop stress y (MPa)

–733.0 –537.0 –640.0

0.0 0.0 0.0

The Fifth Set: Tests on 51.8 mm diameter tubes, Swanson and Colvin, Ref. [16]. x (MPa)

y (MPa)

Remarks

0.0 0.0 0.0 0.0 0.0 –418.0

652.0 761.0 752.0 –404.0 –329.0 –304.0

–638.0

–190.0

–275.0

–301.0

–187.0

–334.0

–188.0

–338.0

h = 1.05 mm thick, internal pressure only h = 1.05 mm thick, internal pressure only h = 1.05 mm thick, internal pressure only h = 2.11 mm thick, external pressure only h = 2.11 mm thick, external pressure only h = 2.11 mm thick, axial compression and external pressure, suspected circumferential buckling failure h = 2.11 mm thick, axial compression and external pressure, failed at grips due to end brooming interaction h = 2.11 mm thick, axial compression and external pressure, suspected circumferential buckling failure h = 2.11 mm thick, axial compression and external pressure, suspected circumferential buckling failure h = 2.11 mm thick, axial compression and external pressure, suspected circumferential buckling failure

75 2.8. Stress–strain curve for quasi-isotropic (90°/ ± 45°/0°)s AS4/3501-6 laminates under uniaxial tension y : x = SR = 1 : 0 The specimens were cut off from five foot long, hand laid up cylinders manufactured by Hercules Incorporated. The tubes were 96.5 mm inside diameter, approximately 1 mm thick, 80 mm gauge length and 419 mm total length. The tubes were similar to those employed by Swanson and his co-workers at University of Utah (USA) in some of the experiments to determine the results used in the biaxial tension failure envelope for this laminate, see the previous section. The stress–strain curves, Fig. 11, under uniaxial tension y : x = 1 : 0, used in the Exercise, originated from the work of Christoforou, Ref. [21]. Christoforou tested a specimen under internal pressure with no axial load. Due to friction at seals, the actual stress ratio was y : x = 20 : 1. The circumferential failure stress was 718 MPa and the hoop failure strain was 1.45% and the axial strain was –0.36%. Table 10 gives the data3 for hoop stress– hoop strain and axial strain used for plotting the stress–strain curve (the circumferential stress was calculated using Equation (3)). The stress–strain curve under y : x = 20 : 1 was published by Swanson and Christoforou, Ref. [20], but the co-ordinates used were  versus ¯   + Q ˜ xx. As pointed out by Swanson and Christoforou, Ref. [20], the slope of such a Q curve is an identity, i.e. the slope makes an angle of 45° to the stress axis, if the material

Fig. 11. Typical stress–strain curves for (90°, ± 45°,0°)s laminate made of AS4/3501-6 under uniaxial tensile loading in y direction (y : x = 1 : 0).

3

The data presented in the Table were obtained through personal communication with Professor S R Swanson who quoted Christoforou MSc thesis (1984), Ref. [21], as a source of information.

76 Table 10 Data for stress–strain curves for quasi-isotropic (90°, ± 45°,0°)s tubes made of AS4/3501-6 carbon/ epoxy material tested under y : x = 20:1. (The results taken from Table 6, page 68 of Christoforou, Ref. [21], for specimen number QIL-3, see also Swanson and Christoforou, Ref. [15]). Hoop stress y MPa

Hoop strain y%

Axial strain x%

0.0 47.1 86.0 108.2 135.8 172.0 191.5 213.6 235.8 257.9 274.7 291.3 305.2 324.8 352.4 371.7 394.0 419.0 444.1 468.9 499.5 530.0 560.5 593.7 624.1 660.4 682.7 702.0 713.1 718.0

0.000 0.084 0.161 0.207 0.257 0.325 0.370 0.414 0.457 0.493 0.530 0.561 0.589 0.624 0.688 0.727 0.773 0.824 0.882 0.933 0.990 1.047 1.117 1.182 1.245 1.335 1.382 1.420 1.452 1.455

0.0000 –0.0175 –0.036 –0.047 –0.059 –0.076 –0.087 –0.098 –0.109 –0.118 –0.127 –0.136 –0.143 –0.152 –0.168 –0.179 –0.190 –0.203 –0.214 –0.227 –0.242 –0.256 –0.273 –0.291 –0.310 –0.331 –0.342 –0.352 –0.360 –0.363

response is linearly elastic. The curve shows a decrease in the slope at around 400 MPa hoop stress indicating a form of initial failure in the laminate. Other tests were carried out at a nominal stress ratio of 1 : 0, and occasionally the stress– strain curves were reported, Swanson and Trask, Ref. [22], and Swanson and Colvin, Ref. [16]. The mean strength reported by those investigators is 713 MPa ( ± 42 MPa) which is close

77

Fig. 12. Typical stress–strain curves for (90°, ± 45°,0°)s laminate made of AS4/3501-6 under biaxial tensile loading with y : x = 2 : 1.

to the strength of 718 MPa used in the Exercise. The scatter in the results is quite small (approximately ± 5%).

2.9. The stress–strain curve for quasi-isotropic (90°/ ± 45°/0°)s AS4/3501-6 laminates under a stress ratio of SR = 2 : 1 The specimens were similar to those used to obtain stress–strain curves under SR = 1 : 0 and also to those used for establishing the tension–tension biaxial envelope, described above. The stress–strain curves under biaxial tension y : x = 2 : 1 were from the work of Trask, Ref. [19] at the University of Utah, USA. Trask tested two specimens under a stress ratio of SR = 2 : 1. The lay-up was described as (90°/ ± 45°/0°)s2 and the exact ply consequence was (90°/ + 45°/–45°/0°/0°/–45°/ + 45°/90°). The stress–strain curves were obtained from the various graphs reported by Trask. The hoop stress versus hoop strain curve was digitised from Fig. 4-14 of Trask, Ref. [19], while the hoop stress versus axial strain curve had to be extracted from Trask’s Figs 4-21 and 4-30. The latter figures were for specimen number LTCU-86-1-#3 and were plotted by Trask as ¯  + Q ¯ xx and  (hoop stress) versus Q ¯ x + Q ¯ xxx x (axial stress) versus Q ¯ , Q ˜ x and Q ˜ xx were measured for that specimen and given where the values of the stiffness Q ¯ ¯ ¯ as Q = 51.3 GPa, Qx = 18.9 GPa and Qxx 60.2 GPa. The measured elastic properties and failure stresses of the two tubes were reported by Trask. The stress–strain curves, back calculated from the graphs, are shown in Fig. 12 and the corresponding data are listed in Table 11. The response is linear at low strain levels but shows

78 Table 11 Data for the stress–strain curves for quasi-isotropic (90°, ± 45°,0°)s tubes made of AS4/3501-6 carbon/ epoxy material tested under SR = 2:1. Data were extracted from the work of Trask, Ref. [19]. Hoop stress y MPa

Hoop strain y%

Axial strain x%

0.0000 137.8 206.7 275.6 344.5 413.4 438.9 454.8 482.3 551.2 620.1 689.1 758.0 808.3

0.0000 0.2343 0.3558 0.4729 0.5923 0.7100 0.7583 0.8389 0.8715 1.0086 1.1359 1.2595 1.3881 1.5607

0.0000 0.0621 0.0814 0.1124 0.1376 0.1669 0.1689 0.1828 0.1939 0.2208 0.2408 0.2777 0.3208 0.3345

significant softening at higher strains. The non-linear behaviour starts at a strain between 0.6 to 0.8%. That corresponds to a change in the slope of stress–strain curves around 450 MPa hoop stress. Further reduction appears to take place at a hoop stress of 750 MPa. The final hoop strengths of two tubes tested by Trask, Ref. [19], were 857 and 847 MPa.

2.10. The stress–strain curve for ± 45° angle ply laminate under biaxial tension y : x = SR = 1 : 1 The specimens were in the form of ± 45° tubes which were made by the same manufacturer (DRA) using the same material as for the ± 55° tubes described earlier (section 2.3). The stress–strain curves under biaxial tension y : x = 1 : 1, used in the Exercise, were from the work of Li, Ref. [23], and Reid et al., Ref. [24], at UMIST. The ± 45° E-glass/epoxy tubes were fabricated by wet filament winding and cured at 120°C. The specimens were two cover filament wound tubes and the tube wall consisted of four layers of E-glass/epoxy oriented at 45°/–45°/45°/–45° to the tube axis. The tubes were 100 mm inner diameter, 310 mm overall length, 60 mm gauge length and typically 1 mm thick. The tubes were end reinforced and tested under internal pressure and axial load using the rig, Fig. 7(b), and test methods described by Soden et al., Ref. [6]. The tubes were tested under equal biaxial tension, i.e. the axial stress was equal to the hoop stress. The stresses were calculated using Equations (1) and (2), i.e. no allowance made for the change of shape of the tube under load. The strains were measured at the mid-section of the outer surface by bonding electrical resistance strain-gauges in the axial and circumferential directions.

79

Fig. 13. Typical stress–strain curves for ± 45° GRP angle ply laminate under biaxial tensile loading with y : x = 1 : 1.

The stress–strain curves chosen for the Exercise are shown in Fig. 13. These were selected from the results of one test on a lined specimen carried out by Reid et al., Ref. [24]. The final hoop stress was around 444 MPa and the corresponding hoop and axial strains were 2.47% and 2.17%, respectively. Table 12 gives the data for hoop stress, hoop strain and axial strain used for plotting the stress–strain curves. Readings from individual strain gauges on 4 different specimens varied by up to 22%. One of the peculiar features of these results is that the hoop strains are larger than the axial strains, although ideally all the strains should be equal to each other, Li, Ref. [23]. Reasons for such divergence between the hoop and axial strains are not clear. In some of these tests, the crack spacing was recorded from a series of photographs as the loading progressed. Cracks were observed for stresses as low as 50–70 MPa. The rate of crack growth tended to level out beyond a stress of approximately 200 MPa, which coincided closely with the observed failure stress (216 MPa) of an unlined specimen tested by Soden et al., Ref. [7]. The specimens failed by extensive cracking parallel to the fibres, presumably due to transverse tension and by fibre tension fractures in the gauge length. The mean stress at which maximum strains were recorded was 419 MPa, but the strength of the tubes was believed to be higher than that. Other tests were carried out by Soden et al., Ref. [7] at the same stress ratio of SR = 1 : 1 on 3 lined tubes. The mean strength of these lined tubes which failed by rupture was 502 ± 35 MPa, Ref. [7]. The volume fraction of the fibres of similar tubes was 0.55. The strength of an unlined specimen, which failed by leakage was reported by Soden et al., Ref. [7] as 216 MPa . These values of hoop failure stresses (502 and 216 MPa) are marked in Fig. 13.

80 Table 12 Data for stress–strain curves for ± 45° angle ply tubes made of E-glass/epoxy material tested under y : x = 1:1. Data for Fig. 21 of the failure Exercise, Reid et al., Ref. [24]. Hoop stress y MPa

Hoop strain y%

Axial strain x%

Hoop stress y MPa

Hoop strain y%

Axial strain x%

Hoop stress y MPa

Hoop strain y%

Axial strain x%

0.0 19.80 29.91 39.66 49.75 59.59 69.52 79.25 89.12 98.88 108.79 118.66 128.41 138.26 148.31

0.0 0.0745 0.1113 0.1479 0.1868 0.2218 0.2601 0.3012 0.3433 0.3939 0.4450 0.4965 0.5498 0.6146 0.6735

0.01 0.0584 0.0911 0.1174 0.1483 0.1785 0.2149 0.2492 0.2871 0.3313 0.3716 0.4226 0.4762 0.5188 0.5670

158.27 168.12 177.78 187.65 197.47 207.34 217.11 227.20 237.01 246.93 256.73 266.57 276.55 286.40 296.25

0.7446 0.8057 0.8680 0.9192 0.9715 1.0328 1.0927 1.1638 1.2155 1.2688 1.3383 1.3872 1.4442 1.4930 1.5636

0.6196 0.6617 0.7037 0.7442 0.7958 0.8437 0.8918 0.9398 0.9869 1.0334 1.0950 1.1411 1.1898 1.2351 1.3021

306.13 316.22 326.11 335.91 345.76 355.57 365.46 375.50 385.27 395.06 404.98 414.77 424.67 434.47 444.42

1.6075 1.6599 1.7172 1.7648 1.8284 1.8723 1.9244 1.9775 2.0287 2.0790 2.1326 2.1982 2.2595 2.3510 2.4750

1.3465 1.3962 1.4503 1.5017 1.5778 1.6183 1.6665 1.7175 1.7745 1.8326 1.8971 1.9589 2.0261 2.0976 2.1780

2.11. The stress–strain curve for ± 45° angle ply laminate under biaxial tension y : x = SR = 1 : –1 The stress–strain curves under biaxial loading y : x = 1 : –1, used in the Exercise, were from the work of Kaddour et al., Ref. [25] at UMIST and were generated from tests on ± 45° filament wound tubes of the same E-glass/MY750 epoxy material and manufactured by DRA. The tubes were 100 mm inner diameter, 370 mm overall length, 120 mm gauge length and typically 5.9 mm thick and had a fibre volume fraction of 0.6. The tubes were end reinforced and tested under internal pressure and axial compression. The test equipment was similar to that described by Soden et al., Ref. [7], but a large (500KN) capacity machine was used to apply the axial loads and the ratio of internal pressure to axial compression was kept constant during the test. No end grips were used. The larger wall thickness for SR = 1 : –1 specimens, compared with the SR = 1 : 1 tubes, was to avoid shell buckling under these axial loads. The inside surface hoop  and axial x stresses were calculated from the measured pressure P and axial load F and the original tube dimensions using thick cylinder theory. The inside surface axial compressive stress was equal in magnitude to the inside surface tensile hoop stress in these tests. The stress–strain curves chosen for the Exercise are shown in Fig. 14 and were selected from the results of one lined specimen. At failure, the computed stress at the inside surface of the tubes was 94.8 MPa and the hoop failure strain was 9.9% and the axial failure strain was –11.2%. Table 13 gives the data for hoop stress, hoop strain and axial strain used for

81

Fig. 14. Typical stress–strain curves for ± 45° GRP angle ply laminate made under biaxial tensile loading with y : x = 1 : –1.

plotting the stress–strain curves. The strains were measured at the inside surface of the tubes.

2.12. Stress–strain curves for cross-ply (0°/90°)s laminate made of E-glass/epoxy material tested under y : x = 1 : 0 Work on characterising the behaviour of 90°/0° cross ply glass/epoxy has been carried out by a number of investigators. Very recently, Eckold, Refs [26–28], and Hinton, Ref. [29], obtained stress strain curves for cross ply laminates using different test specimens. In all of these specimens, particular attention was paid to observing and recording cracking of the laminates. Details are described below for the results of the two groups: 2.12.1. Results for tubes One set of stress–strain curves under uniaxial tension y : x = 1 : 0 originated from the work of Eckold, Ref. [27]4 at AEA Technology (Didcot, Oxon, United Kingdom). The specimens

4

Personal communication from Dr. G C Eckold. The results were obtained as a part of a collaborative project carried out by the National Physical Laboratory and AEA Technology, Harwell, within the “Materials Measurement Program”; a program of underpinning research financed by the UK Department of Trade and Industry.

82 Table 13 Data for stress–strain curves for ± 45° angle ply tubes made of E-glass/epoxy material tested under y : x = 1 : –1. Kaddour et al., Ref. [25]. Hoop stress y MPa

Hoop strain y%

Axial strain x%

0.0 5.59 7.11 10.07 11.13 15.02 18.83 22.30 26.69 32.22 38.22 45.10 50.47 54.57 58.13 62.97 65.37 73.47 76.35 76.48 79.49 81.44 83.81 84.88 88.87 90.97 92.01 92.70 93.92 94.82

0.0 –0.0402 –0.0520 –0.0749 –0.0863 –0.1156 –0.1503 –0.1955 –0.2474 –0.3246 –0.4248 –0.5961 –0.8205 –1.0404 –1.3764 –1.8534 –2.3117 –4.0114 –4.4314 –4.4583 –5.0472 –5.6688 –6.1527 –6.6135 –7.5960 –8.4206 –8.9463 –9.7163 –10.499 –11.190

0.0 0.0363 0.0469 0.0676 0.0772 0.1055 0.1380 0.1775 0.2281 0.3038 0.4032 0.5713 0.7874 1.0001 1.3154 1.7608 2.1856 3.7974 4.2021 4.2266 4.7956 5.3825 5.8412 6.2716 7.1889 7.9416 8.4200 9.1120 9.8910

were in the form of the tubes of 150 mm inside diameter, 1.04 mm thick, 700 mm overall length and 620 mm gauge length. The lay up was 90°/0°/0°/90° where the 90° direction is the hoop direction. The material employed was similar to that used in testing the ± 45° and ± 55° winding angle tubes described above in sections 2.3, 2.5 and 2.6, except that the matrix was MY750/HY 917/DY 070. The fibre volume fraction was typically 0.6. The tubes were end reinforced and tested under internal pressure with no axial load. The end load was carried by

83

Fig. 15. Stress–strain curves for (0°/90°) GRP cross ply tubes under internal pressure, at y : x = 1 : 0. Results supplied by Eckold (1997).

four tie bars and O ring and cup seals were used to prevent leakage from tube ends during pressurisation. No liner was used and the test was stopped when weeping occurred. Whitening did not occur and individual transverse cracks in the 90° plies were formed which extended along the entire axial gauge length of the tubes. These cracks were clearly visible with minimal damage/opacity extending beyond the cracks themselves. Cracks began at a hoop stress of approximately 175 MPa, Eckold, Refs [26,28], see Fig. 15. The last available readings of the strains were at hoop stress of 323 MPa. At that stress, the hoop strain was 1.55% and the axial strain was –0.118%. Failure took place by weeping through a delamination (remote from the ends) at a hoop stress of approximately 400 MPa and no fibre fracture occurred. Further details can be found in the work of Eckold, Ref. [26], and Eckold et al., Ref. [28]. 2.12.2. Results for coupons QinetiQ (formerly known as The Defence Evaluation and Research Agency (DERA), Fort Halstead, Kent, UK) has also been engaged in obtaining a full characterisation of the behaviour of cross ply glass/epoxy laminate under uniaxial tension. The specimens tested were made of four layers oriented at 0°/90°/90°/0° relative to the loading direction (along the 0° direction), with an overall thickness of 1.9 mm and all the layers have an identical thickness of 0.475 mm. The coupons were 25 mm wide, 200 mm long and have a gauge

84

Fig. 16. Typical stress–strain curves for (0°/90°) GRP cross ply coupons under uniaxial tensile loading with x : y = 1 : 0.

length of 100 mm, leaving a 50 mm distance at each end for bonding end taps. The edges of the specimens were polished. The specimens were equipped with strain gauges in directions parallel to and the transverse to the loading direction. The material was similar to that of the 55° winding angle tubes (described in sections 2.3, 2.5 and 2.6 of the data pack for Part B) except that the matrix was MY750/HY917/DY070. The fibre volume fraction was 0.62. A total of five tests were carried out and the mean final failure stress was 590 MPa, with a coefficient of variation of CV = 11.8%. The, mean failure strain in the loading direction was x = 2.69%, with CV = 14.4%, and that in the transverse direction was y = –0.13%, with CV = 8.3%. Typical stress strain curves are shown in Fig. 16. The onset of first crack was recorded at a strain of x = 0.375% which corresponds to a load per unit area 117.5 MPa on the coupon. The onset of longitudinal splitting was observed at a strain of x = 1.3% and that resulted in a small kink, not shown here, in the strain perpendicular to the loading direction. The coupons finally failed by fibre fracture. Table 14 presents data for the stress and both strains, along and perpendicular to the loading direction, measured in one typical specimen. The failure stress was 609 MPa and the failure strains were x = 2.69% and y = 0.12% along and perpendicular to the loading direction, respectively. Eckold, Ref. [28], also tested coupons of the same material and of similar construction and the stress at final failure (fibre fracture) was 598 MPa with a standard deviation of 47 MPa.

85 Table 14 Data for stress–strain curves for cross-ply (0/90) coupons made of E-glass/epoxy material tested under uniaxial tension. Data supplied by Hinton, Ref. [29]. x MPa

y %

x %

x MPa

y %

x %

0.000 6.172 22.082 38.143 53.749 68.146 83.140 98.680 114.62 130.49 145.09 159.26 174.37 189.16 205.01 220.17 236.19 251.86 267.80 283.65 299.29

0.000 0.005 –0.002 –0.009 –0.016 –0.022 –0.028 –0.035 –0.041 –0.048 –0.050 –0.053 –0.054 –0.056 –0.058 –0.062 –0.067 –0.071 –0.074 –0.076 –0.080

0.000 0.004 0.050 0.099 0.147 0.192 0.240 0.290 0.343 0.397 0.463 0.514 0.607 0.684 0.766 0.826 0.890 0.954 1.023 1.099 1.164

314.89 331.07 345.56 360.79 376.65 392.22 422.74 437.38 446.08 460.34 474.86 489.94 499.77 515.10 529.57 538.10 552.77 567.56 581.42 595.42 609.14

–0.083 –0.086 –0.090 –0.093 –0.086 –0.090 –0.093 –0.094 –0.096 –0.099 –0.105 –0.114 –0.107 –0.102 –0.098 –0.092 –0.090 –0.085 –0.087 –0.099 –0.125

1.228 1.303 1.365 1.428 1.496 1.565 1.703 1.768 1.813 1.877 1.948 2.009 2.091 2.166 2.240 2.322 2.399 2.472 2.544 2.613 2.691

This fracture strength is shown in Fig. 15. The full stress strain curves up to final failure were not available at the time of writing. The results of Eckold on coupons are very similar to those of Hinton’s group, Ref. [29], see Fig. 16. 2.12.3. Results to be used for the exercise The original intention was to follow the policy of using the experimental results from tubes, Fig. 15, but because the full stress–strain curves are available for coupons, the organsiers have chosen the results described in Ref. [29] for comparison with predictions. The data are listed in Table 14. 3. Properties for unidirectional layers and constituents Details of the properties used in the laminate predictions were given in Part A, Ref. [30], of the Exercise. Properties for unidirectional layers were usually obtained from the same sources

86 as the experimental data for the laminates used in Part B. In some cases where laminates of similar materials were tested in different laboratories the same set of material properties was employed to minimise the number of data sets issued. Inevitably not all of the required properties were available and then approximate values (sometimes arbitrary values) were employed. Typical values of properties of four different unidirectional (UD) laminae used in Part A the Exercise were given by Soden et al., Ref. [30]. The four (UD) laminae are: 1 – E-Glass/MY750 epoxy (Silenka E-glass/MY750/HY917/DY063) 2 – E-Glass/LY556 epoxy (Gevetex E-glass/LY556/HT907/DY063) 3 – AS4 Carbon/epoxy (AS4/3501-6) 4 – T300 Carbon/epoxy (T300/BSL914C) The data and its sources are discussed below.

3.1. Material I: E-glass/MY750/HY970/DY063 The data presented in Part A for this material were provided by Hinton et al. Refs [31,56], and were based on tests carried out on different specimens depending upon the property sought. The longitudinal tensile properties of the unidirectional fibre reinforced material in the direction parallel to the fibre (E1, 12, u1T, X1T) were measured using thin walled circumferentially wound tubes subjected to internal pressure using a test rig similar to that shown in Fig. 7(a). The transverse (perpendicular to the fibres) tensile properties (E2, 21, u2T, X2T) were measured using thin walled circumferentially wound tubes subjected to axial tension. In both cases, the stress–strain curves were linear up to failure. The longitudinal compressive properties (E1, 12, u1C, X1C) were measured using thick blocks of unidirectional lamina under axial compression. The stress–strain curve is linear up to failure. Early work at QinetiQ, Hinton, Ref. [32], on pultruded bars of 60% volume fraction, showed that the compressive strength was 520 MPa which agrees with typical values reported elsewhere, see Tsai, 33]. However, recently QinetiQ carried out a more thorough investigation on the determination of the longitudinal compressive strength of E-glass/MY750 epoxy. In this recent work, see Hinton, Ref. [31], using improved test rig the results show that for high fibre volume fractions of fibres (65–71%), the strength was about 1150 MPa. The value of longitudinal compressive strength was scaled to 60% volume fraction to give the value of 800 MPa used in this exercise. Further experiments are required to confirm this result. The transverse compressive properties (E2, 21, u2C, X2C) were measured using thick hoop wound tubes under axial compression, Harwood and Hinton, Ref. [34], and rectangular cross section blocks of UD material, Jones, Ref. [35]. The stress–strain curve exhibits a slight nonlinearity as described in Part A of the exercise. The in-plane shear properties including initial modulus G12, the strength S12 and failure strain u12 were measured using circumferentially wound tubes under torsion. The stress–strain curve is highly non-linear and Hinton et al., Ref. [36], described the non-linearity using a fourth order polynomial. The thermal expansion coefficients and stress free temperature were measured by Hinton, Ref. [32], using blocks of UD composite and cross ply laminates, respectively.

87 Parvizi et al., Ref. [37], measured a fracture surface energy for cracks running parallel to the fibres using the work-to-fracture and linear elastic fracture mechanics methods. The specimens used were edge-notched rectangular bars under three-point bending and double notched rectangular specimens under uniaxial tension. They reported a value of 120 ± 30 J/m2 of the fracture energy for initiation of cracks. Eckold, Ref. [26], reported a value of 165–185 J/m2 for the matrix cracking energy for the same composite material. Krawczak and Pabiot, Ref. [38], measured GIC for E-glass/Epikote 828 epoxy materials using DCB and reported values in the range 87 to 224 J/m2 for the initiation mode, depending upon the coupling agent used. Miyase, Ref. [39], measured the same property for E-glass/Epon 828 epoxy using a Width Tapered Double Cantilever Beam (WTDCB) and reported an average value of 214 ± 30 J/m2. For the E-glass/MY750 lamina used in the exercise, the critical strain energy release rate GIC for mode I (delamination mode of deformation) was assumed to be 165 J/m2, based on the various values mentioned. 3.2. Material II: E-glass/LY556/HT917/DY063 Most of the data for this material presented in Part A were reported by in Refs [1,2,8,40]. The types of fibre and epoxy are rather similar to those of the Material I described above and, hence, the properties are also similar. The fracture toughness and thermal expansion coefficients were assumed to be the same as those for material I. 3.3. Material III: T300/BSL914C carbon/epoxy Most of the data for this material presented in Part A were reported by Schelling and Aoki, Refs [3,4]. The shear stress–strain curve was assumed to follow the same behaviour as that of T300/914 epoxy lamina, see Panhwar and Scott [41], Ladeveze and Dantec, Ref. [42], and Lafarie-Frenot and Touchard, Ref. [43], and similar laminae (XAS/914C) reported by Sanders and Grant, Ref. [44]. The thermal expansion coefficients, the stress free temperature and interlaminar fracture toughness were all assumed. 3.4. Material IV: AS4/3501-6 carbon/epoxy The AS4/3501-6 material comes in the form of prepregs. A typical curing cycle recommended by the suppliers (Hercules) of this material is summarised below, see Dickson et al., Ref. [45] Parameters

Details

Initial pressure Initial heat rise Dwell Pressurise Final heat rise Cure Depressurise

56 cm Hg vacum, 568 kPa to 121°C at 2–3°C/min 60 min at 121°C raise pressure to 690 kPa and vent bag 121–177°C at 2–3°C/min 177 to 60°C at 4°C/min under pressure release autoclave pressure

88 Characterisation of the UD properties of AS4/3501-6 was made by a large number of researchers including Swanson and his co-workers, Refs [14–19]. The longitudinal tensile properties (E1, 12, u1T, X1T) were measured by Colvin and Swanson, Ref. [17], and Swanson and Trask, Ref. [19], using coupons under uniaxial tension. The stress–strain curve was reported to exhibit slight stiffening with the modulus varying from E1ini = 126 GPa at small strain to E1sec = 142 GPa at failure. Work reported by other investigators on various types of CFRP materials confirmed such non-linearity and showed that the secant modulus is 5–15% higher than the initial modulus, Ditcher et al., Ref. [46], and Schulein et al., Ref. [47]. The transverse tensile properties (E2, 21, u2T, X2T) were measured by Swanson and Toombes, Ref. [48], using coupons subjected to axial tension. The stress–strain curve is linear up to failure. The uniaxial longitudinal compressive properties (E1, 12, u1C, X1C) for AS4/3501-6 CFRP material were measured by a number of workers and frequently different values of failure strength and failure strain were reported. A scatter as large as 30% in the compressive strength of AS4/3501-6 material obtained from different methods was reported by Abdullah, Ref. [49], although the coefficients of variations were less than 8% within each test method. Wung and Chatterjee, Ref. [50], used three test methods and reported a standard deviation in the mean value of compressive strength reaching 28% within one test method. They also obtained a mean strength values ranging from 820 to 1179 MPa. Swanson and Nelson, Ref. [14] and Swanson and Trask, Ref. [19], reported a strength value of 1190 MPa from axial compression tests carried out on 0°/90° cross ply tubes of 96.5 mm diameter and 1.5–2.11 mm thick. Other workers (e.g. Kim and Castro, Ref. [51], Daniel et al., Ref. [52]) have reported values of the compressive strength ranging from 1450 to 2000 MPa. Sun and Jun, Ref. [53], reported values of X1C of 1100–1750 MPa at various values of volume fractions of fibres (Vf = 0.5–0.67), all collected from other published work. The value of the compressive strength was taken as 1480 MPa, Swanson and Nelson, Ref. [14], for use in the Exercise. The transverse compressive properties (E2, 21, u2C, X2C) were measured by Swanson and Toombes, Ref. [48], using hoop wound tubes under axial compression. The stress–strain curve was reported to be linear. However, Daniel and Ishai, Ref. [54], showed that the stress–strain curve for the same material in non-linear and the shape of the non-linear curve selected in Part A of the exercise was based upon this. In the latter work, the tests were carried out on coupons. Other types of carbon/epoxy materials such as IM7/8551-7 (Colvin and Swanson, Ref. [55]), T300/LY556 and T800/GY281 (Schulien et al., Ref. [47]), HTS/914C (Ditcher et al., Ref. [46]) and T800/924C (Jelf and Fleck, Ref. [57]) also showed non-linear response under transverse compression. For these materials, the final modulus at failure was between 7% to 36% lower than the initial modulus, depending upon the material. The type of matrix and method of curing is likely to have an effect on the degree of non-linearity. It is known that 3501-6 epoxy is brittle, Trask, Ref. [19], while LY556/HT976 epoxy could be fairly ductile, depending upon the curing procedure, Schulein et al., Ref. [47]. The in-plane shear properties including initial modulus G12, the strength S12 and failure strain u12 were measured using different test specimens and the results were presented by Swanson et al., Refs [20,48,58,59], and other investigators. Swanson and his co-workers used three types of specimens under different types of loadings (1) 90° hoop wound tubes under torsion, Swanson et al., Ref. [58], (2) Iosipescu specimens (notched beam under four point bending), Swanson et al., Ref. [58], and (3) ( ± 45°) tubes under internal pressure, Swanson

89 and Christoforou, Ref. [20], and Swanson and Toombes, Ref. [48]. They reported that the stress–strain curve is highly non-linear and that the values of failure strain and strength depend on the test specimen, although the initial modulus was almost the same in all cases. The typical shear stress–strain curve selected for the material taken from Swanson and Toombes, Ref. [48], for tests on unidirectional lamina under torsion. The curve is in agreement with that published by Daniel and Ishai, Ref. [54], which was obtained from tests on ± 45° and 10° off-axis coupons. The failure strain in the work of Daniel and Ishai, Ref. [54], was less than that specified in Part A which was 2%, taken from Ref. [48]. The failure strain specified is smaller than that reported by Swanson et al., Ref. [58]. The value of shear strength for AS4/3501-6 was chosen to be 79 MPa, Swanson and Toombes, Ref. [48]. This value was slightly lower than a previous value of 96 MPa which was obtained from torsion and Iosipescu specimens and was published by Swanson et al., Ref. [58]. Other published values, Refs [51–53,65], are in the range of 71–110 MPa, depending upon the test method. The thermal expansion coefficients and the curing temperature were reported by Kim and Castro, Ref. [51], Lee and Daniel, Ref. [60], Daniel and Ishai, Ref. [54], and others. The critical strain energy release rate GIC for AS4/3501-6 material was taken as 220 J/m2. GIC was measured by many investigators using Double Cantilever Beam (DCB) specimens. The results ranged from 137 to 260 J/m2, see Daniel et al., Ref. [61], Smiley and Pipes, Ref. [62], Rybicki et al., Ref. [63], and Yaniv and Daniel, Ref. [64]. Long and Swanson, Ref. [66], gave a value of 136 J/m2, based upon a communication with Hercules Inc (AS4/3601-6 material manufacturer). 4. Laminate models The composite specimens from which the experimental results were obtained did not have exactly the same structure as the models specified for analysis. All of the models were symmetric laminates with a small number of layers. Almost all of the specimens were tubes and were made by hand lay up or filament winding. The exact dimensions of the test specimens have been described in section 3. Some of the major differences between the models given in Part A and the actual test pieces are highlighted below. The helical filament winding process results in a rather complex interwoven structure (see for example Jones and Hull, Ref. [67]). One complete cover produces one +  layer and one – layer at each point on the tube, but the layer sequence can be reversed at different positions. In a filament wound tube with more than one cover the +  and the – layers occur alternately throughout the wall of the tube. The thicker filament wound tubes ( > 2 covers) have more layers than the corresponding theoretical models. For example, the (90°/ ± 30°/90°) glass/epoxy filament wound tube would have more than two ± 30° covers to make up the required wall thickness giving more than the two + 30° and two –30° layers assumed in the model. The quasi isotropic (90°/ ± 45°/0°)s carbon/epoxy tubes used in the experiments were made by hand lay up. The majority of those tubes had the same 8 layer symmetric laminate arrangement specified for the model but thicker tubes had more (3 or 4) repetitions of the 4 layer sequence. In the experiments the tubular test specimen geometry applies constraints on the deformation of the laminate. Axisymmetric deformation of a thin cylindrical shell under

90 uniformly distributed loading gives rise to the following constraints on bending (Kx and Ky) and twisting (Kxy) curvatures: Kx = d2w/dx2

(4)

Ky = 0

(5)

Kxy = xy/R

(6)

(Where w is radial displacement, xy is the in-plane shear strain and x is axial displacement). The two cover ± 55° filament wound glass/epoxy specimens had a layer sequence of + 55/–55/ + 55/–55, or the reverse, at any given point on the tube (see above). An unconstrained laminated plate of such construction would twist under the action of in-plane normal loading, giving a slightly different stress distribution from the symmetric model. For a balanced laminate (i.e. a laminate with equal +  and – layers) shear strain would only occur if shear or bending loads were applied. For a tubular specimen loaded only by normal loads, the stresses experienced by the tube wall remote from its ends could be simulated by a plate of the same balanced antisymmetric laminated construction, constrained so that the three bending and twisting curvatures (equations (4) to (6)) are zero. To avoid introducing such complications in Part A of the Exercise all laminates were specified as symmetric, which is an alternative way of simulating tube behaviour. In thicker walled filament wound tubes with more covers (e.g. the ± 55° filament wound tubes tested under compression and the ± 45° filament wound tubes tested at SR = 1 : –1) the effects of direct load/twisting coupling would be negligible because their walls were constructed from many layers but the overall wall thicknesses were greater than those of the models. One difference between the symmetric laminate models and the antisymmetric structure of the filament wound tubes which should be noted is the difference between the effective thickness of the middle layers. The 0°/90° specimens had the same arrangement as the theoretical model which had a thick central layer. The antisymmetric structure of the two cover filament wound ± 55° and ± 45° tubes used in the biaxial tension tests gave four layers of equal thickness but the symmetric 4 layer angle ply models specified for those cases had two adjacent layers at the same angle, resulting in an equivalent central layer of double thickness which could be predicted to behave differently by some theories. The ± 55° and ± 45° Glass fiber/epoxy tubular specimens were designed using linear elastic thin shell theory. Recent nonlinear theoretical analysis has indicated that the 60 mm gauge length was too short for the ± 45° specimens. A more rigorous analysis of the stresses in tubular specimens requires allowance for large displacements and non-linear material properties. Appendix I, that was issued with the experimental results, contains a single sample problem which contributors with integrated laminate failure analysis and finite element packages could use to demonstrate a more complete analysis of tubular specimen behaviour. Uniaxial or biaxial loading was specified for analysis of the theoretical models. In the experiments the tubes were frequently loaded under internal or external pressure. The pressure produces radial compressive stress on one surface but the other surface has no radial stress applied. In thin walled tubes (e.g. the 1 mm thick, 100 mm diameter tubes) the effects of the radial stress are assumed to be negligible (Al-Khalil et al., Ref. [5]). Under some conditions (e.g. biaxial compression, Table 6) it was necessary to use thick walled tubes to avoid

91 buckling. In such cases the stresses are not uniform throughout the thickness of the tube. Very high external pressures (up to 200 MPa) were required to fracture some of the tubes but the most severe stress conditions appear to be at the inside surface where the circumferential stress was maximum and the radial stress was zero (Kaddour et al., Ref. [10]).

5. Conclusions 1 The wide range of experimental results selected for comparison with the theoretical predictions in Part B of the Exercise have been presented. 2 We have tried to draw attention to limitations in some of the experiments and results. The results are subject to error and more and better experimental data are desirable in some cases. 3 The source of data and the methods used in deriving the material properties for the unidirectional laminae and constituents use in part A of the exercise have been outlined. 4 Differences between the symmetric laminated plate models used in the theoretical analysis and the tubular test specimens have been discussed. 5 Comparisons between the theoretical predictions and the wide range of experimental data should help identity any major discrepancies, limitations and areas requiring further theoretical and experimental work.

References 1 Hütter U, Schelling H and Krauss H, ‘An experimental study to determine failure envelope of composite materials with tubular specimen under combined loads and comparison between several classical criteria’, in Failure Modes Of Composite Materials With Organic Matrices And Other Consequences On Design, NATO, AGRAD, Conf Proc No. 163, Munich, Germany, 13–19 October 1974. 2 Krauss H and Schelling H, Kunststffe, Bd59, Heft 12, pp 911–917, 1969. 3 Schelling H and R M Aoki (1992), DFVLR, Germany, Personal communication. 4 (ZTL80) Dornier, Fokker, MBB, DLR: ‘Investigations of Fracture Criteria of Laminae’. 1975–1980, BMVg. (multiaxial testing, reports in German). 5 Al-Khalil M F S, Soden P D, Kitching R, and Hinton M J, ‘The effects of radial stresses on the strength of thin walled filament wound GRP composite pressure cylinders’, Int J. Mech. Sci., V38, pp 97–120, 1996. 6 Soden P D, Kitching R and Tse P C, ‘Experimental failure Stresses for ± 55° filament wound glass fibre reinforced plastic tubes under biaxial loads. Composites, V20, 125–135, 1989. 7 Soden P D, Kitching R, Tse P C, Tsavalas Y and Hinton M J, ‘Influence of winding angle on the strength and deformation of filament wound composite tubes subjected to uniaxial and biaxial loads’, Compos Sci Tech, V46, pp 363–378, 1993. 8 Forster R and Knappe W (1970), Kunststoffe, Bd60, Heft 12, pp 1053–1059. 9 Kaddour A S and Soden P D, ‘Design of a high pressure rig for biaxial and triaxial compression testing of composite tubes’, Science and Engineering of Composite Materials, V5, pp 27–38, 1996. 10 Kaddour A S, Soden P D and Hinton M J, ‘Failure of ± 55° filament wound composite tubes under biaxial compression’, J Compos Mater, V32, no.18, pp 1618–1645, 1998. 11 Al-Khalil M F S, ‘Strength Of Filament Wound Structure Under Complex Loading’, PhD thesis, UMIST, England, 1990.

92 12 Al-Salehi F A R, Al-Hassani S T S and Hinton M J, ‘An experimental investigation into the strength of angle GRP tubes under high rate of loading’, J Comp Mater, V23, pp 288–305, 1989. 13 Kaddour A S, Soden P D and Hinton M J (1996), Unpublished work. 14 Swanson S R and Nelson M, ‘Failure properties of carbon/epoxy laminates under tensioncompression biaxial stress’, in Composites’86: Recent Advances in Japan and United States, Proc Japan-US CCM-III, edited by K Kawata, S Umekawa and A Kobayashi, Tokyo, Japan, pp 279– 286. 15 Swanson S R and Christoforou A P, ‘Response of quasi-isotropic carbon/epoxy laminates to biaxial stress’, J Compos Mater, V20, pp 457–471, 1986. 16 Swanson S R and Colvin G E, ‘Compressive strength of carbon/epoxy laminates under multiaxial stress’, Final annual report to the Lawrance Livermore National Laboratory, UCRL-21235, 1989. 17 Colvin G E and Swanson S R, ‘In-situ compressive strength of carbon/epoxy AS4/3501–6 laminates’, J Eng. Mater Tech, V115, pp 122–128, 1993. 18 Swanson S R Christoforou A P and Colvin G E, ‘Biaxial testing of fibre composites using tubular specimens’, Exp. Mech., V28, pp 238–243, 1988. 19 Trask B N, ‘Response Of Carbon/Epoxy Laminates To Biaxial Stress’, MSc thesis, Department Of Mechanical And Industrial Engineering, The University Of Utah, Utah, USA, 1987. 20 Swanson S R and Christoforou A P, ‘Progressive failure in carbon/epoxy laminates under biaxial stress’, Trans ASME, J Engg Mater Techno, V109, pp 12–16, 1987. 21 Christoforou A P, M.Sc. Thesis, Department of Mechanical and Industrial Engineering, the University of Utah, Utah, USA, 1984. 22 Swanson S R and Trask B C, ‘Strength of quasi-isotropic laminates under off-axis loading’, Compos Sci Techno, V34, pp 19–34, 1989. 23 Li S, ‘Modelling Damage In Thin Walled Filament Wound Structures’, PhD thesis, UMIST, Manchester, UK, 1993. 24 Reid S R, Soden P D and Li S, ‘Application of damage models to filament wound tubes’, MoD final report, Contact No 2044/186, 1995. 25 Kaddour A S, Soden P D and Hinton M J, ‘Quasi-Static Behaviour Of ± 45° Glass/Epoxy Filament Wound Composite Tubes Under Equal Biaxial Tension-Compression Loading: Experimental Results’, Composites: Part B, V34, pp 689–702, 2003. 26 Eckold G C, ‘The effect of damage on the mechanical properties of composites and its impact on design’, IMechE Conference Designing Out Failure In Composites, held at London UK, 23 September 1994. 27 Eckold G C, Hancox N L and Lee R J, ‘Application of micromechanics in the prediction of damage initiation and growth in structural composites’, in 3rd Int. Conf. On Deformation and Fracture Of Composites, Institute of Materials, University of Surrey, Guilford, UK, 27–29 March 1995, pp 66– 76. 28 Eckold G C (1997), Personal communication. 29 Connor, P J, Hayman C J, Hinton M J and Ferguson R F, ‘Automated method for measuring crack growth in glass fibre reinforced epoxy laminates’, in 5th Int Conf Deformation and Fracture of Composites, 18–19 March 1999, The Institution of Mechanical Engineers, London UK. pp 179– 180. 30 Soden P D, Hinton M J and Kaddour A S, ‘Lamina propertied, lay-up configurations and loading conditions for a range of fibre reinforced composite laminates’, Compo Sci Techno, V58, pp 1011– 1022, 1998, see also Chapter 2.1 in this book. 31 Hinton M J(1995), Private communication. 32 Hinton M J (1978), RARDE report. 33 Tsai S W(1992), ‘Theory of Composites Design’, Think composites, Palo Alto, 1992. 34 Harwood C K and Hinton M J, ‘Investigation of the transverse compression properties of glass fibre/epoxy resin system using circumferentially wound tubes’, RARDE, Memorandum 38/79 (EM4), MoD, UK, 1979.

93 35 Jones R M, ‘Mechanical Properties Of GRP In The Through Thickness’ MSc dissertation, UMIST, Manchester, UK, 1986. 36 Hinton M J, Soden P D and Kaddour A S, ‘Strength of composite laminates under biaxial loads’, Applied Composite Materials, V3, pp 151–162, 1996. 37 Parvisi A, Garrett K W and Bailey J E, ‘Constrained cracking in glass fibre reinforced epoxy cross ply laminates’, J Mater Sci, V 13, pp 195–201, 1978. 38 Krawczac P and Pabiol J, ‘Fracture mechanics applied to glass fibre/epoxy matrix interface characterization’, Journal of Composite Materials, V29, pp 2230–2253, 1995. 39 Miyase A, ‘Transverse tesnsile strength anisotropy in thick filament wound ring composites’, J Mater Sci, V19, pp 923–928, 1984. 40 Forster R (1972), Kunststoffe, Bd 62, Heft 1, pp 57–62. 41 Panhwar N M and Scott M L, ‘Shear fatigue performance of postbuckling fibre composite panels’, in ICCM-9, Vol VI, pp 527–534, 1993. 42 Ladevese P and Le Dantec E, ‘Damage modelling of the elementary ply for laminated composites’, Compos Sci Techno, V43, pp 257–267, 1992. 43 Lafarie-Fernot M C and Touchard F, ‘Comparative in-plane shear behaviour of long carbon-fibre composites with thermoset or thermoplastic matrix’, Compos Sci Techno, V52, pp 417–425, 1994. 44 Sanders R C and Grant P, ‘The strength of laminated plates under in-plane loading, Part 1: Failure criteria’, Report SOR (P) 130, BAe, Warton Aerodrome, Preston, UK. 45 Dickson T, Munro M and Lee S, ‘Selection of an-plane shear test method based on the shear sensitivity of laminate tensile modulus’, Composites, V26, pp 17–24, 1995. 46 Ditcher A K, Rhodes F E and Webber J P H, ‘Non-linear stress–strain behaviour of carbon fibre reinforced plastic laminates’, J. Strain Analysis, V16, pp 43–51, 1981. 47 Schulein R, Keinzler J, Scharr G, Hackenberg R and Aoki R, ‘Verbesserung der ausnutzbarkeit der neuen hochfesten C-fasern in hochbelastelen verbubdstrukturen und verbesserung des dimensionierungsverfahrens’, VDI Fortuschritt Berichte, Reihe 5, Nr 186, Dusseldorf, VDI-Verlag, 1989. 48 Swanson S R and Toombes G R, ‘Characterisation of prepreg tow carbon/epoxy laminates’, J Engg Mater Techno, Trans ASME, V111, pp 150–153, 1989. 49 Abdullah M G, ‘State of the art of advanced composite materials: compression test methods’, JANNAF, CMGS, and S &MBJ Joint Meeting, 27–30 November 1994, Jet Propulsion Lab., California Institute of Technology, Pasadena, CA, 1984. 50 Wung E C J and Chatterjee S N, ‘On the failure mechanisms in laminate compression specimens and the measurement of strengths’, J Compos Mater, V26, pp 1885–1914, 1992. 51 Kim R and Castro A S, ‘A longitudinal compression test for composites using a sandwitch specimen’, J Compos Mater, V26, pp 1915–1929, 1992. 52 Daniel I M, Hsiao H M, Wooh S C and Vittoser J, in AMD – V162, Mechanics of thick composites, pp 107–126, ASME publication, 1993. 53 Sun C T and Jun A W, ‘Effect of matrix non-linear behaviour on the compressive strength of fibre composites’, in AMD – V162, Mechanics of Thick Composites, ASME, pp 91–105, 1993 54 Daniel I M and Ishai O, ‘Engineering Mechanics Of Composite Materials’, Oxford University Press, Oxford, UK, 1994. 55 Colvin G E and Swanson S R, ‘Mechanical characterization of IM7/8551–7 Carbon/epoxy under biaxial stress’, J Engg Mater Techno, V112, pp 61–67, 1990. 56 Ferguson R F, Hinton M J and Hiley M J, ‘Determining the through-thickness properties of FRP materials’, Compos Sci Techno, V 58, pp 1411–1420, 1997. 57 Jelf P M and Fleck N A, ‘The failure of composite tubes due to combined compression and torsion’, J Mater Sci, V29, pp 3080–3084, 1994. 58 Swanson S R, Messick M J and Toombes G R, ‘Comparison of torsion tube and Iosipescu in-plane shear test results for a carbon fibre reinforced epoxy composite’, Composites, V16, pp 220–224, 1985.

94 59 Swanson S R, Messick M J and Tian Z, ‘Failure of carbon/epoxy lamina under combined stress’, J Comp Mater, V21, pp 619–630, 1987. 60 Lee J,-W and Daniel I M (1990), J Compos Mater, V24, p 1225. 61 Daniel I M, Yaniv G and Auser J W, ‘Rate effects on delamination fracture toughness of graphite epoxy composites’, in ICCS/4, Paisley College of Technology, Scotland, 27–29 July, pp 2.258– 2.272, 1987. 62 Smiley A J and Pipes R B, ‘Rate effects on Mode I interlaminar fracture toughness in composite materials’, J Compos Mater, V21, pp 670–687, 1987. 63 Rybicki E, Herenandez T D, Dileber J E, Knight R C and Vinson S S, ‘Mode I and mixed mode energy release rate values for delamination of graphite/epoxy test specimens’, J Compos Mater, V 21, pp 105–123, 1987. 64 Yaniv G and Daniel I M, ‘High-tapered double cantilever beam specimen for study of rate effects on fracture toughness of composites’, ASTM STP 972, J D Whitcomb, Ed., Philadelphia, USA, pp 241–258, 1988. 65 Sun C T and Zhou S G, ‘Failure of quasi-isotropic composite laminates with free edges’, J Reinf Plast Compos, V7, pp 515–557, 1988. 66 Long B J and Swanson S R, ‘Ranking of laminates for edge delamination resistance’, Composites, V25, pp 183–188, 1994. 67 Jones M L C and Hull D, ‘Microscopy of failure mechanisms in filament wound pipe’, J Mater Sci, V 14, pp 165–177, 1979. 68 Carroll M, Ellyin F, Kujawski D and Chiu A S, ‘The rate dependent behaviour of ± 55° filament wound glass fibre/epoxy tubes under biaxial loading’, Compos Sci and Techno, V55, pp 391–403, 1995. 69 Aliutov N A and Zinoviev P A, ‘Deformation and failure of fibrous composites with brittle polymeric matrix under plane stress’, in Mechanics of Composites, MIR publishers, Moscow, 1982.

Appendix Details of the specimen to be analyzed in Part B As pointed out in the instructions for Part A, the organizers are asking those participants who have integrated failure analyses and structural analysis packages if they wish to analyze a simple structure. The following section gives details of the simple structure proposed. Identical details are distributed to all participants. Structure to be analyzed: Figures (A1) and (A2) show details of the tubular specimen to be analyzed in Part B of the Exercise. Material and dimensions: The tube is 100 mm inside diameter, 1 mm thick and 283 mm long and made of E-glass/ MY750 epoxy material with fiber volume fraction of 60%. The properties for this material (material I) were sent previously to all contributors. Lay-up: The tube is made of four plies oriented at + 55°/–55°/ + 55°/–55° with respect to the tube axis. All the plies have an equal thickness of 0.25 mm.

95

Fig. A1. Geometry of the tubular specimen. (Inner diameter = 100 mm, thickness = 1 mm, overall length = 283.0 mm).

End Reinforcement: Reinforcement is applied along a distance of 111.5 mm from each end of the specimen, leaving a section, 60 mm long, in the middle of the tube without reinforcement. The end reinforcement is in the form of circumferentially wound glass/epoxy material identical to the

Fig. A2. Details of end reinforcement geometry of a ± 55° GRP composite tube. Dimensions are in mm.

96 material of the tube (E-glass/MY750 epoxy). Two shapes are shown in Fig. (A2) for the end reinforcement (i) a stepped shape representing the state of the as-manufactured reinforcement and (ii) an approximate shape, broken line in Fig. (A2), with a radius of 298 mm, tangential to the outside surface of the tube at the end of the 1 mm thick test section. Either of these forms of reinforcement can be used in your analysis. Loading: The tube is subjected to a uniform internal pressure along a length of 213 mm. The tube is free at its two ends and no axial end load or pressure is applied. Results: Please plot a graph showing stress versus circumferential and axial strains on the outside surface of the tube at the center of the gauge length as the pressure increased up to the final failure. Use the scales provided previously in Fig. 16 of Part A. These results can then be compared with your previous prediction and the experimental results provided for Part B, Fig. 8. Also, please report the circumferential and axial stresses at the inside and outside surfaces of the tube at the center of gauge length at the final failure pressure. Contributors may also choose to plot a figure or figures in their favored format to demonstrate the development of failure in the specimen.

Section 3 Description of the Individual Failure Theories by their Originators

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

98

CHAPTER 3.1

Prediction of composite laminate fracture: Micromechanics and progressive fracture P. K. Gotsisa, C. C. Chamisa and L. Minnetyanb a

NASA Lewis Research Center, 21000 Brookpark Road, Cleveland, OH 44135, USA b Clarkson University, Potsdam, NY, USA

Abstract An investigation is described on the prediction of first-ply failure and fracture in selected composite laminates. The laminates are made from glass fibers and graphite fibers in epoxy matrices. Failure envelopes are generated for combined loading of these laminates on the basis of first-ply failure and laminate fracture. The evaluation is performed by a micromechanics-based theory and progressive fracture. Keywords: glass fibers, graphite fibers, epoxy matrices, composite properties, failure envelopes, combined loading, residual stress, pristine properties, in situ properties, stress/strain curves, cross-ply laminates, angle-ply laminates, multidirectional laminates

1. Introduction One of the persistent difficulties in the design and analysis of composites has been prediction of laminate fracture under uniaxial and or combined loading by using either unidirectional composite (ply) data or micromechanics with pristine constituent material properties. The difficulty has been compounded many times over by the availability of many and diverse failure criteria. It became apparent, therefore, that some kind of a formalized comparisons among the various failure theories with measured data would be instructive and very useful. To that end, Hinton and Soden1 organized an exercise to confirm the current state-of-the-art for predicting failure in composites. The objective of this article is to describe the results obtained by the present authors as part of their participation in that exercise. The authors’ participation consists of two major parts: (1) generation of first-ply failure envelopes for combined loading by using the micromechanics-based uniaxial composite strengths, as will be described later, and (2) generation of laminate fracture envelopes and stress/strain diagrams for uniaxial and combined loading by using progressive fracture. The effects of residual stresses on the laminate first-ply failure are also evaluated as the differences between pristine and in situ constituent properties. The pristine constituent material properties were calibrated to the unidirectional composite properties so that the micromechanics used

99 will reproduce the unidirectional composite properties provided. The details are described in the text. The specific theories and computer codes are also described.

2. Computer codes used Two computer codes were used to generate the laminate failure envelopes. These computer codes integrated composite mechanics and progressive structural fracture: (1) Integrated Composite Analyzer (ICAN) and (2) Composite Durability Structural Analyzer (CODSTRAN). These computer codes are briefly described below for completeness.

2.1. ICAN computer code2 The most cost-effective way to analyze/design fiber composite structures is through the use of computer codes. Composite analysis computer codes to date have been based mainly on ‘classical’ laminate theory. Extensive research has been conducted by the authors to develop composite mechanics theories and analysis methods from micromechanics to finite elements. These theories and analysis methods account for environmental effects and are applicable to intraply hybrid composites, interply hybrid composites, and combinations therefrom. Most of these theories are represented by simplified equations that have been corroborated by experimental results and finite-element analysis, as will be cited later. The composite mechanics theories with their respective simplified equations constitute a structured multiscale formalism which is (1) ‘upward integrated’ (synthesis) from material behavior space to structural analysis and (2) ‘top-down traced’ (decomposition) from structural response to material behavior space (see Fig. 1). This structured theory has been incorporated into a computer code called ICAN (Integrated Composite Analyzer). A brief history of the developments pertaining to composite mechanics and related computer codes that led to the evolution of ICAN is described in the following paragraphs. The importance of and need for a multilevel (multi-scale) analysis in designing structural components with multilayered fiber composites were recognized about 30 years ago.3 A multi-level analysis computer code (MFCA: Multilayered Fiber Composites Analysis) that was found to be efficient in predicting the structural response of multi-layered fiber composites (given the constituent materials properties, the fabrication process and the composite geometry) is documented in Ref. 4. Intraply hybrid composites are the logical sequel to conventional composites and to interply hybrid composites. Subsequently, theoretical and experimental investigations have been conducted on the mechanical behavior of intraply hybrids at the Lewis Research Center.5–7 The theoretical methods and equations described in these references, together with those for hygrothermal effects,8 have been integrated into a computer code for predicting hygral, thermal and mechanical properties of, and thereby ‘designing’, intraply hybrid composites. The attendant computer code is identified as INHYD for INtraply HYbrid Composite Design.9 Experimental comparisons are included in Refs 7 and 8. The computer code ICAN is a synergistic combination of the aforementioned computer programs, MFCA and INHYD, together with several significant enhancements. It utilizes the micromechanics embodied in INHYD and the laminate theory of MFCA to build a

100

Fig. 1. Multi-scale composite laminate behavior simulation.

comprehensive analysis/design capability for structural composites. Additional features unique to ICAN are as follows: • • • • • •

stress concentration factors at a circular hole; predictions of probable delamination locations around a circular hole; Poisson’s ratio mismatch details near a straight free edge; free-edge stresses; material cards for finite-element analysis for several commercial computer codes; laminate failure stresses based upon first-ply failure and fiber breakage criteria, with and without hygrothermal degradation; • transverse shear stresses and normal stresses; and • explicit specification of interply layers. In addition, ICAN has its own (dedicated) data bank of material properties for commonly used fibers and matrices. ICAN is primarily designed to analyze the hygrothermomechanical response/properties of fiber-reinforced/resin-matrix type layered composites, given the local membrance loads and bending moments. Three types of layer are recognized by the program. They are: (1) the standard composite system that consists entirely of a primary composite made of one type of fiber and matrix; (2) the intraply hybrid composite system that consists of a primary composite and a secondary composite arranged in a prescribed manner within a layer (for purposes of identification, the primary composite in the hybrid is the one that constitutes the largest volume ratio); and (3) the interply layer that consists of matrix only.

101 The complete details of the equations in the code are given in Ref. 2. Prediction of composite hygrothermomechanical properties is achieved through use of the various micromechanics theories mentioned earlier. Laminate properties are obtained through macromechanics and laminate theory. Classical laminate theory is used to predict local stresses and strains. The free-edge stress calculations are based upon the approach outlines in Ref. 10 with enhancements to accommodate interply layer and local characteristics of the adjacent ply. The stress concentration factors around a circular hole are obtained by using the equations given in Ref. 11. The laminate failure stress analysis utilizes two criteria: (1) the first-ply failure based upon the maximum strength and (2) the first-ply failure based upon the fiber breakage. Complete laminate failure analysis is performed by using two different ply combined-stress failure criteria and one interply delamination criterion for each specified load condition.

2.2. CODSTRAN computer code2,12 CODSTRAN is a new approach for predicting the progressive fracture of composite laminates and structures. It is based on the following observation and/or concepts. (1) Any laminate or structural component can sustain a certain amount of damage prior to structural fracture (collapse). (2) During damage propagation, the laminate or component exhibits progressive degradation of structural integrity as measured by global structural behavior variables such as loss in frequency, loss in buckling resistance or excessive displacements. (3) The critical damage can be characterized as the amount of damage beyond which the degradation of the laminate’s structural integrity is very rapid, induced by either (i) small additional damage or (ii) small loading increase. (4) Laminate damage is characterized by the following sequential stages: (i) initiation, (ii) growth, (iii) accumulation, (iv) stable or slow propagation (up to critical amount), and (v) unstable or very rapid propagation (beyond the critical amount) to collapse. These concepts are fundamental to developing formal procedures to (1) identify the different stages of damage, (2) quantify the amount of damage at each stage and (3) relate the degradation of global structural behavior to the amount of damage at each stage. CODSTRAN is a combination of composite mechanics (ICAN2) with the finite-element method (MHOST12) to permit a formal description of local conditions to global structural behavior. It is accomplished through an integrated computer code as shown schematically in Fig. 2. The bottom of this figure describes the constituent properties as functions of environmental and mechanical loading conditions. Based upon the ply stresses and the ply strength allowables, the criteria for damage initiation, growth and accumulation and propagation are examined. The constituent properties are updated as a result at every load increment. For example, if the ply longitudinal stress exceeds the allowable strength, then the fiber and matrix longitudinal moduli are replaced by negligible values so that the ply essentially does not carry any load and the stresses get redistributed to the surrounding plies. However, if a particular ply’s transverse or shear strength exceeds its allowable, then only the matrix is assumed to have failed and therefore the matrix modulus is replaced with a negligible value. Once the current constituent properties are degraded, as shown in the left part of Fig. 2, through repeated applications of micromechanics, macromechanics and laminate theory the global structural stiffness matrix is assembled and fed to the finite-element

102

Fig. 2. Progressive fracture in composite laminates and structures simulation cycle via CODSTRAN.

analysis. Thus, the left part integrates (synthesizes) local damage conditions to global structural behavior (response). The results of finite-element analysis are the nodal stress resultants. These are used to decompose global response changes (laminate stresses/strains, for example, as a result of any increments in loads or stiffness updates) on the local (micro) material stress/resistance. After an incremental finite-element analysis, the computed generalized nodal stress resultants are supplied to the ICAN module which evaluates the nature and amount of local damage, if any, in the plies of the composite laminate. Individual ply failure modes are assessed by ICAN by using failure criteria associated with the negative and positive limits of the six ply-stress components in the material directions, as follows: SL11C < L11 < SL11T SL22C < L22 < SL22T SL33C < L33 < SL33T SL12(–) < L12 < SL12( + ) SL23(–) < L23 < SL23( + ) SL13(–) < L13 < SL13( + ) The stress limits are computed by the micromechanics equations in ICAN, on the basis of constituent stiffness and strength parameters. The equations used for ply stress limits are given

103 in the ICAN User’s and Programmer’s Manual.2 Ply stress limits are evaluated as follows. Ply longitudinal tensile strength is computed from fiber tensile strength and the fiber volume ratio. Ply longitudinal compressive strength is computed as the minimum of three different criteria which are: (1) the rule of mixtures based on fiber compressive strength and fiber volume ratio, (2) fiber microbuckling based on matrix shear modulus and fiber volume ratio and (3) compressive shear failure or kink band formation, which is mainly based on ply intralaminar shear strength and matrix tensile strength. The ply transverse tensile stress limit is computed from matrix modulus, matrix tensile strength and fiber volume ratio. Similarly, the ply transverse compressive stress limit is computed from the matrix compressive strength, matrix modulus and fiber volume ratio. The ply intralaminar shear strength limit is computed based on matrix shear strength, matrix shear modulus and the fiber volume ratio. The interlaminar shear strength is based on matrix shear strength, matrix shear modulus, fiber volume ratio and fiber transverse shear modulus. In addition to the failure criteria based on the ply stress limits, a modified distortion energy (MDE) failure criterion that takes into account combined stresses is considered as follows:

F=1

冋冉 冊 冉 冊 冉 冊冉 冊 冉 冊 册 L11a SL11a

KL12 =

2

+

L22b SL22b

2

 KL12

L11a SL11a

L22b L12s + SL22b sL12s

2

(1 + 4 L12  L13)EL22 + (1  L23)EL11 [EL11EL22(2 + L12 + L13)(2 + L21 + L23)]1/2

where a and b are tension and compression, respectively. Details of the MDE criterion, as well as other options for the assessment of local failure in composites, are given User’s and Programmer’s Manual.2 The MDE failure criterion is obtained by modifying the usual distortion energy failure criterion that predicts combined-stress failure in isotropic materials. The modification takes into account the significant differences in the stress limits of the longitudinal and transverse directions of an orthotropic composite ply. Each component of ply stress is normalized with respect to its limiting strength. No relationship is assessed between normal and shear strengths. A directional interaction factor defined in the MDE criterion reduces to unity for homogeneous isotropic materials. The MDE criterion has been demonstrated to be a good predictor of combined-stress failure in composites. The MDE failure criterion becomes active in the majority of cases during computational simulation of progressive damage and fracture. If the failure predicted by the MDE criterion is not accompanied by a specific explicit failure mode, then the type of failure is assessed by comparison of the magnitudes of the terms in the MDE criterion. Depending on the dominant term in the MDE failure criterion, fiber failure or matrix failure is assigned. The generalized stress/strain relationships are revised locally according to the composite damage evaluated after each finite-element analysis. The model is automatically updated with a new finite-element mesh having reconstituted properties, and the structure is reanalyzed for further deformation and damage. The load is incremented only if no further damage is noticed as a consequence of changes in ply-level stresses. Otherwise, only the material properties at the constituent level are updated at every iteration until a balance between the applied loading and the local response is reached. Overall structural equilibrium is maintained by iterations around the ‘cartwheel’ until a specified convergence

104

Fig. 3. (a) CODSRTAN load incrementation. (b) Overall CODSTRAN simulation.

is reached. This procedure is illustrated in Fig. 3(a). The final result in terms of load versus global displacement is shown in Fig. 3(b). The schematics in Figs 1–3(a) and (b) collectively summarize the fundamentals and implementation of this computational simulation approach to composite structural fracture and also to structural fracture in composite laminates.

3. Modified constituents The fibers, matrices and UD lamina properties were supplied by the organizers.13 In order to take full advantage of the simulation capabilities in ICAN and CODSTRAN and since both of these computer codes are based on micromechanics (constituent materials), we proceeded to modify the constituent materials, provided by the organizers (compare correspondence values, Tables 1 and 2), to the extent that predicted composite unidirectional (ply) properties matched the corresponding measured data in Table 3. Another good reason to base the ICAN/ CODSTRAN simulations on micromechanics is that constituents’ in situ properties generally

Table 1 Mechanical properties of various matrices. Matrix type:

3501-6 epoxy Pristinea

Manufacturer Modulus (GPa) Shear modulus (GPa) Poisson’s ratio Tensile strength (MPa) Compressive strength (MPa) Shearstrength (MPa) Tensile failure strain (%) Thermal coefficient (10–6°C–1)

BSL914C epoxy ICANb

Pristinea

Hercules 4·2 1·56 0·34 69 250 50 1·7 45

ICANb

DFVLR 5·8 c

0·38 53·75 223·9 94·068 – 43·2

4·0 1·481 0·35 75 150 70 4 55

LYSS6/HTS07/DY063 epoxy

MY750/HY917/DY063 epoxy

Pristinea

Pristinea

ICANb

Ciba Geigy 5·724 c

0·398 30·26 224·13 93·79 – 45

3·35 1·24 0·35 80 120 – 5 58

ICANb

Ciba Geigy

4·772 c

0·4 41·79 137·2 86·89 – 64·8

3·35 1·24 0·35 80 120 – 5 58

4·2827 c

0·4 47·92 173·7 87·5 – 52·38

a

All pristine properties taken from Ref. 13. ICAN calbrated. c Shear modulus = E/2(1 + v) is computed internally in ICAN code. b

105

106

Table 2 Mechanical properties of various fibers. Fiber type:

Longitudinal modulus (GPa) Transverse modulus (GPa) In-plane shear modulus (GPa) Major Poisson’s ratio Transverse shear modulus (GPa) Longitudinal tensile strength (MPa) Longitudinal compressive strength (MPa) Longitudinal tensile failure strain (%) Longitudinal compressive failure strain (%) Longitudinal thermal coefficient (10–6°C–1) Transverse therrnal coefficient (10–6°C–1) a b

All pristine properties taken from Ref. 13. ICAN calibrated.

AS4

T300

Gevetex E-glass 21XK43

Silenka E-glass 1200 tex

Pristinea

ICANb

Pristinea

ICANb

Pristinea

ICANb

Pristinea

ICANb

225 15 15 0·2 7 3350 2500 1488 1·111 –0·5 15

208·34 14·96 14·96 0·2 6·9 3282·7 6455 – – –1·818 16·182

230 15 15 0·2 7 2500 2000 1·086 0·868 –0·7 12

220·68 14·965 14·965 0·2 6·965 2496 2034·48 – – –1·773 12

80 80 33·33 0·2 33·33 2150 1450 2·687 1·813 4·9 4·9

85·862 83·860 33·31 0·2 32·62 1889 1786 – – 6·633 4·86

74 74 30·8 0·2 30·8 2150 1450 2·905 1·959 4·9 4·9

73·1 81·72 30·8 0·2 30·8 2124 1310 – – 6·822 11·16

Table 3 Mechanical properties of various unidirectional composites. Fiber type: Matrix type: Manufacturer:

Fiber volume fraction (%) Longitudinal modulus (GPa) Transversemodulus(GPa) In-plane shear modulus (GPa) Major Poisson’s ratio Through-thickness Poisson’s ratio Longitudinal tensile strength (MPa) Longitudinal compressive strength (MPa) In-plane shear strength (MPa) Transverse tensile strength (MPa) Transverse compressive strength (MPa) Longitudinal thermal coefficient (10–6°C–1) Transverse thermal coefficient (10–6°C–1) a b

AS4

T300

3501–6 epoxy

BSL914C epoxy DFVLR

Hercules

Gevetex E-glass 21XK43 Silenka E-glass 1200 tex LY556/HT907/DY063 epoxy Ciba Geigy

MY750/HY917/DY063 epoxy Ciba Geigy

Pristinea

ICANb

Pristinea

ICANb

Pristinea

ICANb

Pristinea

ICANb

60 126 11 6·6 0·28 0·4 1950 1480 48 200 79 –0·1 26

60 127·309 11 6·289 0·272 0·414 1969 1480 48 200 79·93 –0·1 26

60 138 11 5·5 0·28 0·4 1500 900 27 200 80 –0·1 26

60 140 11 6·248 0·27 0·44 1500 900 27 200 80 –0·940 26·24

62 53 48 17·7 5·83 0·278 0·4 1140 570 35 114 72 8·6 26·4

62 53 37 17·57 6·427 0·28 0·403 1131 570 35 114·48 72·413 8·6 26·26

60 45·6 16·2 5·83 0·278 0·4 1280 800 40 145 73 8·6 26·4

60 45·634 16·206 5·937 0·279 0·415 1295 800 40 144·964 73 8·64 26·28

All pristine properties taken from Ref. 13. ICAN calibrated.

107

108 differ from their pristine state based on laminate data.14 The modified constituent properties obtained and used are shown in Tables 1 and 2. Comparing values from the tables for the matrix properties, matrix strength properties are substantially different. The tensile strengths are about 50% lower, while the compressive strengths are about 30% higher for the DFVLR matrix. The shear strengths are also higher. The interesting point to be noted is that if the in situ matrix retained its pristine tensile strength, then a substantial tensile strength reduction could be caused by cracking during processing owing to thermal mismatch. The fiber properties (compare correspondence values, Table 2) are about the same for both the asprovided and calibrated cases. The unidirectional composite properties (compare correspondence values, Table 3) are practically identical, as they should be.

4. ICAN simulation results Herein we summarize the simulation results obtained for biaxial stress failure envelopes by using ICAN, the modified constituent properties (Tables 1 and 2) and first-ply failure. The envelopes are described in Ref. 13.

4.1. Unidirectional laminate The biaxial failure stress envelope for E-glass/LY556 unidirectional laminate subjected to transverse and shear loading (y versus xy) is shown in Fig. 4. Note only the upper half is shown since this failure envelope is symmetric about the y axis. The biaxial failure stress envelope for the T300/BSL914C unidirectional laminate subjected to longitudinal and shear loading (x versus xy) is shown in Fig. 5. Only the upper half is plotted since it is symmetrical about the x axis. The biaxial failure stress envelope for the E-glass/MY750 unidirectional laminate subjected to biaxial longitudinal and transverse loading (x versus y) is shown in Fig. 6. This failure envelope shows discontinuities from the TC to TT and from the CT to CC quadrants. The discontinuities are accentuated because of the large difference between longitudinal tensile strength and longitudinal compressive strength with transverse tensile strength and compressive strength.

Fig. 4. Biaxial failure stresses for 0° lamina made of GRP material: E-glass/LY556/HT907/DY063.

109

Fig. 5. Biaxial failure stresses for 0° lamina made of CFRP material based on first-ply failure (predicted by ICAN code).

Fig. 6. Biaxial failure envelope for 0° E-glass/MY750 lamina based on first-ply failure (predicted by ICAN code).

The micromechanics equations in ICAN were extensively validated by 3D finite-element simulations.15

4.2. Multi-ply laminates A description of the loading patterns and lay-ups of the multidirectional laminates is provided in Ref. 13. The biaxial failure stress envelope for the E-glass/LY556 [90/ ± 30]s laminate subjected to biaxial loading (y versus x) stress state is shown in Fig. 7. Note this failure envelope is generated by first-ply failure, This laminate is strong in compression but relatively weak in tension. The biaxial failure stress envelope for the E-glass/LY556 [90/ ± 30]s subjected to biaxial loading (xy versus x) is shown in Fig. 8 based on first-ply failure. Note only the upper half is plotted since it is symmetric about the x axis. Note this laminate is weak in both tension and shear but strong in x compression. The biaxial failure stress envelopes for the E-glass/MY750 laminate subjected to x and y combined loading is shown in Fig. 9. This laminate is strong in y tension and in the y, x compression/compression quadrant.

110

Fig. 7. Biaxial failure envelope for 90/ ± 30 E-glass/LY556 lamina based on first-ply failure (predicted by ICAN code).

The biaxial failure stress envelope for the AS4/epoxy 3501-6 [90/ ± 45/0]s laminate subjected to biaxial loading (x versus y) is shown in Fig. 10. Note that this laminate is ultra strong in compression-compression and strong in tension-tension. Note also that these types of combined loading have a substantial synergistic effect in the compression/compression quadrant.

4.3. Effects of thermal stresses All the laminate failure envelopes were generated without accounting for lamination residual stresses. As is well known, these stress results from the curing temperature differential and the thermal expansion coefficient (TEC) mismatch. It is instructive, therefore, to evaluate the

Fig. 8. Biaxial failure envelope for 90/ ± 30 E-glass/LY556 lamina based on first-ply failure (predicted by ICAN code).

111

Fig. 9. Biaxial failure envelope for ± 55 glass/epoxy lamina based on first-ply failure (results predicted by ICAN code).

effect of residual stresses on at least one failure envelope. The failure envelopes for the AS4/ epoxy 3501-6 [90/ ± 45/0]s laminate are replotted in full scale without and with residual stresses in Fig. 11. The residual stresses have dramatic effects on the laminate biaxial failure stress envelope in three quadrants that have tensile stresses. That is an expected result because the plies in quasi-isotropic laminates have relatively high transverse ply stresses owing to curing temperature differential and TEC mismatch. It should also be noted that the compression quadrant part increased in resistance.

Fig. 10. Biaxial failure stresses for [0°/ ± 45°/90°] laminate on CFRP material. The results are for firstply failure and predicted by ICAN code.

112

Fig. 11. Biaxial failure stresses for [0°/ ± 45°/90°] laminate of CFRP material (– – –, with thermal stresses; ——, without thermal stresses). The results are for first-ply failure and predicted by ICAN code.

5. CODSTRAN simulation results CODSTRAN simulation results are presented for one unidirectional laminate and all the multi-ply laminates. Those for the unidirectional laminate are presented for comparison with the ICAN predictions.

5.1. Unidirectional laminate The failure envelope generated by plotting CODSTRAN results is shown in Fig. 12. This biaxial failure envelope is for xy versus y. Although there are not enough points to obtain a smooth function, those points are exactly the same as in Fig. 4. This is an expected result since

Fig. 12. Biaxial failure stresses for 0° lamina made of GRP material: E-glass/LY556/HY907/DY063.

113 both CODSTRAN and ICAN are using the same combined-stress failure criterion which is stress based.16

5.2. Multi-ply laminates The combined-stress failure envelopes predicted by using the progressive fracture feature in CODSTRAN are shown in Fig. 13 for the E-glass/LY556 [90/ ± 30]s laminate subjected to combined x versus y loading. Two failure envelopes are shown: the inner one is for initial damage while the outer one is for laminate fracture. The inner failure envelope in Fig. 13 includes the residual stresses whereas that in Fig. 7 does not. Two important observations are: (1) the residual stresses affect significantly the first-ply failure in quadrants with tensile stresses as was already mentioned, but do not affect the compression/compression quadrant; and (2) the laminate fracture tensile stresses are about 10 times those that cause first-ply failure. Two important conclusions follow: (1) multi-ply laminates designed for first-ply failure under x versus y (tension-tension loading) will be way overdesigned; and (2) multiply laminates for monotonic single load to fracture must be designed by using progressive fracture for cost-effective composite utilization. On the other hand, for unidirectional laminates that are uniformly loaded, the combined-stress fracture envelopes may be assessed without a progressive failure analysis. The combined-stress failure envelopes simulated by using CODSTRAN for E-glass/LY556 [90/ ± 30]s laminate subjected to combined (xy versus x) loading is shown in Fig. 14. The CODSTRAN simulations include the residual stresses which are significant contributors to first-ply failure. Comparing the inner failure envelope in Fig. 14 with that in Fig. 8 shows that the residual stresses decrease the first-ply laminate tensile and shear failures by about three times. Additional important observations are: (1) the residual stress enhances the laminate compression first-ply failure by about 25%; (2) progressive fracture (suitable for monotonic

Fig. 13. Biaxial failure envelope for 90/ ± 30/90 laminate made of E-glass/LY556 (results predicted by CODSTRAN code).

114

Fig. 14. Biaxial failure envelope for 90/ ± 30/90 laminate made of E-glass/LY556 under combined shear and direct stresses (results predicted by CODSTRAN code).

load) dramatically expands the laminate failure envelope in the tensile quadrant; and (3) the progressive fracture combined-stress failure envelope is not a smooth function because (i) of the stress redistribution that follows a local fracture, (ii) the various failure modes are discrete and, therefore, are not continuous, and (c) the jumps occur as the fracture progresses from one failure mode to the next. The important conclusions are: (1) there is a substantial laminate reserve strength from first-ply failure to laminate fracture when the laminate is loaded in combined shear and tension and (2) the laminate combined-stress fracture envelope is not a smooth function. The laminate failure envelopes for combined x with y loading for the E-glass/MY750 [ ± 55]s laminate are shown in Fig. 15. Two failure envelopes are plotted in Fig. 15 as was the case in the previous two figures. The inner envelope is for initial damage and the outer for laminate fracture. Interestingly, the two envelopes are practically identical when x and y are both compressive. The observations and conclusions made in connections with Figs 14 and 13 apply to Fig. 15 as well. It may be noted that only eight points are used to construct each envelope and the straight lines may not be representative in all cases, e.g. in Fig. 15 for stress

Fig. 15. Biaxial failure envelope for ± 55° laminate made of E-glass/MY750 (results predicted by CODSTRAN code).

115 ratio of y /x = 2 : 1. Figure 17 shows that the final failure stress is higher than that shown in Fig. 15 for stress ratio of y /x = 2 : 1. 6. Stress versus strain In this section we describe stress versus strain as determined by progressive fracture via CODSTRAN. The stress versus strain curves for all cases except one were obtained without including effects of residual stresses. For that one case, stress/strain curves are also presented by including the effect of residual stress. It should be noted that the authors have not made use of the nonlinear ply stress/strain data provided. The stress/strain curves (y versus x and y versus y) for the E-glass/MY750 [ ± 55]s laminate loaded under uniaxial tension is shown in Fig. 16(a) and (b). The curve y versus x exhibits a linear relationship, while the curve of y against y exhibits bilinear behavior as follows. The first portion is linear up to a strain of about 0·5%, then the strain increases very rapidly up to 12% or 24 times more with a relatively small increase (only 20%) in stress. The failure strain of about 12% may be misleading because the increase in loading was negligible after the 0·5% strain was exceeded. In other words, laminate fracture was imminent past the 0·5% strain. One important implication is that laminate configurations exist with large strain to fracture. Keep in mind, however, that this large strain is only useful for monotonic loading to fracture, as minor fluctuations in the loading past 0·5% strain would cause immediate fracture of the laminate. The important conclusion is that progressive fracture is a suitable method to generate laminate stress/strain curves to fracture. Two stress/strain curves are shown in Fig. 17, where y is plotted versus y and x for the E-glass/MY750 [ ± 55]s laminate loaded at a stress ratio SR = y /x = 2 : 1. Both curves are practically linear to fracture with the x strain exhibiting a minor deviation from linearity near the high strain region. It is interesting to note that the biaxial stress reduced the fracture strain by a factor of four compared with uniaxial loading shown in Fig. 16, while the corresponding fracture stress increased by a factor of six. The important conclusion is that the stress/strain

Fig. 16. Stress/strain curves for ± 55° GRP laminate made of E-glass/MY750 under uniaxial tension (results predicted by CODSTRAN code).

116

Fig. 17. Stress/strain curves for a ± 55° laminate made of E-glass/MY750 under y/x = 2/1 (results predicted by CODSTRAN code).

behavior of angle-ply laminates is dramatically different under combined loading than it is for uniaxial loading. The stress/strain curves (y versus x and y versus y) for the AS4/3501-6 [0/ ± 45/90]s laminate are plotted in Fig. 18 for uniaxial loading. The y /x curve exhibits bilinear behavior. The y /y curve shows a trilinear behavior, the last portion beyond 1% strain is impractical because it occurs with a very small amount of increase in stress and will happen very fast. The important conclusion is that the AS4/3501-6 [0/ ± 45/90]s laminate exhibits bilinear stress/

Fig. 18. Stress/strain curves for [0°/ ± 45°/90°] laminate made of CFRP material (y/x = 1/0) (results predicted by CODSTRAN code).

117

Fig. 19. Stress/strain curves for [0°/ ± 45°/90°] laminate made of CFRP material (y/x = 2/1) (results predicted by CODSTRAN code).

strain behavior to fracture. The first linear portion ends at about 0·25% strain corresponding to transply fracture in both the 90° and the ± 45° plies, while the second portion ends due to fiber fractures at about 1·0% strain which corresponds to about 70% of the ultimate strain of the specimen (Table 2). It may be noted that if a tensile test were conducted at a constant extension rate, the stress would be reduced where there are changes in the slope of the stress/ strain diagram in Fig. 18. The stress/strain curve for the AS4/3501-6 [0/ ± 45/90]s laminate subjected to combined stress y /x = 2 : 1 loading is shown in Fig. 19. This stress/strain curve is linear to fracture. The

Fig. 20. Stress/strain curves for ± 45° laminate made of E-glass/MY570 material (y/x = 1/1) (results predicted by CODSTRAN code).

118

Fig. 21. Stress/strain curves for ± 45° laminate made of E-glass/MY750 material (y/x = 1/–1) (results predicted by CODSTRAN code).

fracture strain is about 15%, which corresponds to the fracture strain of the AS4 fibers, and the fracture stress (Table 2), is about twice that of the uniaxial case (Fig. 19). The important conclusion is that some biaxial stress states enhance the fracture stress of quasi-isotropic laminates substantially. Unfortunately, the magnitude of the enhancement is load-dependent, which can be taken advantage of only for monotonic loadings to fracture.

Fig. 22. Stress/strain curves for 0°/90° E-glass/MY750 laminate under uniaxial tension (y = 0) (results predicted by CODSTRAN code).

119 The stress/strain behavior of the E-glass/MY750 [ ± 45]s laminate subjected to combined y /x = 1 : 1 loading is shown in Fig. 20. The useful fracture strain for this angle-ply laminate is about 2·5%, which is the end of the first linear portion. Beyond this point the strain increases rapidly to about 6% with a relatively negligible increase in stress. As was mentioned previously, this rapid increase in strain will only be possible under monotonic loading. Any load fluctuation would cause imminent fracture. The important conclusion is that biaxial tensile loading enhances the fracture stress of [ ± 45] angle-ply laminates, as was the case for the [ ± 55]s angle-ply laminate (Fig. 17). The stress/strain curves for the E-glass/MY750 [ ± 45]s angle-ply laminate subjected to biaxial y /x = 1 : –1 loading is shown in Fig. 21. Both stress/strain curves are linear to fractures at about 0 4% strain. This strain is considerably smaller than the biaxial tension loading case (Fig. 20). The important conclusion is that, for biaxial loads with mixed signs (tension-compression), the fracture stresses of angle-ply laminates are substantially lower than the fracture stress of biaxial tension-tension loaded angle-ply laminates. The stress/strain curves (x versus x and x versus y) for the E-glass/MY750 [0/90]s crossply laminate subjected to uniaxial loading y /x = 0:1 are shown in Fig. 22 and they exhibit linear behavior. The x /y curve shows 2% fracture strain, which is about 80% of the fiber tensile fracture strain (Table 2), and about one-half of the fracture stress of the unidirectional laminate (Table 3). Damage initiation by transverse tensile failures that occur relatively early in the 90° plies is not observable from the linear stress/strain relationship. The stress/strain diagram ends when the 0° plies experience fiber fractures. The important conclusions are: (1) cross-ply laminates with equal 0° and 90 plies loaded in uniaxial tension fracture at about one-half the developed tensile fracture stress of the uniaxial laminate and (2) cross-plying does not affect the linear stress/strain behavior of the unidirectional laminate. 7. Summary A theoretical investigation was conducted to evaluate the first-ply failure and laminate fracture of selected laminates made from glass/epoxy and graphite/epoxy polymer-matrix composites and subjected to uniaxial and combined loadings. The evaluation employed laminate theory for first-ply failure and progressive fracture for laminate fracture. Both are based on micromechanics and require constituent material (fiber and matrix) properties as inputs as well as fiber volume ratio, void volume ratio (if any) and cure temperature. The important results from this investigation are as follows. 1. All problems set in the laminated composites failure exercise were solved. 2. Unidirectional composite (ply) first-ply failure and laminate fracture were identical. Laminate theory was sufficient to evaluate behavior for uniaxial and combined loadings. 3. Multi-ply laminate (angle-ply or multidirectional) laminate theory is sufficient for first-ply failure. However, progressive fracture is required for laminate fracture. 4. Residual stresses dramatically reduce first-ply failure in angle-ply and multidirectional laminates in predominantly tensile stress states. However, their effects are more benign on the laminate fracture stress. 5. Combined tensile loadings can dramatically enhance the laminate fracture stress of angleply laminates as compared with uniaxial loadings. However, these loadings had a

120

6. 7.

8. 9.

negligible effect on the fracture stress of cross-ply laminates loaded in directions parallel to the fibers. Combined tension with compression loadings had negligible enhancement effect on the laminate fracture stress compared with uniaxial loading. The angle-ply laminates loaded in uniaxial tension as well as those loaded in biaxial tension that were investigated exhibited substantial strain to fracture beyond the initial linear portion. The ultimate strain occurred at relatively very small load increases. Cross-ply laminates fracture at a ply stress which is about the same as that for the unidirectional laminate in the load direction. The pristine strength of the fibers was not achieved in any of the laminates. Fiber fractures occurred at about 70–80% of their respective pristine fracture strains.

References 1 Hinton, M. J. and Soden, P. D., Predicting failure in composite laminates: the background to the exercise. Compos. Sci. Technol., 1998, 58(7), 1001. 2 Murthy, P. L. N. and Chamis, C. C., ICAN: integrated composite analyzer user’s and programmer’s manual. NASA TP-2515, National Aeronautics and Space Administration, Washington, DC, 1985. 3 Chamis, C. C., Design oriented analysis and synthesis of multilayered-filamentary structural panels. PhD thesis, Case Western Reserve University, Cleveland, OH, 1967. 4 Chamis, C. C., Computer code for the analysis of multilayered fiber composites – User’s manual. NASA TN D7013, National Aeronautics and Space Administration, Washington, DC, 1971. 5 Chamis, C. C. and Sinclair, J. H., Prediction of properties of intraply hybrid composites. NASA TM79087 National Aeronautics and Space Administration Washington, DC, 1979. 6 Chamis, C. C. and Sinclair, J. H., Micromechanics of intraply hybrid composite: Elastic and thermal properties. NASA TM 79253, National Aeronautics and Space Administration, Washington, DC, 1979. 7 Chamis, C. C., Lark, R. F. and Sinclair, J. H., Mechanical property characterization of intraply hybrid composites. NASA TM 79306, National Aeronautics and Space Administration, Washington, DC, 1979. 8 Chamis, C. C., Lark, R. F. and Sinclair, J. H., An integrated theory for predicting the hydrothermomechanical response of advanced composite structural components. NASA TM 73812, National Aeronautics and Space Administration, Washington, DC, 1977. 9 Chamis, C. C. and Sinclair, J. H., INHYD; computer code for intraply hybrid composite design. NASA TP2239, National Aeronautics and Space Administration, Washington, DC, 1983. 10 Pagano, N. J. and Pipes, R. B., Some observations on the interlaminar strength of composite laminates. Int. J. Mech. Sci., 1973, 15, 679–688. 11 Lekhnitskii, S. G. and Brandstatter, J. J. (ed.), Theory of Elasticity of an Anisotropic Elastic Body. Holden-Day, San Francisco, CA, 1963. 12 Nakazawa, S., Dias, J. B., and Spiegel, M. S., MHOST Users’ Manual. Prepared for NASA Lewis Research Center by MARC Analysis Research Corporation, April 1987. 13 Soden, P. D., Hinton, M. J. and Kaddour, A. S., Lamina properties, lay-up configurations and loading conditions for a range of fibre reinforced composite laminates. Compos. Sci. Technol., 1998, 58(7), 1011. 14 Chamis, C. C. and Sinclair, J. H., Durability/life of fiber composites in hygrothermomechanical environments. In ASTM STP 787, American Society for Testing and Materials, Philadelphia, PA, 1983, pp. 498–512. 15 Caruso, J. J. and Chamis, C. C., Assesment of simplified three dimensional finite element analysis. J. Compos. Technol. Res., 1986, 8, 77–83. 16 Chamis, C. C., Failure criteria for filamentary composites. In ASTM STP 460, American Society for Testing and Materials, Philadelphia, PA, 1970, pp. 336–351.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

121

CHAPTER 3.2

Failure criteria for use in the design environment* G. C. Eckold AEA Technology, Harwell, Oxfordshire OX11 0RA, UK Abstract This paper is concerned with the use of composite failure criteria in a design situation. The example chosen is from the process industry which employs composite materials in demanding applications and has requirements for simple, pragmatic and auditable procedures. This is distinct from other environments such as the aerospace industry or in R&D where there are resources available to undertake more sophisticated analyses. In the context of this failure prediction exercise the methods currently adopted by the process sector have been applied, albeit with some modification, since materials of construction are different. Attempts have also been made to predict ultimate strengths, although this is not the prime purpose of a design code, and it is envisaged that there will be some discrepancies owing to the fact that such issues as material non-linearity and differences in tensile and compressive properties are ignored. The more interesting relationship to note will be that between the derived design allowables and the initiation of failure since the basis of chemical process plant design is that normal service loads should be kept within these limits. Keywords: design environment, process industry, design code

Notation Eo Fi, Fij Fx, Fy, Fxx, Fxy, Fyy, Fss R  i, j x, y, s

Unidirectional laminate modules Strength parameters for quadratic interaction criterion Strength terms for quadratic interaction criterion (ASME RTP-1) Design factor (ASME RTP-1) Design strain Winding angle Stress components Axial, hoop and shear stress components

* This article represents the author’s contribution to a worldwide exercise to confirm the state-of-the-art for predicting failure in composites, organised by Hinton and Soden.28

122 1. Introduction The ability to predict failure is a key aspect in the successful design of an engineering structure. It is not that a component will be designed to the limits of ultimate performance because, except in all but rare circumstances, this would clearly be foolhardy, but knowledge of the limit is necessary for a judgement to be made with regard to factors of safety. The first step in establishing a basis for design is to consider what is meant by failure. For a metallic structure it will almost certainly be related to the onset and development of yield or rupture. With composites, on the other hand, the situation can be more complex and is more closely associated with the particular needs of the component of concern. For example, in a tension element it could be breakage through the failure of fibre reinforcement, for a pipe under internal pressure it could be ‘weeping’ of fluid through the development and propagation of microcracks; for a vehicle suspension member (leaf spring) it could be gross delamination leading to the collapse of bending stiffness; and for a stressed aircraft skin it could be instability initiated through the delamination of plies at a free edge. As a result, it is unlikely that a single approach to failure prediction will meet the needs of all potential applications and materials of construction, and this should be a consideration at the outset of any design process. The attributes of a failure criterion in the design environment are somewhat different from those required for R&D purposes. In the latter it is absolute predictive capability which is a goal, whereas in the former, needs are more pragmatic, specifically: • A basis on which the designer can ascribe some form of physical (or quasi-physical) meaning. Abstract mathematical formulations are not favoured as they can be difficult to rationalise in engineering terms. • Implementation by the use of easily measured material constants. This is particularly important given that in a true component design situation, validation testing, often undertaken by third parties will have to be carried out. • Capability of providing a predictive method of known accuracy. Although absolute accuracy is desirable it is not necessarily a prerequisite. A known level of error can be accommodated in design, albeit with the introduction of an added level of conservatism. • Known range of applicability which extends to the materials of concern and duties under consideration. Application across the boundaries of industrial sectors is not necessarily of interest. • Ease of use. Sophisticated analyses can be employed, but they must be presented as a series of charts or tables. There may be exceptional circumstances for ‘specials’ and here computer programs, FEA, etc may be needed. However, for what are considered to be standard components, procedures which can be easily used and audited by all parties are necessary. Given a suitable criterion, its function in the design situation is to allow an evaluation to be made as to the risk of failure. Based on this, an assessment is then required on what value is appropriate for factors of safety and this will be based on both technical and commercial considerations. The net result will be a component or structure which is judged to be fit for the purpose. Failure criteria for composites are many and varied.1–6 Indeed, the profusion of such theories has been a feature of composites literature. In their simplest form they are similar, in

123 principle, to those used for isotropic materials. Examples here include variants of maximum stress/strain and distortional energy theories. The prime differences being that stress components parallel and transverse to the fibre are considered as opposed to those in the principal directions. In some cases, criteria consist of a number of expressions, each of which is related to a separate stress/strain term. This allows some consideration of the mode of failure, such as fibre fracture or transverse cracking to be undertaken. For others, where this is considered too elementary a view, stress terms are combined in an attempt to take into account their interaction. Typical amongst the latter is the quadratic interaction criterion’ which has the form: Fij ij + Fii = 1

(1)

where Fij and Fi are strength parameters and i and j are components of stress. For implementation, say for plane stress in two dimensions, six strength parameters are required. Of these, five can be shown to be conventional tensile, compressive or shear strength terms which can be measured in a conventional test programme, but the last is more difficult to obtain, since a biaxial test is necessary, which is not easy to perform. A further apparent difficulty arises as a consequence of the way in which individual terms of eqn (1) interact. For example, it has been reported8 that under certain circumstances eqn (1) would predict that a reduction in one strength value would lead to an increase in overall composite strength in a desirable mode of loading. This clearly cannot be correct. It should not be construed that this observation is intended to single out this approach for particular criticism but to demonstrate that, basic difficulties remain. It is cited because it is one of the most common criteria described in textbooks on the subject and is incorporated within most finite element codes which cater for anisotropic materials. The validation of failure theories with experimental data remains a vexed issue. For uniaxial loads and simple laminates correlation can be good. However, for other situations, some no more complex than biaxial loading on a simple cylinder with angle ply or quasiisotropic laminates, the situation is often unsatisfactory. Good agreement can be achieved, but often only with selected datasets. One reason for this may be that one of the basic assumptions used in composite stress analysis, that of linear behaviour up to failure (non-linearity due to ply discount notwithstanding), is not strictly correct. Comparatively recent work based on damage mechanics and micromechanics has shown that the initiation and propagation of damage, such as transverse cracks, can have a profound effect on structural response.9–12 These effects are akin to plasticity in metals where local redistribution of stress through local stiffness change can result in significant improvements in observed ultimate performance. It may be that continued work in these areas will provide the missing ingredient in failure theory development. The purpose of this paper is not to advocate any one particular approach to composite failure analysis, but to describe how one particular industrial sector, the process-plant industry, has addressed the issue. This industry uses composites in an environment which is both demanding in terms of operating conditions and competitive given that there are alternative material systems that can be used for most duties. It has been necessary to come to terms with the uncertainties described in the preceding paragraphs and to produce methods which are able to be presented in a format suitable for incorporation within formal design codes and which have and are being used with apparent success. Comparison with

124 experimental data as part of the failure-prediction initiative is a valuable exercise in that it will assist in evaluating whether or not the degree of conservatism embodied within the methods is appropriate.

2. Design methods Composite process plant is manufactured from a number of material types, but is almost invariably based on glass reinforcement in random mat, woven cloth or unidirectional form. There are numerous design standards13–19 for both pipe and vessels and the two documents which stand out as being the most comprehensive are BS4994.1987 and the more recent ASME RTP-1. Indeed, these are the only documents which provide explicit guidance on the use of anisotropic materials which in the case of the process plant usually take the form of filament wound angle ply or unidirectional laminates. In design, the overall objective is to produce a laminate which at operational loads will not be subject to microcracking at any time during its service life. This is considered important so as to prevent ingress of process fluid into the laminate whereupon it may precipitate rapid deterioration. It should be noted that in many cases the chemical environment can be very aggressive. Concentrated mineral acids, alkalis and oxidising agents are not unusual. Furthermore, loads tend to be applied continuously over extended timescales and creep rupture is a mode of failure which must be prevented. As a result, design margins, ie the ratio of short-term static strength to the design allowable, tend to be in the region of 8–10. This is based on early work with chopped-strandmat laminates which indicated that an upper design strain of 0·2% is appropriate20 and has been substantiated by subsequent creep rupture data from laminates exposed to various chemical media.21,22

2.1. BS4994.1987 The design basis in BS499413 is one of limited strain and a detailed procedure is given for the calculation of an allowable value based on the consideration of the method of manufacture, fatigue, operating temperature, environment and curing procedure. The minimun design margin (defined above) is 8, although consideration is being given to reducing this to 6. From the derived strain value an allowable load is determined by the use of the tabulated modulus values, although a supplier may use higher measured values by agreement (note that BS4994 employs the concept of unit load and modulus expressed as kg/m2 of glass, but for the purposes of this paper the discussion is presented in conventional stress terms). For filamentwound laminates a further factor is applied to take into account the effects of anisotropy.23 Figure 1 shows the variation in axial modulus as a function of winding angle as given in the standard. The data shown are for E-glass fibres and a vinylester resin.24 Most of the process plant is fabricated with styrenated resin systems, e.g. polyester, vinylester, etc, and the graph was originally calculated by using properties appropriate to these materials and simple laminate theory. Modulus values would be low if an epoxy-based system is to be considered. There is scope for adjusting the data to take the variation in fibre fraction into account since, in the document itself, the information is given per unit weight of glass. In Fig. 1 the data

125

Fig. 1. Variation of modulus with winding angle.

corresponds to a glass content of 75% by weight ( ~ 60% by volume). Also shown are measured modulus values from nozzle cutouts (material removed when a circular hole is cut for the attachment of a nozzle) taken from large GRP tanks and the agreement is good given that-a design code should be based on lower bound or deviated data. The factors which are applied to these modulus values to determine allowable stresses are shown in Table 1. At 0·2% strain the allowables as calculated from this method are shown in Fig. 2. The discontinuities arise as a result of the step changes in the applied factor for ± 15° and ± 75° laminates. There are two key assumptions in this process: • strains in all directions (axial, transverse, shear) are equally important and should be limited by the same upper limit; • transverse strains for high winding angles ( > ± 75°) should be further limited to a lower overriding maximum of 0·1%. This is because of the low absolute value of transverse

Table 1 BS4994 factors for calculating longitudinal and circumferential design stresses. Fibre angle to longitudinal axis 0 <  15° 15° <  75° 75° <  90°

Longitudinal factor

Circumferential factor

1·0 0·5 0

0 0·5 1·0

126

Fig. 2. Allowable stresses for laminates.

strength and the fact that at high angles the transverse direction is not supported to any great extent by the reinforcement. For transverse strains there is an inconsistency in the document because the data in Table 1 indicate that the design value should be zero as opposed to the 0·1% limit. The BS4994 procedure is still workable, however, as the intent is that the transverse strength should be ignored in terms of structural contribution when deriving the basic laminate construction, but then to go back and check that the overall strain is less than the 0·1% level. The calculation on which the Table 1 parameters were based again employed simple laminate analysis to evaluate strains in each of the primary material directions.23 This results in a continuous design curve (as an alternative to Table 1) for allowables an example of which, for 0·2% strain, is also shown in Fig. 2. One of the most significant question marks over the method is whether or not shear and normal strains should be limited by the same value. A criticism of the current BS4994 approach is that the procedure does not allow the full benefits of the reinforcement for combined loading. This is not a issue for storage tanks as the loading is dominated by the static head pressure. Axial loading is small and variable. It would not be practical to advocate that all possible load permutations are considered, as there would be little added benefit. The exception is for pressure vessels where there is a 2 : 1, hoop : axial pressure stress. It is straightforward to generate an additional curve for this case and this is also shown in Fig. 3. Note that it is the allowable stress in the axial direction which is shown. Again the 0·1% transverse strain limit has been imposed to ensure consistency with the uniaxial case. With this approach a simplified biaxial load envelope such as that shown in Fig. 4 can be developed. The values on the ordinate and abscissa are given by the uniaxial curve in Fig. 2 and the combined 2 : 1 point is given by the biaxial plot in Fig. 3. Also shown

127

Fig. 3. Allowable stresses for laminates.

in Fig. 4 are the results of a netting analysis where the stress values at the point where the loads combine to act in the fibre direction are given by: x = Eo cos2 

(2)

y = Eo sin2  where x and y are axial and hoop stress components for the 2 : 1 condition,  is the design strain (0·2%), Eo is the unidirectional laminate modulus and  is the winding angle.

Fig. 4. Stress envelope for ± 55° winding angle.

128 As can be seen the netting analysis provides a more conservative result for the winding angle used for the calculation ( ± 55° to the axial direction), although the curves are of a similar shape. For other laminate configurations this will not be the case as the values given by eqn (2) will not be for the 2 : 1, hoop : axial stress ratio. This is not a major issue for process plant as the main design conditions are static head pressure (uniaxial load) and internal pressure which is addressed by the above discussion. A point of interest is that the approach of evaluating allowables on the basis of limiting maximum strain does not apparently result in the ± 55° winding angle being optimum for a 2 : 1 load case as is customarily assumed (see Fig. 3). A somewhat higher angle is indicated. The corollary to this is that a 2 : 1 loading situation is not the best for a ± 55° angle ply. There is some supporting evidence for this from biaxial test results where the performance of such a laminate is noted to be better at a stress ratio of nearly 3 : 1.25 The physical significance of this has not been explained, but clearly the assumptions within the netting analysis are in difficulty. It is worth noting that the stress ratio at which the in-plane shear stress, in the material’s coordinate axis becomes zero, is 2·8 (the winding angle at which the shear stress becomes 0 for a 2 : 1 loading condition is ~ ± 60°).

2.2. ASME RTP-1 The ASME document18 describes a two-tier approach and the supplier is free to chose which to use for the design. In the simpler of the methods the determination of allowables is carried out as follows: • in the hoop direction the allowable stress is calculated by using a maximum strain limitation of 0·1% regardless of the winding angle; • in the axial direction the allowable stress is calculated by dividing the ultimate strength by a factor of 10 again regardless of winding angle. For the second method an altogether more complex approach is taken. Within the code are appendices which give a description of the mechanics of laminate theory and the user is invited to carry out a full analysis. It is arguable as whether or not this is appropriate for a design code as if not correctly interpreted its use will cause problems. For example, the calculation of the coupling [B] matrix is described, but no guidance is given as to what to do with it. Coupling may give rise to twist in an unconstrained flat plate, but such deformations cannot exist in a closed cylindrical surface! Once the stress analysis is complete the assessment continues by using the quadratic interaction criterion described in the proceeding section. It is presented in a modified form as follows: R 2(Fxx 2x + 2Fxy x y + Fyy 2y + Fss2s) + R (Fsx + Fy y)  1 = 0

(3)

Measured values can be used for the strength terms (Fxx, Fxy, etc.) and if these are not available, strains are given, which vary with the direction and mode of loading, to be used in conjunction with the relevant stiffness (Table 2). The coefficient, R, is the design factor for the laminate and is specified as being between 8 and 12 for the inner surface adjacent to the process fluid, depending on the criticality of service etc. Away from the surface a lower figure may be used, but for simple membrane stresses there is no benefit to be gained. Allowable stresses as calculated by the ASME method are compared with BS4994 values in Fig. 2.

129 Table 2 ASME RTP-1 strain values. Stress component Longitudinal tension Longitudinal compression Transverse tension Transverse compression Shear

Strain value (%) 2·00 1·20 0·15 0·80 1·50

2.3. Performance-based approach It is worth noting that there is another approach to process plant design, but one which is really only suitable for products manufactured on a commodity basis such as standard pipework systems. Here the objective is to measure long-term performance of typical components (not laminates) under conditions similar to those experienced in service. Creeprupture testing of pipework components over durations > 10 000 h is one of the established methods.26 The key difference with the approach is that knowledge of how laminates behave is not necessary as actual components are tested. The supplier then has a responsibility to establish quality assurance and quality control procedures which ensure that future manufactured items are, for all practical purposes, identical to those which have been qualified. Another problem is that testing usually takes place under a single load condition, most commonly internal pressure with closed ends, and the use of the arising data for other situations is not straightforward. There has been some progress in addressing this aspect of design through making assumptions regarding the shape of the failure envelope.27 The strains at the design stresses tend to be higher ( ~ 0·3–0·4%) than those calculated by the design-byrule approach described in the vessel standards. This is appropriate given that the latter employ short term static strength values as a basis and some conservatism is necessary. 3. Failure prediction The procedures described in the preceding sections are intended for the evaluation of design allowables for GRP process-plant materials and to be commensurate with the needs of a design environment. This poses certain difficulties when attempting to offer predictions of behaviour for the materials included within this failure exercise. (Details of the lamina properties and lay-up configurations and loading of the laminates analyzed are provided in Ref. 29.) These problems have been accommodated as follows: • Material properties. Tabulated material properties in BS4994 are intended for resin systems of the polyester type and their use for epoxy systems would not be appropriate. However, the standard does allow alternative values to be used provided they are based on test data. In this exercise the as supplied property values for unidirectional laminae have been used together with simple laminate analysis to give off-axis information. Table 3 gives the values as calculated by this process. A difficulty also arises in the specification of an allowable

130 Table 3 Calculated modulus values. Laminate

E-Glass/LY556/HT907/DY063 ± 30° angle ply Hoop Axial E-glass/MY750/HY917/DY063 ± 55° angle ply Hoop Axial Cross-ply ± 45° angle ply

•

•

•

•

Calculated modulus (GPa)

1·58 30·5 23·5 1·54 31·1 17·7

maximum strain. For polyesters this is currently 0·2%, but this may be considered low for epoxy systems. The results from performance-based testing, most of which relates to epoxy components, indicates that values in the range 0·3–0·4% are satisfactory. A strain limit of 0·4% has been used in all of the calculations described in the following sections, except for transverse tension where the 0·1% limit is maintained. The selection of 0·40% is, in a sense arbitrary, but it provides a basis for this exercise. Prediction of ultimate strength. The design code is not concerned with the prediction of ultimate performance, and, furthermore, there is no mechanism for doing so. For the purposes of this paper the method which has been adopted is that which would be taken by an engineer given the as supplied properties and the general approach to laminate design described in BS4994. Calculations have been performed at selected points and straight lines have been drawn between them, as opposed to trying to use a formal composite failure criterion, which will be addressed by other contributors to this exercise. Thermal stresses. Most GRP process vessels are fabricated by using room-temperature cured materials. Sheer size effectively precludes systems which need elevated temperatures during processing and as a result the effects of thermal stresses are not relevant. However, in the following calculations a simple manual correction has been applied by using given thermal expansivities and the stated stress-free temperatures. Compressive properties. BS4994 makes no attempt to differentiate between tensile and compressive strengths. The reality is that if there is a significant compressive loading component, e.g. vacuum, it will be structural instability which governs the design. This is especially true for glass-reinforced composites where modulus values are relatively low. The introduction of additional properties would be an unnecessary complication. In this exercise no attempt has been made to allow for differences in strengths in the two modes of loading. Non-linear properties. Non-linearity, either through a material characteristic or ply discount, does not feature in the calculation method, since allowables are generally based on low strains prior to any form of initial failure. No attempt has been made to accommodate this in the following predictions. Calculated stress/strain curves have been limited (arbitrarily) to 0·5% strain.

131 • Carbon-fibre materials. CFRPs do not yet feature in the catalogue of material systems employed by the process industry. A basic premise behind the arguments expressed in this paper is that design methods should be appropriate for the industry and materials for which they are intended. More advanced materials (and their associated cost) require, in principle, more sophisticated design methods if the advantages of their use are to be realised. On this basis no attempt has been made to offer predictions for CFRP laminates.

3.1. Unidirectional laminate Predictions for biaxial stress envelopes for unidirectional laminae are shown in Figs 5 and 6. In Fig. 5 the envelope depicting ultimate strength is based on the as supplied strength values which are then treated as per BS4994. The design envelope was derived by using a strain limitation of 0·1% in the transverse direction and 0·4% (as discussed above) for the longitudinal direction. The ratios between ultimate strength and design are broadly in line with the intent of the code. The quadrant where there is compressive stress is, again, treated by following prevailing design code practice. The prediction will be conservative where loads are primarily uniaxial (compression) and in the transverse direction owing to the higher strengths in this direction. The converse will be true for compressive loads which will be predominantly in the fibre direction. However, as has already been discussed, this is unlikely to be a design issue as structural instability will almost certainly become the governing concern. The envelopes in Fig. 6 have been calculated using a similar process, but here it is expected that shear non-linearity will have a marked effect on the results. As the shear modulus falls a greater proportion of the load will be carried in other directions and this may have the effect of reducing observed strength. On this basis it would be expected that the simple linear calculation would represent an upper bound. The combinations of load as represented by Fig. 6 do not feature in process-plant design where stresses are primarily membrane or, in the case of local features such as nozzles, discontinuities, lifting brackets etc,

Fig. 5. Stress envelope for 0° laminate (E-glass/MY750/HY917/DY063).

132

Fig. 6. Stress envelope for 0° laminate (E-glass/LY556/HY907/DY063). Note that structural instability will govern design in compressive quadrants.

bending. Although pipework systems do experience some torsional loading it is generally the case that these are small. Large in-plane shear stresses are not seen. In BS7159 the combination of normal and shear stresses is considered using the conventional Tresca criterion. However, it must be said that this was probably adopted directly from the equivalent steel codes and its applicability to composites, especially those which are anisotropic, was not thoroughly considered. To adopt this approach in this exercise to shear loadings would perhaps be pushing the codes beyond their intent.

3.2. (90/ ± 30°/90) laminate The use of combined winding angles is not an uncommon occurrence in storage/pressure vessels where an angle-ply laminate (usually > ± 45°) is used to support basic loadings in the axial direction and additional hoop winding is applied to accommodate pressure loads. In some design examples the two cases are kept entirely separate, i.e. angle-ply laminates are used to carry axial load only and 90° windings are employed to carry circumferential load only. Whilst this may be construed to be conservative, the fact that the role of each layer is unambiguous means that the design is ‘conceptually’ simple and this has the benefit that approval by third parties, who will be competent in terms of process plant but not necessarily well versed in composites, will be straightforward. The ramification of time saved in this part of the procurement process can outweigh the penalty of added material cost. In terms of design, the approach taken for a hybrid laminate construction is to consider each type of layer in turn and to establish which has the lowest allowable strain. This value is then used as a basis for further evaluation. The first stage in the analysis is to determine the behaviour of each laminate layer and this is shown in Figs 7 and 8. These were derived as follows:

133

Fig. 7. Stress envelope for 0° laminate (E-glass/LY556/HT907/DY063).

Unidirectional envelope: • from treatment for the unidirectional laminate described in the preceding section. ± 30° envelope: • axial and hoop strengths were determined by the ultimate transverse (0·197%) or shear (0·38%) strains as appropriate; • strength at the 2 : 1 ratio was determined by netting analysis limited by the ultimate longitudinal strain (2·13%). From these results, associated modulus values (Table 3) and the layer thicknesses (90°–0·35 mm, ± 30°–1·65 mm) the various contributions of each layer type can be assessed. The results of this calculation are shown in Fig. 9. The envelope for ultimate strength takes

Fig. 8. Stress envelope for ± 30° laminate (E-glass/LY556/HT907/DY063).

134 into account redistribution of stress after the ‘weaker’ layer has failed. For the design envelope all values were determined by using a design-strain limitation of 0·4% (0·1% for transverse tension) generally in accordance with the methods described in Section 2.1. For the axial directions the design values look particularly pessimistic. This is due to the 0·1% transverse strain value. In principle this could be increased if it were felt that cracks in this layer could be tolerated. The values for the compressive quadrants were evaluated by assuming that the simple use of the equivalent tensile data was appropriate. The comments on structural instability made in material Section 3.1 for unidirectional material are equally applicable for the more complex laminates. It is acknowledged that the ultimate envelope may represent an upper bound as a consequence of material non-linearity which will result in local redistribution of load giving rise to higher strains in other directions. Discrepancies will also occur because the compression strengths have not been used. For the design envelope the important considerations are not limited to its relationship to static strength, but more importantly how it compares with the initiation of damage. It is essential that there is a sufficient margin here so that effects such as long term loading, the chemical environment and variability of material properties can be accommodated.

3.3. Angle ply ( ± 55°) laminate Figure 10 shows the biaxial stress envelope for a ± 55° laminate. Calculations were carried out as follows.

Fig. 9. Stress envelope for ± 30°/90° laminate (E-glass/LY556/HT907/DY063).

135

Fig. 10. Stress envelope for ± 55° laminate.

Ultimate strength envelope: • axial and hoop strengths were determined by the ultimate transverse (0·246%) or shear (4%) strains as appropriate; • strength at the 2 : 1 ratio was determined by netting analysis limited by the ultimate longitudinal strain (2·13%). Design strength envelope: • Comments with regard to the ultimate envelope being an upper bound because of material nonlinearity and the effects of the higher compressive strengths are as per previous predictions. Stress/strain curves, to 0·5% strain, are shown in Figs 11 and 12. These are based on simple laminate calculations using the provided material constants. It is anticipated that at high strains significant non-linearity will be noticeable. This is not necessarily relevant in design if it is concerned solely with allowables limited by lower strains.

3.4. Cross-ply laminate Figure 13 shows a stress/strain curve for the cross-ply laminate subjected to uniaxial loading. Again it is linear and no attempt has been made to identify the ‘knee’ which occurs as a consequence of transverse cracking.

136

Fig. 11. Stress envelope for ± 55° laminate under a stress ratio of 1 : 0.

3.5. Angle ply ( ± 45°) laminate Figures 14 and 15 show the stress/strain characteristics for the ± 45° laminates. These were calculated as for those described above and similar comments apply.

Fig. 12. Stress/strain curve for ± 55° laminate under a load ratio of 2 : 1.

137

Fig. 13. Stress/strain curve for 0°/90° laminate under a load ratio of 0 : 1.

Fig. 14. Stress/strain curve for ± 45° laminate made of GRP material (y /x = 1/–1).

138

Fig. 15. Stress/strain curve for ± 45° laminate made of GRP material (y /x = 1/1).

4. Conclusions This paper considers the attributes of composite failure criteria necessary for use in the design environment where the need for reliable prediction of performance must be balanced by simplicity and ease of use. The prime purpose of a failure theory is as a tool in the assessment of whether or not a design is ‘fit for purpose’. It may be that calculations for ultimate strength are necessary, but more likely it will be some other facet of behaviour which provides a baseline for design. This will certainly be the case where loads are applied continuously and over a long term. This is not to say that knowledge of maximum load capability is not important, as it can provide a backdrop against which upset conditions can be considered and give confidence to the user with respect to performance in service. The industrial sector which has been chosen as an example, process plant manufacture, has at its disposal a number of design standards for use which, in the main, meet their needs. However, it is acknowledged that their treatment of composite materials behaviour can be criticised as not being technically robust and certainly would not be considered satisfactory in other areas where there is scope to back up more rigorous approaches with extensive testing programmes. Participation in this failure prediction exercise will be of benefit in putting this and other methods into an overall design and engineering context.

References 1 Azzi, V. D. and Tsai, S. W., Anisotropic strength of composites. Exp. Mech., 1965, 5, 283. 2 Norris, C. B., Strength of orthotropic materials subjected to combined stresses. Forest Products Labj FPL 1816, 1950. 3 Puppo, A. H. and Evensen, H. A., Strength of anisotropic materials under combined stresses. AIAA J., 1972, 10, 468.

139 4 Hutter, U., Schelling, H. and Krauss, H., An experimental study to determine the failure envelope of composite materials with tubular specimens under combined loads and comparison between several classical criteria. AGARD-CP-163, 1975. 5 Gol’denblat, I. I. and Kopnov, V. A., Strength of glass reinforced plastics in the complex stress state. Mekhanika Polimerov, 1965, 1, 70. 6 Huang, C. L. and Kimser, P. G., A criterion for strength for orthotropic materials. Fibre Sci. Technol., 1975, 8, 103. 7 Tsai, S. W. and Wu, E. M., A general theory of strength for anisotropic materials. J. Compos. Mater., 1971, 5, 5882. 8 Hart-Smith, L. J., The role of biaxial stresses in discriminating between meaningful and illusory composite failure theories. Compos. Struct., 1993, 25, 3–20. 9 Talreja, R., Stiffness properties of composite laminates with matrix cracking and interior delamination. Engng Fract. Mech., 1986, 25, 751–762. 10 Joshi, G. P. and Frantziskonis, G., Damage evolution in laminated advanced composites. Compos. Struct., 1991, 17, 127–139. 11 McCartney, L. N., The prediction of non uniform cracking in biaxially loaded cross ply laminates. NPL Report DMM(A) 142, 1994. 12 Eckold, G. C., Hancox, N. L. and Lee, R. J., Application of micromechanics in the prediction of damage initiation and growth in structural composites. Deformation and Fracture of Composites, Surrey, March 1995. 13 Specification for design and construction of vessels and tanks in reinforced plastics, BS4994. BSI, 1984. 14 Specification for reinforced plastics pipes, fittings and joints for process plants, BS6464. BSI, 1984. 15 Code of practice for design and construction of glassreinforced plastics (GRP) piping systems for individual plants or sites, BS7159. BSI, 1989. 16 Specification for low pressure fibreglass line pipe, API 15LR. API, 1990. 17 Specification for high pressure fibreglass line pipe, API 15HR. API, 1988. 18 Reinforced thermoset plastic corrosion resistant equipment, ASME RTP-I. ASME, 1992. 19 Pressure vessels in glass fibre reinforced thermosetting plastics. AD-Merkblatt Nl. 20 Smith, T. R. and Owen, M. J., The progressive nature of fatigue damage in glass reinforced plastics. In Proc. 6th Int. Resins and Plastics Conf. of the British Plastics Federation. British Plastics Federation, London, 1968, paper 27. 21 Roberts, R. C., Reinforced Plastics Congress. BPF, 1978, p. 145. 22 Hogg, P. J., Hull, D. and Legg, M. J., Composite Structures, ed. I. M. Marshall. Applied Science Publishers, London, 1981, p. 106. 23 Eckold, G. C., A design method for filament wound GRP vessels and pipework. Composites, 1985, 16, 41–47. 24 AEA Technology, Internal Report, 1995. 25 Tolhoek, P., Composite Materials in the Offshore Industry. Aberdeen, November 1995. 26 Practice for obtaining hydrostatic or pressure design basis for fibreglass pipe and fittings, ASTM D2992. 27 Eckold, G. C., A performance based design methodology for GRP pipework and fittings. Proc. Inst. Mech. Eng., 1995, 209, 41–50. 28 Hinton, M. J. and Soden, P. D., Predicting failure in composite laminates: the background to the exercise. Compos. Sci. Technol., 1998, 58(7), 1001. 29 Soden, P. D., Hinton, M. J. and Kaddour, A. S., Lamina properties, lay-up configurations and loading conditions for a range of fibre reinforced composite laminates. Compos. Sci. Technol., 1998, 58(7), 1011.

140

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

CHAPTER 3.3

Stress-based Grant–Sanders method for predicting failure of composite laminates* E. C. Edge British Aerospace Defence Ltd, Warton, Preston PR4 1AX, UK Abstract The stress-based Grant–Sanders method developed at British Aerospace Defence is described. It has been applied to a number of examples to produce envelopes of initial and final failure and stress/strain curves, all of which are illustrated in this paper. Unlike some other current methods, each strength prediction is accompanied by predictions of the mode and location of failure. The use of discrete failure criteria gives an abrupt appearance, with sharp corners, to many of the predicted failure envelopes, in contrast to the elliptical shapes produced by techniques based on tensor polynominals. Keywords: composite laminates, strength prediction, Grant–Sanders method, discrete criteria, non-linear stress–strain

Notation Note – in this work allowable stress is to be interpreted as strength, i.e. no safety factor is included. E1 E1t E1c E2t E20 E2c G12 G120 2t 0 20

Longitudinal modulus (in fibre direction) Longitudinal tension modulus (in fibre direction) Longitudinal compression modulus (in fibre direction) Transverse tension modulus (at 90° to fibre direction) Initial value of E2t for uncracked material Transverse compression modulus (at 90° to fibre direction) In-plane shear modulus Value of G12 with no matrix cracking Transverse tensile strain (at 90° to fibre direction) Value of 2t at which reduction of E2t commences (in general case) Value of 2t at which reduction of E2t commences for D = 1

* This article represents the authors’ contribution to a worldwide exercise to confirm the state-of-the-art for predicting failure in composites, organised by Hinton and Soden.1

141

2m 12 120 21 1 2 12 F1t F1c F2t F2c F12 te cr H1c, H12 D D* R f

Value of 2t at which E2t vanishes Major Poisson’s ratio Value of 12 for uncracked material Minor Poisson’s ratio Longitudinal stress (in fibre direction) Transverse stress (at 90° to fibre direction) Shear stress Allowable longitudinal tensile stress (in fibre direction) Allowable longitudinal compressive stress (in fibre direction) Allowable transverse tensile stress (at 90° to fibre direction) Allowable transverse compressive stress (at 90° to fibre direction) Allowable shear stress Effective ply thickness Delamination criterion (N mm–1) (see Section 3.2.5) Parameters for shear-compression interaction formula (see Section 3.2.4) Parameter indicating strain at which reduction of E2t commences Parameter defined in Appendix A (12/F12)2 E2t/E20

Values assumed for variables in this work The values quoted in this section do not include the data supplied by the organisers. • • • •

H1c = F1c and h12 = 1·5
F12; see Section 3.2.4. cr = 26 N mm–1 (for all materials); see Section 3.2.5. te is ply thickness as supplied by the organisers; see Section 3.2.5. D = 1 (for all laminates and layers); see Section 2.6.2 and Appendix A.

1. Introduction This paper describes the stress-based Grant–Sanders method developed at British Aerospace for the prediction of failure in composite laminates. The model is an update of that described by ESDU.2 The method is based on a ply-by-ply analysis with discrete failure criteria, the only interactions considered being shear-tension for matrix failure and shear compression for fibre failure. Hence each failure prediction is accompanied by a prediction of the mode and location of failure. Nonlinear stiffnesses are accommodated by inputting curves of secant modulus versus strain. BAe Defence have never used the ply-discount technique: instead the shear and transverse tension stiffnesses have been taken to reduce gradually as strain increases above a threshold value.3 A recently developed procedure for the interaction of the matrix-dominated non-linearities, described in this paper, has been incorporated into the model. The technique has been applied to a number of examples to produce envelopes of initial and final failure and stress/strain curves, all of which are illustrated in this paper. The lamina properties and details of the laminates analysed are described in Ref. 4. Some difficulty has been experienced in some cases in deducing the stress/strain behaviour in transverse tension

142

Fig. 1. Variation of secant transverse modulus with transverse strain for different laminae.

from the data supplied. The curves which were used in the analyses are shown in Fig. 1. The use of discrete failure criteria gives an abrupt appearance, with sharp corners, to many of the predicted failure envelopes, in contrast to the elliptical shapes produced by other methods which are based on tensor polynominals. The corresponding non-linear curve used at BAe for XAS/914C for shear modulus is shown in Fig. 2. Similar curves have been used in this work.

Fig. 2. Assumed variation of secant shear modulus with shear strain for XAS/914C material.

143 2. Description of the method An updated version of the Grant–Sanders method, based on stress criteria, has been used in this work. In the following description, the suffixes 1 and 2 refer to loading parallel and perpendicular to the fibre direction, respectively (the conventional notation). The failure criteria used in the method are described in Section 3. The notation used is given in the list above. Values of material and interactive constants used in this work, other than those supplied by the organisers, are also given above. Details of the interactive degradations assumed between E2t, G12 and 12 are given in Section 2.6.2 with a worked example in Appendix A. 2.1. Ply-by-ply analysis Classical thin-laminate theory is used to determine the loading induced into each individual ply by the current level of applied loading. These ply loadings are then resolved into ply axes. The failure criteria are then applied to each ply in turn, producing an array of reserve factors. At the end of each iteration the location of the minimum reserve factor is determined, giving both the location and mode of failure predicted at this stage of the calculation. Both the location and mode of the predicted failure may, of course, change from one iteration to the next. Note, however, that each failure prediction is accompanied by definitive information on the mode as well as the location of failure. 2.2. Convergence procedures In the earlier version, iteration proceeded by multiplying the initial input value of the applied mechanical load by the minimum reserve factor. This method gives rapid convergence for cases where non-linear effects are relatively small and has been retained for the initial failure calculations. For calculations where non-linear effects are dominant, the initial input value of the applied mechanical loading is incremented by a prescribed amount after each iteration. The initial sign of this increment is determined after the first calculation. When two successive minimum reserve factors are on opposite sides of unity, the increment is halved and its sign is reversed. The calculation proceeds until the minimum reserve factor differs from unity by less than the prescribed amount, or the increment becomes vanishingly small with respect to the precision of the arithmetic, or a prescribed number of iterations have been performed. The Grant–Sanders technique has been developed in an aircraft design environment and consequently is not designed with matrix-dominated situations in mind. The deficiencies of the convergence procedures are recognised and caused some problems in this work, e.g. the stress/strain curves for ± 55° and ± 45° laminates. 2.3. Initial and final failure In-plane shear failure is regarded as a final and not an initial failure, although in an experiment to failure the test piece may be able to take further significant increases in load. This is

144 because of the low stiffnesses and large distortions which can accompany shear failure; these are unacceptable from an aircraft structural viewpoint. For similar reasons the calculations do not proceed beyond the first predicted fibre failure; in this case final failure can be expected at a load level at most only marginally higher than the first fibre failure. Transverse tension failure has been regarded as initial failure in this work, except for the case of the unidirectional (0°) lamina (Figs 3–5). Following failure, the loading carried by the failed plies is gradually diffused into those of other orientations. In 0/90° and quasi-isotropic configurations under unaxial tension in the 0° direction, the 0° plies take most of the load even in the undamaged state, so that the increase in their share due to failure of the 90° plies is small. In these cases the loading at final failure is generally much higher than at initial failure. This situation is reasonably close to that in the majority of aircraft structures. Note, however, that damage-tolerance situations have not been addressed in this work. Transverse compression is also treated as initial failure, but the much higher strength in the compression mode means that this failure occurs much less frequently. 2.4. Hygrothermal loading Hygrothermal loading has been included in the calculations. In the absence of any information, a condition of 20°C and zero moisture content has been assumed. This assumption will probably overestimate the hygrothermal components of the stresses. For the case of hygrothermal loading only, different convergence algorithms are of course required if any induced stresses are in the non-linear range. 2.5. Out-of-plane stresses and the prediction of delamination Evaluation of out-of-plane stresses is not included in the method as it currently stands. While out-of-plane edge stresses may well have been present in the experimental work, of sufficient value to have caused local layer separation, delamination failure will not propagate across a section until a critical level of interlayer loading is reached away from the edge boundary layer. A crude delamination criterion therefore features in the method, and is described in Section 3.2.5. Edge delamination ‘failure’ is therefore not included among the initial failure conditions evaluated. 2.6. Non-linear stress/strain behaviour accommodated The method can accommodate non-linear stress/strain behaviour, which is input into the program in the form of curves of secant modulus versus strain. There was no problem in accepting the non-linear data for E1t E1c, E2c and G12 supplied for use in this work in the form of stress/strain curves. There has, however, been a problem with the transverse tension modulus, E2t, for which only initial values have been supplied. While these values have been taken to be constant below transverse tension failure, it is considered to be totally unrealistic to retain these values at higher strains.

145 2.6.1. Problems with transverse tension stiffness in this work The Grant–Sanders method does not use the ply-discount technique. When the applied loading is increased beyond the level at which initial failure occurs, the calculation proceeds with the values of E2t and G12 reducing gradually rather then being suddenly annihilated. In the case of transverse tension, the initial value of E2t is retained until ‘failure’. Beyond that point, for XAS/914C or T300/914C prepreg material, the value is reduced in the manner shown in Fig. 1. In Fig. 1 the reduction is shown as starting at a strain 2t, of 0·6%, the maximum value for the material. If for any reason (e.g. ply blocking or simultaneous shear loading) a lower starting point is taken, the whole curve is shifted bodily to the left by the appropriate amount. The curve shown in Fig. 1 is based on a study of the difference between stress/strain responses of 0° and 0/90° XAS/914C coupons. This work was performed under UK Ministry of Defence funded contracts and has been reported elsewhere.3,5 No comparable data have been supplied for the materials which have been used in this work. This did not matter for the filament-wound T300/914C material. For the other three materials it was noted that transverse tension failure occurred at lower strains than with XAS/ 914C. It has been assumed that this indicates the greater brittleness of these materials compared with XAS/914C. For AS4/3501-6, the data supplied indicate failure at 0·436%, a reduction by a factor of 1·375 compared with XAS/914C. Therefore the starting point of the reduction of E2t is reduced to 0·436% and the point at which E2t vanishes is reduced in the same ratio, to 1·455% compared with the corresponding XAS/914C value of 2%. The height of the curve has been multiplied by 1·222 to allow for the greater initial E2t specified for AS4/ 3501-6 (11 GPa against 9 GPa). Consequently at intermediate points the slope of the assumed AS4/3501-6 curve is greater than that of the XAS/914C curve by a factor of 1·681. For the two glass/epoxy materials the assumed brittleness is even greater. The assumptions for AS4/3501-6, while arbitrary, are unlikely to lead to significant errors. This is much less likely to be the case for the glass/epoxies. The rapid collapse of E2t assumed may well be unrealistic and has certainly been an important factor in the convergence problems experienced. Figure 1 illustrates the assumed E2tsec versus strain curves for AS4/3501-6 and the two glass/epoxy materials and compares them with each other and with that for XAS/ 914C. The corresponding non-linear curve used at BAe for G12sec for XAS/914C is shown in Fig. 2. Similar curves have been deduced for use in this work from the information supplied.4 2.6.2. Interaction between E2t, G12 and 12 non-linearities An algorithm has been derived for the interaction between the E2t, G12 and 12 non-linearities. This is based on, but not the same as, that proposed by Engblom and Yang.6 The interactive algorithm is described in this section. A worked example is given in Appendix A. 1. The non-linear curve of E2t versus strain for initially uncracked material and the nonlinear G12 versus strain curve for uncracked material are retained. For XAS/914C material these are illustrated in Figs 1 and 2 respectively. 2. Values are assigned to the parameter D for each ply. (It is not required that D be the same for each ply.) If D = 1 cracking, and hence the start of the reduction of E2t with strain, will start when 2 20 provided there is no shear loading. (For XAS/914C, 20 = 0·6% (see Fig. 1)). D = 0 means that cracking is assumed to start at zero transverse strain regardless

146 of any other applied loading. The interpretations of 0 < D < 1 and D > 1 follow logically. In this work D = 1 was taken for all laminates and in all the layers. 3. Calculate the ratio R = (12/F12)2 4. Calculate

0 = 20
兹(D  R) if D > R 0 = 0 if D ≤ R 5. Calculate E2t by using the input curve shifted bodily to the left or right, as appropriate, so that the reduction of E2t with increasing strain commences at a strain of 0. 6. Calculate G120, the uncracked value of G12, as a function of shear strain in the usual way. 7. Factor G120 to give G12 = G120
E2t/E20 8. Calculate D* = 0/(2m  20 + 0) 9. Factor 120, the uncracked value of 12, to give 12 = 120
f
[2  f
(1  D*)]/(1 + D*) where f = E2t/E20 ( = G12/G120). 10. 21 = l2
E2t/E1.

2.7. Shear-compression interaction We are concerned here with the interaction within a ply between direct compression loading in the fibre direction and shear. As the shear loading is increased the failure level of the compressive load will decrease, because of the associated reduction in matrix stiffness, but the mechanism remains a fibre-dominated one at least until the ratio of shear to compressive load becomes quite large, although changes in the mode of compressive failure may be witnessed. The criterion is described in Section 3.2.4.

3. Failure criteria These are divided into initial and final failure criteria. In many cases first failure is in a final failure mode and the distinction disappears. In the case of unidirectional (0°) material all failures are classed as final irrespective of their nature (e.g. the 90° tension test where initial transverse tension failure precipitates catastrophic collapse), so again the distinction disappears. The computer program does not output intermediate failure points. Failure is predicted if the following conditions are obtained.

147 3.1. Initial failure 3.1.1. Transverse tension 2 ≥ F2t 3.1.2. Combined shear and transverse tension

冉冊冉冊 2 F2t

2

+

12 F12

2

≥1

3.1.3. Transverse compression 2 ≤ F2c (both 2 and F2c negative)

3.2. Final failure 3.2.1. Longitudinal tension 1 ≥ F1t 3.2.2. Longitudinal compression 1 ≤ F1c (both 1 and F1c negative) 3.2.3. In-plane shear | 12 | ≥ F12 3.2.4. Combined longitudinal compression and shear 1 | 12 | + ≥1 H1c H12

(1 and H1c both negative)

Note that H1c and H12 may be the same as F1c and F12 respectively, but are not necessarily so. The decision is generally based on what is known about the material performance. 3.2.5. Delamination 12
te cr 4. Significance of initial and final failures in this work In the failure envelopes presented in this work (Figs 4–8 and 11), final failure means that the laminate is either considered incapable of taking further load or a fibre failure has occurred.

148 (The former covers shear failure when the laminate may take more load but is considered to be unacceptable from a structural viewpoint. The structure may also take some more load after fibre failure, although this additional amount will practically always be very small). Initial failure is a non-catastrophic event following which the failed plies gradually shed load. It is induced in transverse tension (perhaps in combination with shear) and in transverse compression. In some cases final failure, as defined above, is predicted before initial failure. When this happens, the initial and final failure envelopes coincide, as shown in particular in Figs 8 and 11. In the case of unidirectional (0°) laminates (Figs 3–5) the lack of any transverse plies means that any failure is catastrophic, at least as far as the Grant–Sanders method predicts, and the distinction between initial and final failure disappears.

4.1. General comments on predicted failure envelopes 4.1.1. Figures 3–5 Figures 3–5 relate to unidirectional (0°) material for which any failure is considered to be final. The rectangular shapes of the curves arise from the use of discrete failure criteria, the corners representing changes in failure mode. The only exceptions are where failure is predicted in interactive modes with shear involving transverse tension (part of an ellipse) and longitudinal compression (straight line skew to axes). In Fig. 5 the predictions for y may be conservative. 4.1.2. Figures 6 and 7 For initial failure, for both Figs 6 and 7, the very low transverse tensile strength leads inevitably to very low predicted failure levels (possibly excessively so) under tensile loading. Equally inevitably, in view of the much higher transverse compressive strength, predicted failure levels under compressive loading are much higher, being close to the final failure values.

Fig. 3. Biaxial failure stresses for 0° lamina made of GRP material. Material type: E-glass/LY556/ HT907/DY063.

149

Fig. 4. Biaxial failure for 0° lamina made of CFRP material.

Fig. 5. Biaxial failure envelope of 0° GRP lamina under combined longitudinal and transverse loading.

For final failure, the shape of the predicted envelope in Fig. 6 is not dissimilar to that of the quasi-isotropic laminate (Fig. 11). The deviations from a parallelogram shape are caused by the shear-longitudinal compression interaction and by the influence on load-carrying capacity of the changes in the transverse and shear layer stiffnesses. Figure 7 represents a configuration for which the Grant–Sanders method may not be the optimum, and it is difficult to comment sensibly on the results for final failure. 4.1.3. Figure 8 In Fig. 8 also, much higher initial failure levels are calculated in compression than in tension, and for the same reason as for Figs 6 and 7. If this example is based on pressure vessel design there may be problems in tension if matrix cracking quickly leads to delamination. For final failure this lay-up seems very good for tension-tension loading. Its performance in compression-compression is inhibited by the shear-longitudinal compression interaction.

150

Fig. 6. Biaxial failure envelope for 90°/ ± 30°/90° GRP laminate.

Fig. 7. Biaxial failure envelope for 90°/ ± 30°/90° GRP laminate under combined direct and shear loading.

151 Stress/strain curves for ± 55° laminates in uniaxial tension and biaxial tension are presented in Figs 9 and 10, respectively. 4.1.4. Figure 11 Again, as expected, much higher initial failure levels are calculated in tension than in compression for a 0°/ ± 45°/90° CFRP laminate (Fig. 11). Indeed in compression final failure is uniformly predicted to occur first. This is to be expected for a balanced lay-up such as this one. The logical shape for the final failure envelope is a parallelogram. The deviations from such a shape arise from the shear-longitudinal compression interaction and the influence on loadcarrying capacity of the changes in the transverse and shear layer stiffnesses. The stress/strain curves for this laminate are given in Figs 12 and 13. 4.2. Discrepancy between Figs 8 and 9 In Fig. 8 failure is predicted at y = 266 MPa for x = 0·0. In considering this result it has to be borne in mind that shear failure is considered to be final failure (Section 2.3). However, as also pointed out in Section 2.3, the laminate may be capable of taking significantly more load beyond this point, but usually with a structurally unacceptable degree of deformation. In Fig. 9, which is a stress/strain curve, the point where shear failure is predicted is indicated. Beyond this point the laminate stiffness drops at a rapid rate until the matrix properties are completely annihilated, beyond which point linear behaviour is predicted. This is because the fibre stiffnesses are taken to be constant so that no further reduction in laminate stiffness will be calculated.

Fig. 8. Biaxial failure envelope for ± 55° GRP laminate.

152

Fig. 9. Stress/strain curves for ± 55° GRP laminate under uniaxial tension.

The predicted drop in stress prior to the final linear phase may be a purely numerical feature arising from the convergence problems experienced in this region of the curve. However, it is not inconceivable that such behaviour might be observed in a displacement-controlled experiment.

4.3. Final linear behaviour in predicted stress/strain curves (Figs 9, 10 and 12–16) In many of the stress/strain curves (Figs 9, l0 and 12–16) a final linear portion is predicted at high strains after markedly non-linear behaviour at lower values of strain. This arises because at high strain the matrix dominated stiffnesses are taken to be more or less completely annihilated, whereas the fibre dominated stiffnesses are assumed to be constant. As expected this feature is much more pronounced for matrix dominated configurations than for fibre dominated. Thus for the 0°/ ± 45°/90° CFRP laminate (Figs 12 and 13) and the

Fig. 10. Stress/strain curves for ± 55° GRP laminate under biaxial tension (y/x = 2/1).

153

Fig. 11. Biaxial failure stresses for 0°/ ± 45°/90° laminate of CFRP material.

0°/90° GRP laminated (Fig. 16) non-linearities are on the whole fairly small. In contrast, for the ± 55° and ± 45° GRP laminates (Figs 9, l0, 14 and 15), non-linear behaviour is very pronounced at relatively low strains.

5. Conclusions Few problems were experienced in applying the stress-based Grant–Sanders model to fibredominated situations. There were rather more problems in addressing the matrix-dominated

Fig. 12. Stress/strain curves for 0°/ ± 45°/90° laminate made of CFRP material (y/x = 1/0).

154

Fig. 13. Stress/strain curves for 0°/ ± 45°/90° laminate made of CFRP material (y/x = 2/l).

Fig. 14. Stress/strain curves for ± 45° laminate made of GRP material (y/x = 1/1).

155

Fig. 15. Stress/strain curves for ± 45° laminate made of GRP material (y/x = 1/–1).

configurations. The model has not been developed with such applications in mind and some convergence difficulties arose in regions with a high rate of loss of stiffness with increasing strain. Some gross assumptions had to be made about how the transverse tension modulus reduces with increasing strain, as no information was available for the systems studied. The effect is probably small for fibre domination, but could be quite large in matrix-dominated situations. The use of discrete failure criteria led to abrupt failure envelopes with sharp corners where the failure mode or location changed. Again this is believed to be realistic for fibre-dominated loadings, but may be less so for matrix-dominated ones.

Fig. 16. Stress/strain curves for 0°/90° GRP laminate (y = 0).

156 Acknowledgements All the work on the development of this updated Grant–Sanders model was performed at the Warton Unit of British Aerospace Defence (Military Aircraft), where the author has been employed since 1966. The help given by the Company and the author’s colleagues within British Aerospace is gratefully acknowledged. References 1 2 3 4

5 6

Hinton, M. J. and Soden, P. D., Predicting failure in composite laminates: the background to the exercise. Compos. Sci. Technol., 1998, 58(7), 1001. Failure criteria for an individual layer of a fibre reinforced composite laminate under in-plane loading. ESDU Data Item 83014, June 1986. Sanders, R. C. and Grant, P., The strength of laminated plates under in-plane loading. BAe Report SOR(P)13O, January 1982. Soden, P. D., Hinton, M. J. and Kaddour, A. S., Lamina properties, lay-up configurations and loading conditions for a range of fibre-reinforced composite laminates. Compos. Sci. Technol., 1998, 58(7), 1011. Edge, E. C., Final report on P.V. funded portion of CFC Basic Technology Programme. BAe Report SOR(P)177, October 1987, with Addendum 1, February 1989. Engblom, J. J. and Yang, Q., Dynamic residual property predictions for laminated composite structures: FE-based internal state variable approach. Compos. Struct., 1994, 27, 215–223.

Appendix Procedure for interaction of non-linearities – worked example Take data for XAS/914C material (see Figs 1 and 2). For XAS/914C, E20 = 9 GPa, G120 = 5·73 GPa, 120 = 0·3, F12 = 141 MPa, 20 = 0·6% and 2m = 2·0%. Assume that 12 = 50 MPa and 2t = 0·7%. Assign D = 1 for all laminates and in all layers. Let us now go through the procedure given in Section 2.6.2. 1. The non-linear curves for E2t and G12 given for XAS/914C in Figs 1 and 2, respectively, are used. 2. D = 1, so that in the absence of shear loading, the reduction of E2t with increasing strain commences at 2t = 0·6%. 3. R = (50/141)2 = 0·1257. 4. 0 = 0·6
兹(1  0·1257) = 0·5610%. 5. This can be accomplished by evaluating E2t for a strain of 0·6390% ( = 0·6 + 0·6–0·5610), using the curve for XAS/914C from Fig. 1. This gives E2t = 8·425 GPa. 6. Using the curve from Fig. 2 gives 12 = 1·06% and G120 = 4·724 GPa. 7. G12 = 4·724
8·425/9 = 4·422 GPa. 8. D* = 0·5610/(2·0–0·6 + 0·5610) = 0·2861. 9. f = 8·425/9 = 0·9361 and 12 = 0·3
0·9361
[2–0·9361
(1  0·2861)]/(1 + 0·2861) = 0·2908. 10. 21 = 0·2908
8425/135·2 = 0·0181 if the ply is in longitudinal tension or 21 = 0·2908
8·425/118·4 = 0·0207 if the ply is in longitudinal compression.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

157

CHAPTER 3.4

Predicting transverse crack formation in cross-ply laminates* L. N. McCartney Centre for Materials Measurement & Technology, National Physical Laboratory, Teddington TW11 0LW, UK Abstract As a contribution to the exercise to compare methods of predicting damage in composites and subsequent failure, this paper describes a method of predicting the values of the applied stresses for which transverse cracks form in a cross-ply laminate subject to biaxial loading, taking full account of both anisotropy and thermal residual stresses. The analysis is based upon an accurate technique for determining the stress and displacement distributions in multiple-ply cross-ply laminates having ply cracks in one orientation. The resulting relationships governing crack fonnation are exceedingly simple in form involving parameters that may be defined and measured at the macroscopic laminate level, thus offering engineering methods that have great potential for use in design. The methods are then applied to the prediction of transverse cracking in a specific glass-reinforced plastic system for the case of the uniaxial loading of a [0/90]s GRP laminate, and for the equi-biaxial loading of a [ ± 45]s laminate. Keywords: cross-ply laminates, ply cracking, stress transfer, thermoelastic constants, crack closure

Notation EA ET Et vA va vt A

Axial Young’s modulus (defined by principal loading direction) Transverse Young’s modulus (in-plane of laminate normal to axial direction) Transverse Young’s modulus (out-of-plane of laminate normal to axial direction) Axial Poisson’s ratio (in-plane) axial Poisson’s ratio (out-of-plane) Transverse Poisson’s ratio (out-of-plane) Axial shear modulus (in-plane)

* This article represents the author’s contribution to a worldwide exercise to confirm the state-of-the-art for predicting failure in composites, organised by Hinton and Soden.1

158 a t A T t T = T  To x, y, z xi N hi 2h(0) 2h(90) ixx, ixy, iyy, izz ixx, ixy, iyy, izz ui, vi, wi Ci Si, Ui, Vi Fij, Gij, Hij  t o *o ¯ o * ¯o  c *c o *o ¯ o ¯ o** o *o 2h 2L n, p k

o c, co  2

2 t

Axial shear modulus (out-of-plane) Transverse shear modulus (out-of-plane) Axial thermal expansion coefficient (in-plane) Transverse thermal expansion coefficient (in-plane) Transverse thermal expansion coefficient (out-of-plane) Where T is current temperature and To is stress-free temperature Cartesian coordinates x coordinate of ith interface Number of interfaces in right-hand half of laminate Thickness of ith ply element Total thickness of all 0° plies Total thickness of all 90° plies Stress components in the ith ply element Stress components in the ith ply element Through-thickness, axial and transverse displacement coefficients in ith ply element Stress transfer function for ith interface Distributions of xx, ui, vi on interface x = xi Coefficients of differential equations governing stress transfer Effective axial applied stress Effective transverse applied stress Effective axial crack closure stress for biaxial loading Effective transverse crack closure stress for biaxial loading Effective crack closure stress for uniaxial axial loading Effective crack closure stress for uniaxial transverse loading Effective axial strain of undamaged laminate Effective axial strain of cracked laminate Uniform transverse strain for the cracked laminate Effective axial crack closure strain for biaxial loading Effective transverse crack closure strain for biaxial loading Effective crack closure strain for uniaxial axial loading Effective crack closure strain for uniaxial transverse loading Axial laminate strain that is zero at current temperature when unloaded Transverse laminate strain that is zero at current temperature when unloaded Total thickness of laminate Distance between two neighbouring transverse cracks Transverse crack densities Constant for inter-relationships between thermoelastic constants Normalising stress Ply crack length, normalised crack length Energy absorbed by transverse crack formation Effective fracture energy of laminate for transverse cracking Fracture energy for transverse cracking

159 G V g go S

Change of Gibbs free energy in region V of laminate Region occupied by laminate Gibbs free energy density Gibbs free energy density when stress field is zero Axial applied stress for transverse cracking simulation

1. Introduction The many failure theories proposed in the literature (see review2 and other papers in this volume for details) offer the user very little guidance with regard to their relative merits, and the regimes in which they can be used with confidence. The prediction of failure in laminated composites is thus an area of materials technology requiring critical appraisal. Macroscopic failure criteria2 used extensively in practice bear very little relationship to the microstructural phenomena that determine the strength of composite materials. For example, these failure criteria do not take into account the effects of localised damage such as ply cracking, delamination and fibre fracture. The criteria appear to be selected as invariant mathematical functions (with regard to spatial coordinates) that involve material strengths under various single-mode loading conditions. Sometimes the criteria are associated with energies.3 The approach taken in this paper is, firstly, to apply to the principles of mechanics at the microstructural level to predict damage formation in a cross-ply laminate subject to biaxial loading and, secondly, to show how the result is a very simple description of damage initiation and growth, involving parameters that can be related to microstructural features, but which may be defined and measured at the macroscopic level. This bridging from microstructural to macroscopic behaviour is rigorous from a mechanics point of view, and consequently has great potential for use in a design environment. The objective of this paper is to describe the approach, to apply it as part of the exercise, and to indicate briefly how the important results might be used in engineering design. Methods of predicting cracking in composites have most frequently used simplified stresstransfer models, e.g. shear-lag theory and its many variants. While shear-lag models are useful for identifying materials parameters that will have an influence on criteria for crack formation in composite materials, they are based on several approximations that render predictions subject to error. To overcome these limitations, some attention has been devoted to developing better models of stress transfer in laminates that are based on variational techniques. The normal method of analysis assumes a stress representation that satisfies the equations of equilibrium and stress boundary conditions. Such representations involve functions of the axial coordinate which are determined by minimising the complementary energy of the system in a variational calculation. For [0/90]s laminates, Hashin4–7 and Nairn8 have developed such solutions. Nairn and Hu9 have derived corresponding solutions for multiple-ply cross-ply laminates. These solutions lead to lower-bound estimates when used to predict the dependence of axial Young s modulus on crack density. Nairn10 has recently developed a displacementbased variational model that predicts an upper-bound estimate for the axial modulus of a cracked laminate. A characteristic of stress-based variational methods is that the corresponding displacement distribution, related by the stress/strain equations, is not specified. However, consistent solutions for both stress and displacement distributions in cross-ply laminates can be derived by analytical methods (see McCartney11,12). Furthermore, the solution developed

160 automatically minimises the Reissner energy functional13 so that a corresponding solution could have been derived by variational methods. The objective of this paper is to describe a relatively new analysis-based method14,15 of calculating the stress and displacement distributions in a multiple-ply cracked laminate that may be used to predict the formation of transverse cracking as a function of biaxial loading, while taking account of the effect of thermal residual stresses. The author has been requested to provide sufficient information in this paper that would enable readers to reproduce the results if they so wished. In consequence, this paper will repeat important analyses and results that have already been published in the literature rather than describe new work. No attempt is made to review the related literature thoroughly.

2. Geometry A set of 2N + 2 plies is perfectly bonded together to form a laminate. Each ply is regarded as being made of a different transverse isotropic material such that the laminate is symmetric about the mid-plane. When subject to biaxial loading only one half of the laminate needs to be considered. The N + 1 plies in this half are regarded as having thicknesses hi, i = 1 . . . N + 1. The superscript i is attached to the thermoelastic constants EA, ET, vA, vT, T, A and T to denote for the ith ply respectively the axial and transverse Young’s moduli, the axial and transverse Poisson’s ratios, the axial shear modulus, and the axial and transverse thermal expansion coefficients. The axial direction for the ply is defined as the direction of the principal applied load, while the transverse direction is normal to this direction in the plane of the laminate. The axial direction for a single ply is defined to be parallel to the fibres in that ply, while the transverse direction is normal to the axial direction in the plane of the ply. The difference between the current temperature and the stress-free temperature of the laminate is

Fig. 1. Schematic diagram of a [0/90] symmetric laminate subject to biaxial loading.

161 denoted by T. The laminate is loaded biaxially, as indicated in Fig. 1 for the case of a [0/90]s laminate. Rectangular Cartesian coordinates (x, y, z) are introduced having origin at the centre of the mid-plane of the laminate, such that x is in the through-thickness direction, y is in the axial direction and z is in the width direction. The interfaces between the plies are defined to be at x = xi, i = 1 . . . N, with x0 = 0 and xN + 1 = h. The 90° plies in the laminate may have cracks (but not necessarily) only on the planes y = ± L ± 2L, ± 3L, etc. The laminate is assumed to be subject to generalised plane-strain deformation such that ui = ui(x, y), vi = vi(x, y), wi = *cz,

i=1 . . . N+1

(1)

where *c is a uniform transverse strain for the cracked laminate whose value is to be determined by requiring that the effective transverse stress applied to the laminate has a specified value. The resulting stress distribution is representative of that found in laminates well away from their edges where transverse stress transfer effects also occur.

3. Interfacial and boundary conditions All tractions and all displacement components must be continuous throughout the perfectly bonded laminate, i.e.

冎

ixx = ixx+ 1, ixy = ixy+ 1 ui = ui + 1, vi = vi + 1

on x = xi,

i=1 . . . N

(2)

Symmetry about x = 0 implies that 1xy ⬅ u1 ⬅ 0 on x = 0

(3)

and, assuming that the outer surface of the laminate is stress-free, Nxx+ 1 ⬅ Nxy+ 1 ⬅ 0 on lx = h

(4)

Any plies that are uncracked on y = ± L are loaded by applying a given uniform axial displacement and have zero shear stress because of symmetry about the crack planes y = ± L, i.e. vi(x, ± L) = ± cL, ixy(x, ± L) = 0

(5)

where c is the effective axial strain applied to the cracked laminate. Any plies that are cracked on y = ± L will be stress free, i.e. iyy(x, ± L) = 0, ixy(x, ± L) = 0

(6)

The assumption eqn (1) of generalised plane strain leads to a transverse stress distribution zz(x, y) that is non-uniform, from which the effective transverse stress t is calculated by averaging this stress distribution over the laminate edge as follows

冘冕冕 N+1

1 t = 2hL

i=1

L

xi

 L xi1

izz dx dy

(7)

162 where izz is calculated using the following transverse stress/strain relation izz = vitixx + viA

E iT i yy  E iTiT T + E iT*c E iA

(8)

4. Governing differential equations The representation for the stress and displacement field,11,12 defined in Appendix A and B, automatically satisfies the equilibrium equations and all but one of the relevant stress/strain relations in each ply for any functions Ci(y), i = 1, . . . N. These functions characterise stress transfer between the plies of the laminate provided that the recurrence relationships given in Appendix B are obeyed, ensuring that the continuity conditions (eqn (2)) are satisfied. The following axial stress/strain relationships are not satisfied by the representation

iyy ⬅

vi v¯ ia 1 E iT =  i ixx + i iyy + ˜ iA T  viA I *c, y E˜ A E˜ A EA

i=1 . . . N+1

(9)

When a tilde ( ˜ ) is attached to an elastic constant, this denotes that the elastic constant relates to in-plane deformation for which generalised plane-strain conditions prevail such that the transverse strain has a uniform value everywhere in the laminate. To proceed, the stress/strain relations eqn (9) are averaged through the thickness of each ply so that

¯ iyy ⬅

¯vi v¯ ia 1 E iT =  i ¯ ixx + i ˜ iyy + ˜ iA T  viA i *c, y E˜ A E˜ A EA

i=1 . . . N+1

(10)

where the overbar denotes the through-thickness average. By demanding that each of the relations eqn (10) is satisfied exactly, it can be shown12 that the unknown functions Ci(y), i = 1 . . . N, of the representation must satisfy the following N homogeneous simultaneous differential equations

冘 N

i=1

冘 N

Fij Ci (y) +

i=1

冘 N

Gij Ci(y) +

Hij Ci(y) ⬅ 0,

j=1 . . . N

(11)

i=1

It is worth noting that the Reissner energy functional has a stationary value when eqn (10) is satisfied. The coefficients Fij , Gij and Hij may be calculated numerically. Equation (11), when subject to appropriate boundary conditions (specified in Ref. 12) derived from eqns (5) and (6), may be solved using standard techniques (see Ref. 15 for specific details regarding the application of standard techniques to solving the system given by eqn (11)). 5. Thermoelastic constants of cracked laminates Consider a cracked laminate containing a uniform distribution of n cracked planes per unit length such that the separation of the cracked planes is 2L = 1/n. In each crack plane the same 90° plies (but not necessarily all) are assumed to have cracked so that the region between any two neighbouring planes can be taken as the representative volume element V for the cracked laminate. Accurate stress and displacement solutions in the representative volume element can

163 be obtained by representing the 0° plies by a series of refined layers all having the properties of the 0° ply, and similarly for the 90° plies. Each ply of the laminate is divided into a number of layer elements of equal thickness (20 layers to be used in calculations for [0/90]s laminates). The elements adjacent to the ply interfaces are then halved successively (three times for calculations for [0/90]s laminates). Such layer refinement near the interfaces resolves the stress singularity that exists at the transverse crack tips. The effective stress/strain relations for the laminate are of the form

(n) =

 vA(n)  t + A(n) T EA(n) EA(n) vA(n) t + + T(n) T EA(n) ET(n)

*(n) = 

(12)

(13)

where (n) = c and *(n) = *c are respectively the effective axial and transverse strains of the laminate containing a uniform distribution of n cracked planes per unit length, and where EA(n), ET(n), vA(n), A(n) and T(n) denote the corresponding effective thermoelastic constants of the cracked laminate. In eqns (12) and (13) the parameter  is the effective applied axial stress defined by

冘 N+1

1 = h

hii

(14)

i=1

where i is the uniform axial stress in the ith ply when the laminate is uncracked. By making use of the analysis presented in Ref. 12, together with the numerical method15 of solving the simultaneous ordinary differential eqn (11) subject to the 4N boundary conditions specified in12,16, it is possible to confirm the validity of the relationships (12) and (13), and determine the values of the effective in-plane thermoelastic constants. For a cracked laminate the parameters *(n),  and T are regarded as being specified independent parameters. On using eqn (13) the effective transverse stress at applied to the cracked laminate may be written as

冋

t = ET(n) *(n) +

册

vA(n)   T(n) T ⬅ f (*(n), , T ) EA(n)

(15)

On substituting eqn (15) into eqn (12) the effective axial strain for the cracked laminate is given by

(n) =  vA(n)

ET(n) 1 *(n) + EA(n) EA(n)


T ⬅ g(*(n),

冋

1  (vA(n))2

册 冋

ET(n) EA(n)

 + A(n) + vA(n)

, T )

ET(n) T(n) EA(n)

册

(16)

The thermoelastic constants may then be determined from the relationships ET(n) = f (1, 0, 0)

(17)

164 vA(n) f (0, 1, 0) g(1, 0, 0) = = EA(n) ET(n) ET(n)

EA(n) = T(n) = 

1 g(0, 1, 0) + [vA(n)/EA(n)]2ET(n)

f (0, 0, 1) vA(n) , AA(n) = g(0, 0, 1)  ET(n)T(n) ET(n) EA(n)

(18)

(19)

(20)

Thus the effective thermoelastic constants of the cracked laminate are readily calculated by solving the cracked laminate problem for three independent cases specified by the following values of the parameters *(n),  and T

*(n) = 1,  = 0, T = 0 *(n) = 0,  = 1, T = 0

(21)

*(n) = 0,  = 0, T = 1 For each of these sets of values, the transverse stress t = f (*(n), , T ) is calculated from the solution by using eqn (15), and (n) = g(*(n), , T ) is calculated by using eqn (16). The thermoelastic constants for the cracked laminate are then calculated by using eqns (17–20). It should be noted that the same value of vA(n)/EA(n) is obtained when using f (0, 1, 0) and g(1, 0, 0) in eqn (18), indicating that the stress-transfer model is leading to a unique value for the Poisson’s ratio of the cracked laminate. The stress-transfer model described in this paper thus automatically leads to stress/strain relations of the form (12) and (13) where the coefficient of at in eqn (12) is identical to the coefficient of  in eqn (13), i.e. the tensor for the elastic constants of a cracked laminate is symmetric; an essential property if a cracked laminate having non-uniform stress and displacement distributions is to behave as the corresponding effective uncracked laminate having reduced properties that take account of the presence of transverse cracks. Most stress-transfer models used in the literature, if extended to deal with biaxial loading, would not lead to this essential symmetry.

6. Fundamental inter-relationships for thermoelastic constants Unpublished work has shown from the numerical solution of the stress-transfer problem for uniformly spaced crack planes that the following inter-relationships between the effective thermoelastic constants of the laminate are satisfied for any type of cracking behaviour consistent with the boundary conditions eqns (5) and (6) (vA/EA)  (vA(n)/EA(n)) (1/ET(n))  (1/ET) = = (1/EA(n))  (1/EA) (vA/EA)  (vA(n)/EA(n))

冑

(1/ET(n))  (1/ET) T(n)  T = =k (1/EA(n))  (1/EA) A(n)  A

(22)

165 where

k=

1 1  vAv(90) A

冉

冊

EA (90) vA  vA ET

(23)

where EA is the axial Young’s modulus of the laminate in the undamaged state. A similar notation is used for the other thermoelastic constants. The parameter v(90) A is the axial Poisson’s ratio of the material in the 90° ply. The results eqns (12), (13) and (22) are expected to be valid also for multiple-ply cross-ply laminates having non-uniform distributions of cracks. Numerical methods of solving accurately the stress transfer problem for such crack distributions have yet to be developed. However, the results eqns (12), (13) and (22) have been established analytically17,18 for both uniform and nonuniform crack distributions only for the case of a simple [0/90]s cross-ply laminate represented by just two layers; the 0° and 90° plies. The relations (22) ensure that the properties of a cracked laminate correspond exactly to those of an uncracked laminate when the cracks just close during compressive loading. 7. Prediction of transverse cracking The mechanics of the formation of fully developed transverse cracks in cross-ply laminates has been studied in some detail17,18 for the case when the laminate is modelled by just two plies, one representing the 0° ply and the other the 90° ply. It is useful here to summarise the essentials of the approach. The condition for the unstable growth of an isolated transverse crack in the 90° ply of a cross-ply laminate has been investigated.19,20 In Ref. 20 a master curve relates the normalised applied stress (  ˜ o)/ o (allowing for the effects of thermal residual stresses through the crack closure stress ¯ o for the case of uniaxial loading) to the normalised crack length c/co, where o and co are normalising parameters that can be derived from the properties of the plies in the laminate and from the transverse fracture energy of a single ply. A key characteristic of this master curve is that the cracking stress reduces as the crack length increases to an asymptotic limit having the value 兹( /2), independent of the crack length 2c. The asymptote is a very good approximation to the curve for values c/co > 10. As the value of a/co (where a is the thickness of the 90° ply) has values in the region of 45 and 5 for typical carbon- and glass-fibre-reinforced composites respectively, it is concluded that fully developed crack formation in the transverse ply may be considered to be governed by the asymptotic value of cracking stress, particularly for carbon-fibre-reinforced laminates. As discussed in Ref. 21, this assumption is equivalent to assuming that crack formation may be predicted by considering the energy changes that occur when a fully developed transverse crack forms instantaneously in a laminate subject to fixed applied tractions and a fixed temperature. This approach leads to conservative predictions for the stresses for crack formation. Consider now the formation of p new cracked planes per unit length in a laminate that already contains n cracked planes per unit length. The crack planes are assumed to form under conditions of fixed applied tractions and fixed temperature. From energy balance considerations, the fracture criterion for crack formation under these conditions has the form21  + G < 0

(24)

166 where the energy absorbed by the formation of p new cracked planes per unit length of laminate is given by  = 2 p

where =

h(90)

t h

(25)

and where the corresponding change of Gibbs free energy is G =

1 V

冕

[g(n + p)  g(n)] dV

(26)

V

In eqn (25) the parameter t is such that 2 t denotes the fracture energy for transverse cracking in the 90° plies having total thickness h(90), and in eqn (26) g(n) denotes the Gibbs free energy per unit volume when the laminate contains n cracked planes per unit length. With tensor notation for the stress ij, the strain ij and the thermal expansion coefficients ij, the Gibbs free energy per unit volume has the form 1 g =  ij(ij + ij T ) + go(T ) 2

(27)

and, on integration over the region V between the planes y = ± L, it follows that 1 V

冕

g dV = 

1 2V

V

冉冕 V

It can be shown that 1 V

冕

ijij dV 

冕

冊

ijij T dV + go(T )

V

(28)

ijij dV = (n) + t*(n)

(29)

(90) ijij dV = (  o)A(n) + (t  *o)T(n) + o(0) A + * oA

(30)

V

and 1 V

冕 V

where o and *o are crack closure stresses which are specified18 by o =

(90) (0) (90) (0) (90) h(0)E (0) A T  A  vA [T  A ] T (0) (90) h 1  v A vA

(31)

*0 =

(0) (90) (0) (90) (0) h(90)E (90) A T  A  vA [T  A ] T (90) h 1  v(0) A vA

(32)

where h(0) and h(90) denote the total thickness in the laminate of all 0° plies and all 90° plies respectively, and where the superscripts 0 and 90 refer the thermoelastic constants to the properties of the 0° and 90° plies respectively. The result, eqn (30), is the extension of Levin’s theorem22 taking account of the effects of thermal residual stresses.

167 It can be shown by using eqns (12), (13), (22), (24)–(26), (29), (30) that the criterion for the formation of p new cracked planes in unit length of laminate already having n existing cracked planes, may be written

冉

冊

冉

冊

1 1 1 1 1 1 2 p < (  o)2  + (t  *o)2  2 EA(n + p) EA(n) 2 ET(n + p) ET(n)

 (  o)(t  *o)

冉

冊

vA(n + p) vA(n)  EA(n + p) EA(n)

(33)

It is worth noting that the result (33) is independent of the thermal expansion coefficients of the cracked and uncracked laminate; a property that arises from the introduction of the crack closure stresses o and *o and the use of the inter-relationships (22). On using eqns (12) and (22) the result eqn (33) may be written in the following alternative simple form 2 p =

[(n + p)  (n)]2 1 1  EA(n + p) EA(n)

1 2

(34)

which can also be written, on using eqns (13) and (22), in the form 2 p =

[*(n + p)  *(n)]2 1 1  ET(n + p) ET(n)

1 2

(35)

The results, eqns (34) and (35), indicate that the use of a sufficiently accurate stress-transfer model of a cracked cross-ply laminate has led to exceedingly simple results that involve only macroscopic properties defined at the laminate level, and which take full account of the effects of biaxial loading and of the residual thermal stresses.

8. Crack closure for uniaxial, axial and transverse loading Consider now an undamaged laminate that is subject to uniaxial loading in the axial direction that leads to the formation of n uniformly spaced cracked planes such that the same transverse crack pattern occurs in each crack plane. For the case of uniaxial axial loading the transverse stress at t = 0 and the stress/strain relationship (12) for the cracked laminate reduces to the form

(n) =

 + A(n) T EA(n)

(36)

The point of transverse crack closure occurs when  = ˜ o and (n) = ¯ o where

冉

冊

1 1  ¯ o =  [A(n)  A] T EA(n) EA

(37)

168 For the case of uniaxial transverse loading the axial stress  = 0 and the stress/strain relationship (13) for the cracked laminate reduces to the form

*(n) =

t + T(n) T ET(n)

(38)

The point of transverse crack closure occurs when t = * ¯ o and *(n) = ¯ *o where

冉

冊

1 1  * ¯ o =  [T(n)  T] T ET(n) ET

(39)

It follows from eqns (22), (37) and (39) that ¯ o = k* ˜o

(40)

9. Prediction of cracking for biaxial loading For general biaxial loading conditions it can be shown, on using eqns (12), (22) and (37), that

(n)   =

冉

冊

1 1  [ + kt  ¯ o] EA(n) EA

S =  + kt >

冑

4p

1 EA(p)

 EA 1

+ ¯ o

(41)

(42)

where  = (0) is the axial strain in an uncracked laminate subject to the same applied stress and temperature. On substituting eqn (41) in eqn (34) with n = 0, the following fracture initiation criterion is obtained where S is a single parameter, having the dimensions of stress, that is used to characterise the degree of biaxial loading applied to the laminate, and where k is a material constant defined by eqn (23) in terms of the elastic constants of the plies and of the laminate. If the negative root sign had been used in eqn (42) then predicted critical stresses could not be realised in practice as crack closure would first have occurred and the relation (42) would then be invalid. The critical values of S for transverse crack initiation predicted by eqn (42) are denoted by S(p). When determining the values of S(p) for a range of values of p > 0 and a particular crack pattern, it is found that S(p) has a minimum value at p = po, independent of the value of . This is interpreted as implying that transverse crack initiation is characterised by a specific initial crack plane density po, also independent of laminate length provided the laminate is long enough, rather than by the formation of a single crack plane. Given that the fracture criterion (42) predicts that transverse cracking is initiated by the simultaneous formation of a characteristic uniform initial transverse crack plane density po, subsequent cracking will occur only if the applied stresses are increased. It is assumed that additional crack planes will form at the mid-points between those uniformly

169 distributed crack planes that formed at the point of transverse crack initiation; a situation that is repeated as the applied stresses continue to increase. It follows from eqn (41) that

(2p)  (p) =

冉

冊

1 1  [ + kt  ¯ o] EA(2p) EA(p)

(43)

On substituting eqn (43) into eqn (34) with n = p it follows that the formation of additional transverse crack planes at the mid-points between existing crack planes, such that the resulting crack density is always uniform, is governed by the fracture criterion

S =  + kt >

冑

4p

1 EA(2p)

 EA(p) 1

+ ¯ o

(44)

from which critical values of S for the successive doubling of the crack density can be determined for values p = po, 2po, 4po, 8po, . . . .

10. Application to prediction of crack initiation in GRP The analysis described in this paper will now be applied to the prediction of transverse cracking in a glass-fibre-reinforced epoxy cross-ply [0/90]s laminate specified in the exericse.23 The thermoelastic constants for filament-wound individual plies of Silenka E-glass 1200tex reinforced with MY750/HY917/DY063 epoxy are: EA = 45·6 GPa

ET = 16·2 GPa A = 8·6
10–6°C–1

vA = 0·278

vT = 0·4 A = 5·83 GPa

T = 26·4
10–6°C–1

The temperature difference, T has the value –120°C, which means that the laminate has been cooled 120°C from its stress-free temperature. The strain-energy release rate for matrix fracture in the direction of the fibres is 165 J m–2. Thus, the value of t = 82·5 J m–2, where 2 t is the fracture energy for transverse cracking. Three loading cases are considered here. For the first case the loading is uniaxial where the plies have thickness 0·26 mm such that the 0° plies are on the outside of the laminate. For such laminates the cracks must form in the inner 90° plies so that there is only one crack pattern that can arise if orthogonal cracking is avoided. This loading case corresponds to one in the specification for the exercise.23 By carrying out an accurate stress-transfer analysis using the methods described above, it is found that during the uniaxial loading of an uncracked laminate, the characteristic initial crack density p0 has the value 0·52164 mm–1. The corresponding value of the applied stress for transverse crack initiation is 0·12288 GPa, obtained by making use of eqn (42) using the value p = po and setting at t = 0. The stresses at which the crack density suddenly doubles are obtained by making use of eqn (44) with t = 0. The values of the axial and transverse strains can be calculated at any stage of loading by making use of eqns (12) and (13) on setting at t = 0. The

170 Table 1 Data defining the axial and transverse stress/strain curves during transverse crack formation in a [0/90]s GRP laminate subject to uniaxial loading.  (GPa)

(n) (%)

*(n) (%)

0 0·12288 0·12288 0·13398 0·13398 0·22030 0·22030 0·48485 0·48485

0 0·39514 0·44691 0·48607 0·53460 0·86271 0·92800 1·99901 2·06419

0 –0·05759 –0·05045 –0·05518 –0·04849 –0·08199 –0·07299 –0·16661 –0·15763

strains (n) and *(n) are defined to be zero when the laminate is unloaded and T = 0, i.e. at the stress-free temperature. The values of axial strain, (n), and transverse strain, *(n), measured experimentally are obtained from the relationships

(n) = (n)  A T, *(n) = *(n)  T T

(45)

The values of applied stress and corresponding axial and transverse strains that define stress/ strain behaviour are given in Table 1 and plotted in Fig. 2.

Fig. 2. Axial and transverse stress/strain curves for a uniaxially loaded [0/90]s GRP laminate.

171

Fig. 3. Axial and transverse stress/strain curves for an equibiaxially loaded [0/90]s GRP laminate.

The second application, not corresponding to one specified in the exercise,23 is for a similar [0/90]s laminate subject to equi-biaxial loading where at t = . For this case the ply thicknesses are taken to be 0·25 mm and cracking is considered to occur only in the inner 90° ply. The calculations above are repeated with the new ply thicknesses and with the value at t = . The value of the characteristic initial crack density po has the value 0·54249 mm–1 and the applied stress for transverse crack initiation is 0·11095 GPa. The values of applied stress and corresponding axial and transverse strains that define stress/strain behaviour are given in Table 2 and plotted in Fig. 3. The third application is for another case specified in the exercise23 where a ± 45° laminate having ply thicknesses of 0·25 mm is subject to equi-biaxial loading such that  = t. Consider first of all the case when cracking occurs only in the inner plies. The solution to this loading case is identical the second case considered above. The only modification that needs to be made is to transform the axial and transverse strains given in Table 2 by a rotation of 45° (as would be applied for an uncracked laminate) so that the transformed strains correspond to the axial and transverse strains for an equi-biaxially loaded ± 45° laminate with cracking only in the inner plies. The results of such calculations are shown in Table 3 and Fig. 4. It should be noted that, as to be expected for the case where orthogonal cracking is ignored, the axial and transverse strains in the [ ± 45]s laminate have the same value for all stages of cracking in the inner plies. The shear strain (values not shown) are zero only when the laminate is uncracked. Such shear deformation arises because of the asymmetry induced by cracking in just one orientation. Now consider a [90/0]s cross-ply laminate subject to equi-biaxial loading where cracks are allowed to form only in the outer 90° plies. In contrast to the case of the [0/90]s laminate, the cracks that form have half the length of those that form in the [0/90]s laminate. Calculations

172 Table 3 Data defining the axial and transverse stress/strain cunes during transverse crack formation in a [ ± 45]s GRP laminate subject to equi-biaxial loading with regular cracking only in inner plies.  (GPa)

(n) (%)

* (n) (%)

0 0·11095 0·11095 0·12092 0·12092 0·19824 0·19824 0·43547 0·43547

0 0·30479 0·33481 0·36421 0·39235 0·63400 0·67187 1·45134 1·48914

0 0·30479 0·33481 0·36421 0·39235 0·63400 0·67187 1·45134 1·48914

show that the value of the characteristic initial crack density po has the value 0·36719 mm–1 and the applied stress for transverse crack initiation is 0·10122 GPa. The stress for crack initiation is just less than that required for crack initiation in a corresponding [0/90]s laminate. The initial crack density in the [90/0]s laminate is, however, significantly less than that in the corresponding [0/90]s. This arises because the cracks in a [90/0]s laminate are one half the length of those forming in a [0/90]s laminate. It is concluded that a [0/90]s laminate subject to equi-biaxial loading will initiate cracks in both 0° and 90° plies at about the same level of applied stress but that the crack density in the 0° plies will be larger than that in the 90° plies. By rotating the predicted axial and transverse strains through 45°, the applied stress and corresponding axial and transverse strains that define the stress/strain behaviour of an

Fig. 4. Coincident axial and transverse stress/strain curves for an equi-biaxially loaded [ + 45/–45]s laminate.

173 Table 4 Data defining the axial and transverse stress/strain curves during transverse crack formation in a [ ± 45]s GRP laminate subject to equi-biaxial loading with regular cracking only in the outer plies.  (GPa)

(n) (%)

* (n) (%)

0 0·10122 0·10122 0·10396 0·10396 0·14506 0·14506 0·31604 0·31604

0 0·27805 0·29979 0·30775 0·32908 0·45455 0·48763 1·04054 1·09241

0 0·27805 0·29979 0·30775 0·32908 0·45455 0·48763 1·04054 1·09241

equibiaxially loaded [ ± 45]s laminate, with regular cracking only in the external plies, can be calculated as before, and the results are given in Table 4 and plotted in Fig. 4. The calculations for equi-biaxial loading clearly do not take account of the interaction effects between the orthogonal crack systems, and such interaction could have a significant effect on stress/strain behaviour. Also, it should be noted (see Nairn8,9) that when a [90/0]s laminate is stressed uniaxially with the 90° plies on the outside of the laminate, the transverse cracking occurs in a staggered manner (i.e. cracks in one of the 90° plies are offset by half of the crack spacing from those appearing in the other 90° ply). This could also occur when such a laminate is biaxially loaded. The results presented here for the equi-biaxial loading of a [ ± 45]s laminate can thus only be regarded as being tentative. 11. Discussion The analysis described in this paper assumes that the deformation in the laminate can be described by linear elasticity theory. It is also assumed that cracking occurs only in one orientation, and that the fracture of the transverse plies is governed by energy balance principles where the resistance to fracture is characterised by a fracture energy (equivalent to an energy release rate). Furthermore, when considering all possible sites at which transverse fracture could occur, it is assumed that at each site the fracture energy has the same value. Within the limitations of these assumptions the modelling described here is virtually exact, so that any deviations of predictions from experimental results must be the result either of the material behaving in a manner that differs from that assumed (see, for example, the non-linear lamina shear properties provided for the exercise), or from errors of measurement when providing model data. The assumption that the fracture energy at each fracture site has a unique value is thought to be unreasonable as it leads to stress/strain behaviour that exhibits very few cracking events, owing to the fact that a single cracking event (following initiation) must be characterised by a doubling of the crack density. Experimental stress/strain curves do not exhibit such discrete behaviour. The modelling of crack formation needs to introduce some statistical variability so that cracks can form one at a time, leading to non-uniform crack

174 distributions and smoother stress/strain curves. The approach described in this paper has in fact been applied in such a manner.18 The methodology described applies only to multipleply cross-ply laminates subject to inplane biaxial loading for which cracks form only in one orientation, i.e. only in the inner ply for [0/90]s laminates. At the time of writing this paper, software was not available to enable predictions to be made for [ ± 45]s laminates subject to an applied stress ratio of 1/–1, or to quasi-isotropic laminates. However, with the same methodology described above, models are currently being developed for in-plane shear loading of cross-ply laminates, and for general in-plane loading of general symmetric multiple-ply laminates where the plies can be made of different materials and have any orientation with respect to the principal loading direction. Extensions of the methodology to the case of orthogonal cracking are also in progress. The methodology described, while very complex when carrying out calculations for the stress and displacement distributions within the laminate, does lead to a relatively simple description of crack formation under biaxial loading conditions even when thermal residual stresses are present. In particular, the results (34) and (35) are simple in form, they are defined at the macroscopic level, and they take full account of material anisotropy, thermal residual stresses and biaxial loading. Furthermore, the methodology is believed to produce exact results so that they have good potential for being used with great confidence by engineers when designing composite components. The detailed mechanics are needed only to calculate the effective thermoelastic constants of cracked laminates. The results are of immediate value to the designer wanting to determine the stress state for which transverse cracking will initiate. Longer term, it is expected that designers will want to take account of microstructural damage growth that leads to a change in mechanical properties of composite materials during loading. It is hoped that the methodology described in the paper, together with developments currently in progress, will enable the designer to take such effects into account. Strength prediction is the composite designers’ real need. Predicting failure, defined as the separation of a component into at least two separate pieces, is not currently achievable using micromechanism-based models. The microstructural damage modes that occur in a laminate prior to failure are exceedingly complex and highly interactive, with the result that reliable failure prediction under general conditions of loading is exceedingly difficult. The pragmatic approach is to recognise that, while reliable predictions of failure cannot be made using micromechanics-based models, reliable predictions should be possible for the initiation of damage and its growth during the early stages of deformation. The failure concept needs to be redefined so that it is predictable and takes account of damage growth beyond the point of initiation. Predictions of the localised change in mechanical properties of a laminate resulting from damage formation will enable the localised load transfer in composite components to be predicted. Such load transfer is responsible for composite components often out-performing expectations that arise when designs are based on coupon data obtained experimentally in the laboratory. While the approach described in this paper has not attempted to predict laminate failure, it is based upon the application of the principles of mechanics and thermodynamics at the microstructural level where an attempt has been made to model transverse cracking in individual plies. Recent unpublished work has shown that the methodology can be applied to more general laminates, more general loading, and has some potential for being able to take account of cracking in more than one orientation. The simplicity of the key results, the rigour of their derivation, and the fact that parameters describing behaviour are defined at the

175 macroscopic laminate level, means that the time is ripe for their serious exploitation in design methodology for composite components, although it is clear that development of the methodology is currently in its infancy.

Acknowledgements Research in this paper arises from: (1) work carried out for the NPL Strategic Research Programme; (2) collaboration with AEA Technology, Harwell, within the ‘Materials Measurement Programme’ of underpinning research financed by the UK Department of Trade and Industry; and (3) collaboration within the PREDICT project (Prediction of Damage Initiation and Growth in Composite Materials) which is a collaboration between British Aerospace (Sowerby Research Centre), Ciba Geigy Plastics, Tenax Fibres, National Physical Laboratory, Universities of Bristol and Surrey (project led by British Aerospace and funded under the LINK Structural Composites Programme of the DTI’s Research and Technology Initiative-IED Grant Ref. RA 6/25/01).

References 1 Hinton, M. J. and Soden, P. D., Predicting failure in composite laminates: the background to the exercise. Compos. Sci. Technol., 1998, 58(7), 1001. 2 Rowlands, R. E., Strength (failure) theories and their experimental correlation. In Handbook of Composites, Vol. 3: Failure Mechanics of Composites, ed. G. C. Sih and A. M. Skudra. North Holland, Amsterdam, 1985, Ch. 2. 3 Whitworth, H. A. and Yin, S.-W., A failure criterion for laminated fibre reinforced composites. Compos. Eng., 1991, 1, 61–67. 4 Hashin, Z., Analysis of cracked laminates: a variational approach. Mech. Mater., 1985, 4, 121–136. 5 Hashin, Z., Analysis of stiffness reduction of cracked cross-ply laminates. Eng. Fract. Mech., 1986, 25, 771–778. 6 Hashin, Z., Analysis of orthogonally cracked laminates under tension. J. Appl Mech., 1987, 54, 872–879. 7 Hashin, Z., Thermal expansion coefficient of cracked laminates. Compos. Sci. Technol., 1988, 31, 247–260. 8 Nairn, J. A., Strain energy release rate for composite mcirocracking: a variational approach. J. Compos. Mater., 1989, 23, 1106–1129. 9 Nairn, J. A. and Hu, S., The formation and effect of outer-ply microcracks in cross-ply laminates: a variational approach. Eng. Fract. Mech., 1992, 41, 203–221. 10 Nairn, J. A., Some new variational mechanics results on composite microcracking. In Proc. 10th Int. Conf. on Composite Materials, Whistler, BC, Canada, 14–18 August 1995. 11 McCartney, L. N., Theory of stress transfer in a 0°–90°0° cross-ply laminate containing a parallel array of transverse cracks. J. Mech. Phys. Solids, 1992, 40, 27–68. 12 McCartney, L. N., A recursive method of calculating stress transfer in multiple-ply cross-ply laminates subject to biaxial loading. NPL Report DMM(A)150, National Physical Laboratory, Teddington, January 1995. 13 Reissner, E., On a variational theorem in elasticity. J. Math. Phys., 1950, 29, 90–95. 14 McCartney, L. N., Predicting non-linear behaviour in multiple-ply cross-ply laminates resulting from microcracking. In IUTAM Symposium on Non-linear Analysis of Fracture, ed. J. R. Willis. Kluwer Academic Publishers, Dordrecht, 1997, p. 379.

176 15 Hannaby, S. A., The solution of ordinary differential equations arising from stress transfer mechanics. NPL Report DITC 223/93, National Physical Laboratory, Teddington, November 1993. 16 McCartney, L. N., Hannaby, S. and Cooper, P. M., Effects of cracking in multi-ply cross-ply laminates. In Proc. of 3rd Int. Conf. on Deformation and Fracture of Composites, University of Surrey, 27–29 March 1995, pp. 56–65. 17 McCartney, L. N., The prediction of cracking in biaxially loaded cross-ply laminates having brittle matrices. Composites, 1993, 24, 84–92. 18 McCartney, L. N., The prediction of non-uniform cracking in biaxially loaded cross-ply laminates. NPL Report DMM(A)142, National Physical Laboratory, Teddington, July 1994. 19 McCartney, L. N., Mechanics for the growth of bridged cracks in composite materials: Part I. Basic principles. J. Compos. Technol. Res., 1992, 14, 133–146. 20 McCartney, L. N., Mechanics for the growth of bridged cracks in composite materials: Part II. Applications. J. Compos. Technol. Res., 1992, 14, 147–154. 21 McCartney, L. N., Prediction of microcracking in composite materials. In Fracture: A Topical Encyclopaedia of Current Knowledge Dedicated to Alan Arnold Griffith, ed. G. P. Cherepanov. Krieger Publishing Company, Melbourne, USA, 1997. 22 Levin, V. M., Thermal expansion coefficients of heterogeneous materials. Mech. Solids, 1967, 2, 58–61. 23 Soden, P., Hinton, M. J. and Kaddour, A. S., Lamina properties, lay-up configurations and loading conditions for a range of fibre-reinforced composite laminates. Compos. Sci. Technol., 1998, 58(7), 1011.

Appendix A Representation of stress and displacement field The stress/strain relations for the ith ply or elemental layer of the laminate are of the form

ixx =

1 i via i vit i     zz + it T xx yy E it E iA E iT

(A1)

iyy = 

viaa i 1 viA xx + i iyy  i izz + iA T i EA EA EA

(A2)

izz = 

vit i viA 1 xx  i iyy + i izz + iT T i ET E AA ET

(A3)

ixy =

ixy , 2ia

ixz =

ixz , 2it

iyz +

iyz 2iA

(A4)

defining 12 independent thermoelastic constants. The following representation for the stress and displacement field has been derived from the equilibrium equations, strain/displacement and stress/strain relationships by using the methods described in.12 The fundamental assumption is that the shear stress component axy has the following form which is linear in the through-thickness coordinate x, for i=1 . . . N+1 1 ixy = [C i  1(y)(xi  x) + C i(y)(x  xi  1)] h

(A5)

177 Functions Ci(y), i = 1 . . . N + 1, corresponding to the values of the shear stress on the interfaces x = xi, i = 1 . . . N, depend only on the axial coordinate y, and are for the moment regarded as being arbitrary. It is to be understood that C0(y) and CN + 1(y) are identically zero. On substituting eqn (A5) into one of the equilibrium equations, it can be shown that, for i = 1 . . . N + 1, ixx(x, y) =

x  xi [(x  xi)C i 1(y)  (x  xi + 2hi)C i(y)] + Si(y) 2hi

(A6)

where Si(y) is defined in Appendix B, while the second equilibrium equation leads to iyy(y) =

1 [Ci  1(y)  Ci(y)] + i, hi

(A7)

i=1 . . . N+1

indicating that the axial stress in each layer is independent of x. The parameter i denotes the uniform axial stress that would arise in the ith layer if the laminate were undamaged and subject to the same applied tractions and temperature. By making use of eqns (A5) – A7) in conjunction with all but one of the relevant stress/ strain equations it can be shown,12 for i = 1 . . . N + 1, that the displacement components are of the form ui(x, y) =

(x  xi)2 x  xi v˜ ia [(x  x )C  (y)  (x  x + 3h )C  (y)]  [Ci  1(y)  Ci(y)] i i1 i i i 6E˜ ithi hi E˜ iA

+ (x  xi)

冋

册

1 Si(y) + i + Ui(y) E˜ it

(A8)

where v˜ i i =  ˜ ai i + ˜ it T  vit*c EA

vi(x, y) =

x  xi 2hi

冋 冉 冊 冋 (x  xi)

(A9)

冉

冊 册

v˜ ia 1 v˜ ia 1  C  (y)  (x  xi)  i (x  xi + 2hi) C i(y) i1 ˜E iA ia ˜E iA a



(x  xi)3 24E˜ ithi



(x  xi)2 S i(y)  (x  xi)U i(y)Vi(y), 2E˜ it

(x  xi)C i 1(y)  (x  xi + 4hi)C i(y)

册

i=1 . . . N+1

(A10)

In eqns (A8)–(A10), the elastic constants having a tilde are those defined in12 for an uncracked laminate subject to generalised plane-strain conditions. The functions Ui(y) and Vi(y) appearing in eqns (A8) and (A10) respectively are defined in Appendix B.

178 Appendix B Recurrence relationships By imposing the continuity of the traction xx and the displacement components ui and vi at the interfaces x = xi, i = 1 . . . N, by using eqns (A6), (A8), (A10), (2), (3) and (4) it can be shown that Si  1 = Si +

Ui = U i  1 +

Vi = Vi  1 +

+

hi [C i  1 + C i], 2

冋 册 册 冋冉 冊 冉 冊 册

h2i v˜ ia 1 [C  + 2C  ]  [Ci  1  Ci] + hi Si + i , i1 i 6E˜ it E˜ iA E˜ it

h3i 24E˜ it

冋

C i  1 + 3C i 

h2i S i  hiU i, 2E˜ it

(B1)

SN + 1 = 0

hi 2

i=1 . . . N+1

u0 = 0

(B2)

v˜ ia 1 v˜ ia 1  i C i  1  + C i i E˜ A a E˜ iA ia

(B3)

where V1(y) is regarded as being specified for the operation of the recurrence relationship eqn (B3), but which is eliminated when developing the solution of the problem.12

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

179

CHAPTER 3.5

Predictions of the original and truncated maximumstrain failure models for certain fibrous composite laminates L. J. Hart-Smith* Douglas Products Division, Boeing Commercial Airplane Group, Long Beach, CA, USA Abstract This article represents the author’s contribution to a world-wide exercise originated in UK and aimed at confirming the state-of-the-art for prediction of failure in composites. The use of the original and truncated versions of the maximum-strain failure criterion for fibre/polymer composites is illustrated by sample solutions of specific problems provided by the organizers of the exercise. The intent of this exercise is to compare the independent predictions for these same problems made by several originators of composite failure models – both with each other and with the experimental data to be supplied later. The organizers’ requests covered both failure envelopes and stress/strain curves. Only the former are addressed here. Keywords: composite laminate strength, lamina failure criteria, maximum-strain model, truncated maximum-strain model

1. Introduction and historical background While the first use of the maximum-strain failure model in assessing its suitability for use with fibre-reinforced composites dates back to the early 1960s, when all classical failure theories * The present author was not involved in the development of the original maximum-strain failure model (it was before his time), and is but one of several who contributed to the truncated maximum-strain failure model, mainly to justify it rather than to invent it. These are not ‘his’ theories in the same sense as the Ten-Percent Rule or the generalization of the Tresca criterion to fibre/polymer composites are. The reason for preparing this paper is that the original and truncated maximum-strain failure models are too significant in the US aerospace scene to have been omitted from the comparison between failure theories for composite laminates. The hope had been that either the originators, or visibly impartial champions, would demonstrate the capabilities and limitations of every significant composite failure theory, good and bad. Not every theory has been included in the exercise but, even after the test results have been released, the author feels that it would be worthwhile to assess some further theories, if only to demonstrate how simple or complicated it is to match known results. The theories with which the contest organizers are associated, and which were withheld because they had access to the test data, could then also be assessed in the same vein.

180 developed for metals were being evaluated, the first recognition that it was the best such theory for advanced fibre-reinforced polymer composites appears to have been by Waddoups,1 while he was working at General Dynamics, Convair, in Fort Worth, USA. The theory contains five noninteractive strain limits to characterize the failure of a unidirectional lamina; longitudinal and transverse direct strains in tension and compression, combined with an inplane shear strain limit. With and without various additional cut-offs, this model has formed the basis for the majority of aerospace design in the USA for boron- and carbon-reinforced polymer composites. The need for certain improvements, beyond the original maximum strain model, was recognized by several other researchers, working independently. In particular, most US aerospace companies have added some form of truncation for fibre-dominated tensioncompression (shear) loads. The need for doing so arises from a generally unrecognized omission from the set of properties characterizing the laminae for these and all other macrolevel failure models. Whether or not there is any influence of transverse stresses on the longitudinal strength of fibres, the basic set of five lamina strength measurements cited above permits neither its detection nor characterization. Lacking theories with which to allow for this effect, once it had been observed in carbon-epoxy structural laminates, various empirical truncations have been incorporated in these analyses to limit excessive predictions of fibredominated in-plane shear strengths. For this reason, the predictions of strength in this article cover the use of both the original (untruncated) and the most widely used of the truncated models.2 Possible limits on laminate in-plane-shear strengths arising from primary failures in the matrix are customarily ignored by most people in the US aircraft industry because epoxy and polyester matrices are sufficiently ductile in shear that the matrix in 0° and 90° layers, for example, will not fail until long after the fibres in nearby ± 45° layers in the same laminate. (This simplifying assumption has been justified by empirical experiences with carbon-fibrereinforced laminates in which the plies are thin and not clustered together, with the layers in the various directions well interspersed. It also requires fibres in a sufficient number of directions to create a structure, rather than a mechanism. It would not be expected to apply in the case of the ± 55° glass/epoxy laminate, but should for the 90°/ ± 30° and 0°/ ± 45°/90° laminates. Neglecting matrix failures is known not to be justified for carbon-fibre-reinforced laminates in which parallel plies have been blocked together, reducing the number of effective matrix interfaces, and for many laminates reinforced by glass fibres, which can withstand far higher tensile strains-to-failure than can typical polymer matrices. Nevertheless, the solutions attempted here are governed by this simplification, which can be evaluated against the results of other analyses, particularly for fibreglass-reinforced laminates.) A more serious problem, with this and other macrolevel failure theories, in regard to the predicted fibre/polymer composite strengths, concerns the treatment of transverse-tension failures in the matrix. In the longitudinal direction, because these composites of materials are highly orthotropic, there is little difference between the predictions of the maximum-strain and maximum-stress failure models, for what are the fibre-dominated strengths. However, matrix cracking under transverse-tension loads is a fracture-mechanics problem, involving possible interactions not only with other stresses in the same lamina, but definitely with the stiffness and thickness of adjacent plies. Matrix yielding in shear can be characterized by a material strength, but matrix cracking CANNOT be predicted on the basis of material properties alone. Virtually all popular composite laminate strength-prediction models and

181 computer codes are seriously deficient in this regard, as is explained in Ref. 3. Because this paper represents analyses as they are customarily performed, the traditional (rather than best possible) coverage will be provided, in the form of a constant-strain cut-off, for both failure models. This is set by the strain-to-failure of orthogonal fibres, whether they are present or not. The transverse strain-to-failure is set minutely above the longitudinal strain-to-failure of the transverse ply. If there are no transverse plies, the same effect can be achieved by including zero-thickness plies in all directions. Doing so will affect neither the laminate stiffnesses nor the fibre-dominated strengths, but will impose strain limits to protect the matrix. Further, since the intent of this process is to suppress the prediction of physically unrealistic matrix failures, which are commonly predicted by other analyses, additional solutions are provided in which even this limit is overridden whenever the author thinks it appropriate. The solutions to the editors’ problems, described in Ref. 4, other than the quasi-isotropic carbon/ epoxy laminate show that an absence of orthogonal fibres permits transverse strains sufficiently high to exceed the limits with which the empirical techniques described above can cope. Actually, for a given geometry (i.e. laminate) under various states of in-plane stress, the failure of individual unidirectional plies under transverse-tension loads is better characterized by a maximum-stress criterion within each affected ply. This is because the many closely spaced fibres impede the propagation of matrix cracks in any direction other than parallel to the fibres and, as is characteristic of all crack-propagation problems, any spreading of the crack is governed by the stress perpendicular to the direction in which the crack is growing. (In homogeneous solids, additional stress components could turn the crack to grow perpendicular to the highest principal tensile stress. However, just as discrete crack-arrestors modify this behaviour for isotropic metals, the fibres in composite laminates also influence the directions in which matrix cracks can grow.) For this reason, the truncated maximum-strain failure model used here should be combined with a local transverse-tension stress-failure model for when the fibres do not dominate the behaviour, as suggested in MIL-HDBK-17.5 Unfortunately, the value of the transverse stress at failure will vary with the fibre pattern and stacking sequence, but not with any additional stress components which may be present in the lamina being assessed. There is usually no way to directly measure this in-situ transverse tension strength; that measured on an isolated 90° lamina is of no relevance, applying only to that lamina in isolation, and NOT when it is embedded in a multidirectional laminate. Such analyses for matrix cracking would be exactly the same as in the companion paper.6 Therefore, they are not repeated here, since they do not yet represent widespread industrial practice.

2. Definition of the original and truncated maximum-strain failure models for composite laminates Figure 1 characterizes the original (untruncated) maximum-strain model for a unidirectional lamina, with fibres in the 0° (1) direction, showing how the five flat orthogonal facets of the failure envelope can be established independently. The same model has also been used for woven fabric ‘laminae’, with fibres in the 0° and 90° (1 and 2) directions. Each facet in Fig. 1 is customarily located by one measured strain-to-failure in the middle of each facet. Since

182

Fig. 1. Non-interactive maximum-strain failure model for unidirectional fibre/polymer composites.

the mode of failure is specified by the model, only one measurement is needed for each of these noninteractive criteria. Significantly, for unidirectional laminae, there is no difficulty in identifying which constituent of the composite is failing for each state of combined stresses. Each facet is associated with a unique failure mechanism in a single constituent, once the specific composite material has been specified, even though this can change with different fibres and resins. For example, for typical carbon/epoxy unidirectional laminae, longitudinal tension is associated with fibre failures, longitudinal compression with fibre instability, transverse compression and in-plane shear with ductile matrix failure,* and transverse tension with matrix cracking, which is catastrophic in isolated unidirectional laminae. Conventional treatment of a bi-directional (cloth) lamina would have all in-plane strains-to-failure fibredominated, with the only limit set by the matrix being in-plane shear. The possibility of predicting matrix cracking between fibres is therefore precluded, even though it is known to sometimes happen, even if not catastrophically. (Such bidirectional ‘laminae’ should really be replaced by two equivalent unidirectional plies, even though it is not yet normal practice to do so.) Figure 2 shows how two such failure envelopes, for any mixture of 0° and 90° unidirectional laminae, are combined. (Fibres at other angles, such as ± 45°, are covered by rotating the reference axes, which does not correspond with rotating the failure envelope. Any pair of orthogonal fibre directions can be covered simultaneously, but other fibre patterns may require the application of the failure envelope to each and every fibre direction in turn. The laminates with fibres at ± 55° and at 90°/ ± 30° fall in the latter category. The 0°/ ± 45°/90° quasi-isotropic laminates, on the other hand require the application of the failure envelope only twice.)

* At the micromechanical level, this may well better be characterized by noncatastrophic brittle fracture, followed by elastic behaviour at a lower stiffness, which should stabilize after a limited number of load cycles. (Thanks to Professor Alfred Puck for explaining this to the author.) But this is beyond the prevailing macro-level of thinking.

183

Fig. 2. Combination of maximum-strain failure envelopes for 0° and 90° plies.

It is clear that, based on typical measurements for isolated carbon-epoxy laminae, transverse-tension failure of the matrix is predicted to precede longitudinal failure of the orthogonal fibres. Whether or not this actually happens in the manner assumed is a matter for debate, based on quantified assessments of relevant experimental data. All that is significant here is that straightforward application of this, and many other, composite failure criteria will predict this sequence of failures. Such a prediction is even more likely for fibreglassreinforced polymer composites because these fibres have an inherently higher strain-to-failure than carbon fibres. In reality, transverse matrix cracking never occurs in this manner when there are adjacent layers with orthogonal fibres to prevent the kind of cracking which does occur in their absence. Unfortunately, this difference is masked by the traditional but unjustifiable homogenization of fibres and matrix and the assumed lack of interaction with adjacent plies. For this reason, industrial users of the maximum-strain failure models, with and without truncations, now prefer to ignore the possibility of matrix failures in most cases, because this creates less of a bother than knowingly predicting them wrongly. The issue of when matrix failures occur and whether or not they are significant is complicated by either the neglect or consideration of thermal stresses associated with laminates cured at a high temperature and operated in far colder environments. Most, if not all, published macro-level theories ignore the residual thermal stresses within isolated

184 unidirectional laminae. The theories provide for estimation of residual thermal stresses associated with laminates containing fibres in multiple directions. However, this capability is rarely used within the aerospace industry, because of obviously unacceptable and inconsistent predictions whenever it is. This practice of omitting consideration of thermal stresses is followed throughout the present paper. (As an example of the confusion aroused by incomplete accounting of residual thermal stresses, consider the case of a simple balanced 0°/90° laminate. According to conventional wisdom, there would be no thermal stresses within each ply, if they were cured separately – because each ply is erroneously defined to be homogeneous. The lamination process, however, is believed to develop very high transverse tension stresses in each layer, because of the combination of a high transverse coefficient of thermal expansion for the unidirectional ply with a virtually zero coefficient in the longitudinal (fibre) direction. In other words, each ply, after lamination but before the application of any mechanical loads, has a far higher transverse tension stress in it than the prescribed zero stress in it while it could be considered an isolated entity. Standard computer codes for laminated composites therefore predict that the matrix will crack at a far lower strain in a 0°/90° laminate than in an isolated 90° ply cured and tested at the same temperatures. It does not, of course. Rather than acknowledge that there must therefore be something wrong with the modelling of fibre/polymer composites at the macro level, it is customary to simply omit calculation of residual thermal stresses, so as not to expose the error (whatever it might be). In a similar vein, a 0°/90° laminate made by curing two unidirectional pre-preg plies is considered to have high residual thermal stresses, while an equivalent laminate made from adding resin to an uncrimped 0°/90° fibre preform is presumed to be stress free, because each such bidirectional layer is erroneously treated as an individual homogeneous lamina with fibre-dominated strengths in both directions. (The same absurdity arises in the case of woven fabrics, pre-pregged or resin-transfer-moulded.) Omission of residual thermal stresses in this paper should not be considered a justification that they should be ignored. It is only an acknowledgment that conventional usage of the maximum-strain failure models for fibre/ polymer composites which are assumed to be homogeneous, whether rightly or wrongly, leads to so many dilemmas that it has become standard practice in the aerospace composites industry to ignore the issue. Doing so avoids confronting the experts who derived these and other theories and who have made available easy-to-use computer codes for all those who have elected to use them without questioning their validity for what are, in reality heterogeneous composites of materials. There have been two widely different traditional approaches to the issue raised in Fig. 2. One, followed by the McDonnell Aircraft Company in St. Louis, USA, and many other aerospace organizations, in their proprietary analysis methods, is to disbelieve the predictions of matrix transverse-tension failures for carbon/epoxy-type laminates, when they are based on properties measured on isolated unidirectional laminae, and to arbitrarily raise the transverse strain-to-failure and set it sufficiently high that the longitudinal failure of transverse fibres will govern the predictions instead. (The author is in favour of this approach, with a refinement of reducing the transverse ply stiffness from the initial tangent value to a lower secant-modulus value more appropriate for the apparently nonlinear behaviour of ductile matrices near failure. There is strong experimental evidence to support this philosophy. When the 90° carbon/epoxy plies have been supported by low-modulus longitudinal carriers, such as 0° glass fibres or ± 45° carbon plies, the 90° plies do not crack until far higher strain levels than those needed

185

Fig. 3. Typical tensile load/strain curve for 90° carbon/epoxy tapes stabilized by ± 45° cross-plies.

to fail isolated 90° plies, as is shown in Fig. 3. In some instances there have been no such cracks until strain levels far in excess of what a 0° carbon fibre could have withstood.) The other approach, to which the author objects most strongly, is to believe the predictions of matrix failures based on measured unidirectional lamina properties and then seek to modify the predictions in such a way as to imply that they never were in any way erroneous. Many in academia, far too many to cite, have followed this approach as also have some in industry who have accepted available computer codes without attempting to understand what they really represent. This second approach is most commonly referred to as progressive failure theory, or ply discounting. First-ply-failure predictions in tape laminates are over-ridden by subsequent re-analyses with arbitrarily changed matrix-dominated lamina properties until a last-ply failure prediction is reached such that the matrix properties no longer determine when failure occurs. (This stratagem does not work so well with cloth laminates, in which the fibres are predicted to fail first, because in the absence of any cue to differentiate between the kinds of plies, at least one computer code proceeds to compute a last-ply failure strength on the basis of matrix-dominated properties instead.) There is a conspicuous absence of explanations as to how to identify whether the matrix or fibres fail first in an allegedly homogeneous ‘composite material’ with no identifiable separate constituents. Some may argue that, for tape laminates at least, there is little difference of consequence between these two approaches, since both ultimately predict laminate failure at fibredominated strain levels while rejecting any and all predictions of earlier failure based on exceeding matrix-strain limits for isolated unidirectional laminae. However, this is true only for the monotonic application of a single load until failure. For a complete structure, capable of being subjected to subcritical biaxial loads in varying sequences before final failure, the issue of whether or not the matrix really cracks is of vital significance. If the cracks in the matrix were real, they would necessarily decrease the compression strength of fibres parallel to those cracks for any subsequent loads. This is true both for real microcracks which stabilize

186

Fig. 4. Constant-transverse-stress cut-offs for intralaminar matrix cracking in unidirectional laminae.

once they have reached a density matching the geometry and applied loads, and for cracks which are detectable at the macro level. Consequently, since there can be no control of the sequence in which loads can be applied to a structure as complex as an aircraft or rocket, it is necessary to set the design ultimate strength at the level of the first-ply failure prediction. No credit can legitimately be taken for any higher predicted last-ply failure strengths. The most curious thing about the predicted first-ply failures in the form of matrix cracks is that they are customarily presumed to miraculously heal themselves before the application of the next load case. Neither of these two approaches to intralaminar matrix cracking (ignoring the issue or calculating fictitious strengths) can be scientifically rigorous. The traditional constanttransverse-strain model of matrix cracking with the original maximum-strain failure model should be replaced by a constant-transverse-stress limit as part of both the original and the truncated maximum-strain failure models. Such a constant-stress cut-off is acknowledged to not be universal but to change with the fibre pattern, as shown in Fig. 4, which addresses cracking between 0° fibres in a ply which can either be isolated or embedded in various multidirectional laminates. This cut-off is a straight line, with a slope defined by the longitudinal Poisson’s ratio vLT of the unidirectional lamina. (There is zero transverse stress on such a lamina when subjected to purely longitudinal stresses. Any variation in longitudinal stress while maintaining a constant transverse stress is characterized by a parallel line.) The need to use fracture-mechanics equations to analyze such matrix failures, rather than continuing with the far simpler (but physically unrealistic) strength-of-materials approach,

187 was recognized by diligent researchers on both sides of the Atlantic many years ago, as noted in Ref. 7. Unfortunately, this need has yet to be addressed by most of those who have developed strength-prediction computer codes for composite materials. The need to truncate the fibre-dominated predictions of in-plane shear failures, when using the original maximum-strain failure model for composites, has been recognized by many researchers over the years. Not all such modifications are identical, but the need for such modifications has been accepted at the following American aerospace companies: Northrop, Douglas, McDonnell and Lockheed, at least.* (Curiously, the reasons are not all the same. At Douglas, the cut-off was imposed in the belief that the fibres actually failed under biaxial loading; at Northrop and at Lockheed it was introduced to limit the shear strain in the matrix for laminates without fibres in sufficient directions to do so.)9 The most widely used truncation is that shown in Fig. 5, with a 45° sloping line passing through the greater (tensile or compressive) strains-to-failure for unidirectional laminae subjected to uniaxial stress. The other 45° sloping line is equidistant from the origin. (Although the author has published several papers justifying why the slope should be 45° (at least for carbon-epoxy type composites), in the context of the truncated maximum-strain failure model the 45° slope for the truncated maximum-strain failure model is purely the result of empirical wisdom; the 45° cut-offs had been established before he formulated his generalization of the maximum-shearstress failure criterion on the strain plane.) While not undercutting predicted uniaxial strengths or biaxial strengths for stresses of the same sign, it imposes a limit on fibre-dominated laminate strengths for biaxial stresses of opposite sign. It can also impose a limit on the design ultimate shear strain in the matrix for any laminate lacking fibres in a sufficient number of directions to be totally fibre-dominated. Otherwise, the matrix could be subjected to such high stresses as to develop fatigue cracks – a situation which cannot arise in a totally fibredominated laminate. The need for some cut-off of this nature results from the fact that, without it, the in-plane shear strengths of ± 45° carbon-fibre-reinforced polymer composite laminates predicted by the maximum-strain failure model are almost twice as great as the highest measured strengths. Given the difficulty of performing good shear tests, the discrepancy is even greater in most cases. This cut-off is consistent with the author’s generalization of the maximum-shear-stress (Tresca) criterion to carbon fibres and the like,10 which can be looked upon as a scientific justification of why such a truncation was needed in the first place. It should be noted that the same 45° cut-off in the tension-compression quadrants on the strain plane applies to the 0° fibres in either a unidirectional or bidirectional lamina, no matter what the total laminate fibre pattern. The cut-off is located by the Poisson’s ratio for an axially loaded isolated unidirectional lamina because this particular point represents the absence of both transverse (2nd direction) and normal (3rd direction) stresses in the fibres, at least in the absence of thermal stresses. (When a unidirectional lamina is loaded longitudinally, parallel to the fibres, there can be no stresses in either the transverse or normal direction.) It is incorrect to use the very low * The model developed even earlier, at Grumman, was based on the maximum-stress, rather than maximum-strain model but, like those developed at these other companies, was noninteractive and similarly selectively truncated. This pioneering work was performed by Dick Hadcock and his colleagues (see, for example, Ref. 8). Significantly, as long ago as 1969, they published work on the difficulties of making and testing advanced composite coupons.

188 Poisson’s ratio of a 0°/90° cloth to establish the corner points, even for laminates made exclusively from biwoven fabrics, because a unidirectional load on such a laminate actually creates a biaxial stress in all of the fibres. Indeed, it is scientifically incorrect to regard a 0°/90° layer as a lamina for any state of stress, even though doing so overcomes the even greater error of homogenizing fibres and matrix in unidirectional laminae. (The use of bidirectional laminae works well for predicting the fibre-dominated strengths of multidirectional laminates made from such fabrics, and several visibly empirical analysis models benefit from having fibre failures in all four uniaxial load cases. However, it precludes all possibility of predicting any real matrix failures whenever the analysis is based on the measured uniaxial strengths of such ‘laminae’.) Figure 5 includes the same level of transverse strain-to-failure as is used with the original (untruncated) maximum-strain failure model, introduced by the properties of transverse plies. It also shows the next level of transverse strain-to-failure limits imposed by fibre failures for the longitudinal ply under consideration. The 45° cut-offs must be equidistant from the origin

Fig. 5. Truncations of original maximum-strain composite failure model for tension-compression (shear) loads.

189 and are passed through the higher of the measured longitudinal strains-to-failure in tension and compression. The reason for this is that, with the newer small-diameter carbon fibres, failure under axial compression usually occurs because the matrix is incapable of stabilizing the fibres sufficiently well to develop their intrinsic strength. Older carbon fibres, of greater diameter, had the same tension and compression strengths, as one would expect for sheardominated failures.11 It is possible that, for fibres failing by brittle fracture under tensile loads, the shear-failure cut-offs should be moved even further apart. However, in the absence of a pure in-plane shear test justifying raising this strain-to-failure, the best that can be done is to base the cut-offs on the highest test result which can be substantiated. In the same vein, it should be noted that omitting the 45° cut-offs, which some regard as empirical, can be justified only by measured in-plane shear strengths sufficiently high to prove that they do not apply. Since this is a fibre-dominated cut-off, it requires shear on a ± 45° laminate (or equivalent orthogonal tension and compression on a 0°/90° laminate). The theories described above are applied to predict the behaviour of a number of laminates provided by the organizers of the failure exercise.4,12 Full details of these laminates, materials, lay-ups and loading conditions are described in Ref. 4. In solving the problems cited by the editors, nonlinear effects have been accounted for by using secant values of the lamina stiffnesses appropriate for each strain level. Thermal stresses are not considered; this omission is customary in the US aircraft industry. All solutions proceed through a strain-based formulation first, even though the answers have been requested to be plotted in the stress space, because only by this means is it possible to identify which constituent of which ply fails first. The strain-transformation and stress/strain equations used are those given by Jones.13 3. Test problem No. 1: Biaxial (␶xy ⴚ ␴y) failure envelope for unidirectional glass/epoxy lamina The request by the editors to ‘predict’ the strength of a unidirectional lamina under biaxial loads may seem inappropriate because, at the macro level, the lamina properties are the data needed to analyze laminates. Nevertheless, it is appropriate to at least indicate the form of interaction which would be expected for these two failure models. (In the world of micromechanics in which some of the participants in this exercise have specialized, the goal is to predict this failure envelope, hopefully accounting for effects which are traditionally overlooked at the macro level.) The particular section cut through the failure envelope, in-plane shear as a function of transverse tension or compression, should be entirely matrix dominated. There should be evidence of two distinct failure modes – matrix cracking between the fibres under predominantly transverse-tension loads, and ductile shear failure under the influence of inplane-shear and transverse-compression (or small transverse-tension) stresses. The failure envelope for the basic and truncated maximum-strain failure models is easily established, since both of these models are noninteractive. This particular portion of the failure envelope is not affected by the truncation for fibre failures under orthogonal tensile and compressive stresses of opposite signs. Figure 6 shows the failure envelopes for these two models, using the data taken from Ref. 4. It should be noted that these are for isolated unidirectional laminae and bear little relation to the models needed to characterize the behaviour of the same laminae when embedded in

190

Fig. 6. Solution to test problem No. I (unidirectional E-glass/LY556 epoxy lamina).

multidirectional laminates. Figure 7 shows the corresponding in-situ failure envelopes which would be used as part of laminate analysis. The in-plane shear strength is unchanged, but the longitudinal tensile and compressive strengths are increased to tL
(ET/EL) = 377 MPa and cL
(ET/EL) = 187 MPa, respectively. (What the author would actually predict on the basis of a mechanistic failure model allowing for interactions between the stress components affecting

Fig. 7. In-situ properties for analyzing laminates containing unidirectional glass/epoxy layers.

191 ductile shear failures, and a separate constant-transverse stress cut-off matrix cracking, is given in a companion paper.6 See also Ref. 7.) The substantial differences between Figs 6 and 7, in regard to transverse-tension stresses, reflect what is needed to prevent any strength-prediction methods (not just these two) from yielding obviously unacceptable first-ply failure predictions through applying Fig. 6 outside its domain of relevance, for an isolated lamina not embedded in any laminate. The transverse strain limits in Fig. 7 have been established for assumed transverse fibres the same as the 0° reference fibres, but matching different values could be deduced to match the strains-to-failure of nonidentical fibres as necessary. 4. Test problem No. 2: Biaxial (␶xy ⴚ ␴x) failure envelope for unidirectional carbon/ epoxy lamina Like the first, the second problem posed by the organizers is unaffected by the distinction between the original and truncated maximum-strain failure models, because it is a single cut through the failure envelope which does not involve transverse in-plane stresses of the opposite sign. According to these models, the in-plane shear strength of the matrix interacts with neither of the longitudinal lamina strengths. (Again, what the author would predict on the basis of his own deliberations is reported in the companion paper.5 However, whereas there is a substantial difference for the preceding problem, as explained in Ref. 7, there is virtually none for this one.) Figure 8 shows the failure envelope on the stress plane, for these failure models, using data supplied by the organizers of the failure exercise.4 Whereas Fig. 6 needed to be extensively modified to represent the in-situ properties of such a lamina, the information in Fig. 8 can be used directly for both isolated and embedded laminae. Figures 6–8 are precisely rectangular on the stress plane, because the criteria are noninteractive. However, there are slopes for some of the lines on the stress and strain planes for the next problem. These slopes are caused by Poisson effects, one set of which is very small because of the typically very low values of the transverse Poisson’s ratio vTL.

Fig. 8. Solution for test problem No. 2 (unidirectional T-300/914C carbon/epcxy lamina).

192 5. Test problem No. 3: Biaxial (␴y ⴚ ␴x) failure envelopes for unidirectional glass/epoxy lamina While generating the L  T ‘failure envelope’ for a 0° lamina is again more like establishing a theory for analyzing laminates than it is a ‘prediction’ for laminae, at least in the world of macro-mechanics, it is necessary to show how these failure models are actually applied. In this case, both of the distinctions between the original and truncated maximum-strain failure models are evident and, as in Figs 6 and 7, a distinction must be drawn between the characterizations of these laminae in isolation and when embedded in a multidirectional laminate. (The author’s more precise model discussed in the companion paper,6 is bounded between the two results given here, for both carbon and glass fibre reinforcement.) Figure 9 contains the predicted failure envelopes for unidirectional laminae according to these models, using data from Ref. 4. The differences between the isolated and in-situ properties are substantial, but the further differences resulting from the truncations for shear failures of the fibres are quite small, being confined to the lower right corner of the envelope. The 45° slope of the cut-off on the lamina strain plane would be modified for the lamina stress plane in the ratio of the transverse to longitudinal moduli, with allowance for different unidirectional strains-to-failure. In this case, the transverse lamina strength, in tension and compression, per the truncated maximum-strain failure model, would be (16.2/45.6)
(1.278/1.09)
1280 = ± 528.90 MPa. Strictly, a lower secant modulus should be used for the transverse stiffness, as is done here in subsequent exercises. There is no sloping cut-off for

Fig. 9. Solution for test problem No. 3 (unidirectional E-glass/MY750-epoxy lamina).

193 compressive loads in this case, because the glass fibres are far weaker in compression than in tension, so the shear failures cannot be developed. (It is established in Ref. 5 that this particular cut-off should be applied to the carbon/epoxy laminates but not to those reinforced by glass fibres. This cut-off is applied to all problems considered in this paper, to show the effects rather than to justify the procedure.) The transverse in-situ properties have been deduced by combining the actual lamina transverse stiffnesses with the longitudinal strain-to-failure of assumed orthogonal fibres of the same kind as in the reference lamina. Effectively, the longitudinal strengths have been scaled by the ratio ET/EL = 0.355. Hence the 455 and –284 MPa transverse strengths in place of the measured 40 and –145 MPa strengths. As constant-transverse strain lines, the isolated and in-situ transverse strength limits have a small positive slope of vTL = 0.099. Since the actual failure mechanism under transverse-tension loads is by matrix cracking, however, a constant-transverse-stress cut-off would be more realistic. Nevertheless, the objective of this exercise is to characterize the popular maximum-strain failure models as they are used, even if some details of the basic model could be improved upon. In concluding this characterization, rather than prediction of the failure envelope for a 0° glass-epoxy laminate, the author would remind the reader that it is NECESSARY to distinguish between fibre and matrix analyses by using SEPARATE equations. Interactive failure models for homogenized ‘equivalent’ materials are inappropriate for fibre/polymer composites. Only a mechanistic failure model can suffice – and the governing phenomena for laminae when they are isolated and when they are embedded within a laminate are very different. This need for separate characterization of fibre and matrix failures was recognized long ago by Puck (e.g. Ref. 14), who has made invaluable contributions to predicting the strength of fibreglassreinforced composite laminates in particular. His extensive testing has also confirmed the presence of the two distinct failure mechanisms for the matrix.

6. Laminate failure envelopes on the strain plane Solution of the remaining problems is facilitated by first characterizing the relevant failure envelopes on the laminate strain plane. These strains can then be converted to laminate stresses to depict the solutions in the form requested. The need for the intermediate step derives from the far greater clarity in identifying which constituent becomes critical first, and in which layer of the composite. There is no straightforward way of telling whether a transverse stress in a matrix is more or less critical than a longitudinal stress in a fibre oriented in a different direction, even for uniaxial loads. The situation for biaxial loads is far more indeterminate. Figure 10 shows the failure envelope for the original maximum-strain model, as used in industry, for any combination of 0° and 90° plies, whether a combination of unidirectional layers or parts of bidirectional woven fabrics. It also includes the 45° cut-offs in the lower right and upper left corners for the truncated maximum-strain failure model. Every facet of the failure surface is a flat vertical plane through these lines, with the exception of two flat horizontal planes defining the lamina in-plane shear strain limit, one above the base plane and the other an equal distance below. The entire envelope can be defined by only three measured strains-to-failure; the longitudinal strains in tension and compression along the length of the fibre, and the in-plane shear strain between the fibre axis and its normal. The truncations need, in addition, the major Poisson’s ratio vLT for the unidirectional lamina.

194

Fig. 10. Original and truncated maximum-strain failure models for fibre-polymer composite laminae and laminates.

Even if the laminate contains only layers of bidirectional woven fabric, the Poisson’s ratio needed for each such layer is that for an equivalent unidirectional ply, NOT that for a cloth. This distinction is absolutely vital for both the truncated maximum-strain failure model and for the author’s generalization of the maximum-shear-stress failure criterion. The issue does not arise for the maximum-strain failure criteria as used for woven fabric plies by much of the aerospace industry, which usually works well for fibre-dominated failures in laminates containing no unidirectional tape plies. Better yet, since it automatically suppresses the prediction of matrix failures (because none can be detected while measuring the bidirectional ‘lamina’ properties) such a technique actually yields far fewer erroneous answers than are generated for laminates made entirely or partially from tape layers. Decomposing a cloth layer into its equivalent unidirectional laminae is straightforward. If the cloth has a plain weave, each equivalent layer is located at the same mid-plane. If there is a bias between the 0° and 90° plies on each surface of the cloth, as with 5- and 8-harness weaves, for example, the two tape layers are located in the appropriate top or bottom half of the cloth layer. The in-plane shear properties measured for the cloth are used unchanged for the two tapes. Since the

195 crimping of the fibres along their length does not bend them in the transverse direction, the 90° properties for each equivalent tape layer are set equal to those actually measured on a real tape ply made from the same fibres and resin, except for the need to distinguish between the isolated and in-situ strains-to-failure. The longitudinal strain-to-failure, a fibre-dominated property, is taken to be the same as measured on the cloth layer. The effective longitudinal stiffness E0 is factored from the measured E0/90 modulus by a calculated back-out factor from lamination theory. If E90 is known, or assumed, and E0/90 has been measured, E0 can be deduced by using lamination theory in reverse. Typical back-out factors for carbon-epoxy laminates are in the range 1.80 to 1.85.15 Expressing the failure criteria in Fig. 10 with respect to different axes, using the straintransformation equations given by Jones and others, is almost equivalent to rotating this failure envelope around the equal-biaxial strain axis, through an angle twice as great as between the lamina and laminate reference directions. A minor modification is needed because the radial distances from this axis do not match the same scale in the direct- and shear-strain axes. A 0° lamina-is defined here to have fibres in only the ‘1’ axis direction. A bidirectional woven cloth layer is not really a lamina, even though it is often regarded as such, but as long as no attempt is made to predict matrix failures it can be considered as having fibres in both the 0° and 90° directions. Defining by  the angle traversed moving from the laminate axes (x–y) to those in the lamina (1–2), the necessary strain transformation relations can be derived from Jones’s work. On the base plane of the laminate failure envelopes, there is zero shear strain xy and, on parallel planes, xy = constant = . Using the straintransformation equations in Ref. 13, it follows that the in-plane shear strain acting on the lamina is given by

12 =

1 [ ¯  2 sin  cos (1  2)] (cos   sin2 )

(1)

2

Substituting this equation back into the strain-transformation relations yields the following:

再冎

x 1 = 2 y (cos   sin2 )

冋

cos2   sin2 

 sin2  cos2 

 sin  cos  sin  cos 

册

1 2

¯

(2)

for all angles except  = ± 45°, for which the denominator would be zero. Only the normal lines through points B and E in Fig. 10 can be transformed to lie on the base plane for a ± 45° laminate, as lines through those same points perpendicular to the equalbiaxial-strain axis (i.e. sloping down to the right). All other points on the failure envelopes in Fig. 10 lie off this base plane for a ± 45° laminate. Vertical lines through any point on the equal-and-opposite (–45°) strain diagonal GH in Fig. 10 (sloping down to the right) will intercept the xy axis at the points

xy = ±

 , xy = ± 2 when  = 45° sin  cos 

Here,  is the magnitude of the 1 and 2 strains on that diagonal.

(3)

196 Additional formulae are needed to locate the intercepts on the xy = LT = 0 laminate plane equivalent to planes for which 12 = constant in the lamina reference system. These can be shown to be at

y = x ±

LT 2 sin  cos 

(4)

The lamina shear-strain plateaux are thus transformed into two lines parallel to the equalbiaxial-strain axis (i.e. sloping up to the right). This same lamina matrix shear-failure plane would intersect the laminate shear failure axis at a height given by

xy = ±

LT cos2   sin2 

(5)

With reference to Fig. 10, points B and E (and any other point on the same diagonal) on the lamina strain plane will remain at the same location in the laminate strain plane. The vertical lines BG and EI in Fig. 10 can be shown, by means of eqn (2), to cross the x axis at laminate strains of 0/cos2  and the y axis at 0/sin2 , where 0 has the value tL or cL, as appropriate. (Here, cL is defined to be the positive numerical value of the compressive strain.) The before and after co-ordinates of all other points can be characterized similarly. However, the task of actually solving these problems has shown that not all are necessary. Each facet of the failure envelope in Fig. 10 is defined by straight lines, for which any two points will suffice to define them. The combined failure envelope for laminates with fibres in several directions will be established by the intersections between these sets of lines for each fibre direction, and the resulting corner points will usually not be shown in Fig. 10. The most useful points with which to construct these grids of intersecting lines are those passing through either of the ± 45° axes in Fig. 10, i.e. points B, G, E and H. Points A and D are also of use. The co-ordinates of these points in the rotated co-ordinates are as follows: Point A:

x = tL[cos2  + vLT sin2 ]/(cos2   sin2 ) y =  tL[sin2  + vLT cos2 ]/(cos2   sin2 )

(6)

Point B:

x = y = tL

(7)

Point C:

x =  tL[vLT cos2  + tL sin2 ]/(cos2   sin2 ), y = tL[vLT sin2  + tL cos2 ]/cos2   sin2 )

Point D:

(8)

x =  cL[cos2  + vLT sin2 ]/(cos2   sin2 ), y = cL[sin2  + vLT cos2 ]/(cos2   sin2 )

Point E:

x = y =  cL

Point F:

x = cL[(1vLT)cL cos2 ]/(cos2   sin2 ), y = cL[(1vLT)cL sin2 ]/(cos2   sin2 )

(9) (10)

(11)

197 Point G:

x =  y = tL/(cos2   sin2 ),

xy = ± tL/(sin  cos )

Point H:

x =  y =  tL/(cos2   sin2 ),

xy = ± tL/(sin  cos )

Point I:

(19)

x = (1 + vLT)tL sin2 /(cos2   sin2 ), y =  (1 + vLT)tL cos2 /(cos2   sin2 )

Point S:

(18)

x =  (1 + vLT)tL cos2 /(cos2   sin2 ), y = (1 + vLT)tL sin2 /(cos2   sin2 )

Point N:

(17)

x =  (1 + vLT)tL sin2 /(cos2   sin2 ), y = (1 + vLT)tL cos2 /(cos2   sin2 )

Point M:

(16)

x = (1 + vLT)tL cos2 /(cos2   sin2 ), y =  (1 + vLT)tL sin2 /(cos2   sinj2 )

Point L:

(15)

x =  cL  [(1 + vLT)tL sin2 ]/(cos2  sin2 ), y =  cL + [(1 + vLT)tL cos2 ]/(cos2   sin2 )

Point K:

(14)

x = [tL cos2  + cL sin2 ]/(cos2   sin2 ), y = [tL sin2  + cL cos2 ]/(cos2   sin2 ),

Point P:

(13)

x =  [cL cos2  + cL sin2 ]/(cos2   sin2 ), y = [cL sin2  + tL cos2 ]/(cos2   sin2 ),

Point J:

(12)

(20)

x =  y = ± LT/(4 sin  cos ),

xy = ± LT/(cos2   sin2 )

(21)

The constant shear strain plane in Fig. 10, containing point S, will transform as two parallel 45° sloping lines, for ±  laminates, intersecting the x and y axes at four equal points given by eqn (4) as Intercept on x and y axes = ± LT/(2 sin  cos )

(22)

(For 0° and 90° plies, there will be no such intercepts; the shear-strain limits will be parallel to the base plane, as shown in Fig. 10.)

198 Other than sometimes making special provision for the fact that the lines BI and EJ in Fig. 10 are empirically located, and cannot be established by direct measurement, the use of the preceding strain-transformation data is straightforward and unambiguous. 7. Test problem No. 4: Biaxial (␴y ⴚ ␴x) failure envelopes for (90°/ ± 30°)s glass/epoxy laminate The derivation of the solution to this problem must start on the strain plane, using the results of the previous section, to ensure compatibility of deformations while identifying which constituent fails, and how, for each state of biaxial stress. The first problem posed is the section cut through the complete failure envelope for which there is no in-plane shear stress. The results for both failure models can be combined in a single diagram, to show clearly whatever differences exist between the two. Fortunately, in this case, only the diagonal cutoffs for the 90° plies are operative, so the added complexity in the final stress-based failure envelopes is not great. Figures 11 and 12 show, to the same lamina-strain scale, the failure possibilities for 90° and ± 30° layers, in both the fibres and the matrix. Figure 12 includes two additional views of what is really a three-dimensional failure surface for the ± 30° plies. The in-plane shear strains associated with in-plane direct strains for certain modes of failure are needed for the solution of the next problem, a particular section cut through the failure envelope for this laminate. The heights ( xy values) in these additional views have been established by use of eqn (3), for equal and opposite strains  at points G and H (Fig. 10), and eqns (4) and (5) in conjunction with point S in Fig. 10. There is no such need for additional views for the 90° plies in Fig. 11, because all walls are vertical and the ‘roof’ is horizontal. It is clear that the empirically set transverse strain lower limits for matrix failures of the 90° plies (equal to the strains-to-failure of a zero-thickness 0° ply) truncate all predictions of failures of the ± 30° fibres which are not already preceded by failures of the 90° fibres, themselves. This observation is true for both failure models and creates a curious dilemma, because the transverse matrix strains have been set at this limit only in order to prevent computer analyses from predicting premature failures which would not occur. This setting of the matrix strains-to-failure equal to the longitudinal fibre strains-to-failure does not actually imply that the matrix really will fail then. This situation cannot arise for any laminate in the standard 0°, ± 45°, 90° family for which there are fibres in all of these directions, so the issue has not been addressed in preparation of the usual computer codes. On balance, it seems to the author better to allow the laminate failure envelopes to be truncated by real fibre failures in the ± 30° plies rather than by an arbitrary estimate of matrix failures in the 90° plies, even though available computer codes in the aerospace industry would do the opposite. A very real concern about this, however, is that 0° tensile loads on the laminate would induce axial compression in the 90° fibres which, once the matrix between them really was extensively cracked, would be unable to withstand such loads. With proportionally so few 90° fibres, these are predicted to become critical before the ± 30° fibres for a 0° tensile load. The laminate would then be reduced to the lesser stiffness of the ± 30° fibres alone. Figure 13 shows the single strain-based failure envelope for the combination of these plies, with definitely unattainable failure deleted. The truncations for 90° fibre failure under orthogonal loads of opposite sign could cause instantaneous failure of the remainder of the

199

Fig. 11. Failure envelope for 90° unidirectional laminae.

laminate, depending on whether the situation was load- or displacement-controlled. The results of calculations in this regard are reported later in the paper. Shear failure in the matrix between the fibres in the ± 30° plies are predicted to lie just outside the predictions of fibre failures by the original maximum-strain failue model, near the upper-left and lower-right corners of the basic parallelogram.. Had Fig. 13 been completed according to the customary use of these failure criteria in the US aerospace industry, the right and left sides of the envelope would not have been closed by the sloping lines shown, representing failure of the ± 30° fibres. Instead, there would have been vertical cutoffs, through the equal-biaxial strain points B and E, for matrix failures in the 90° plies. These would represent the traditional arbitrary limits set on the transverse strengths of the 90° plies according to the original maximum-strain failure model. They would have been even more restrictive had the measured transverse ply strains-to-failure been used instead. It is the author’s belief that the intent of setting higher transverse strain limits has always been to avoid predicting premature matrix failures. On the other hand, the

200

Fig. 12. Failure envelope for ± 30° laminate (unidirectional tape plies). (a) Failure envelope in strain plane for ± 30° laminate (unidirectional tape plies). (b) Section A-A of the failure envelope for ± 30° laminate (unidirectional tape plies). (c) Section B-B of the failure envelope for ± 30°C laminate (unidirectional tape plies).

201 corresponding limits for matrix failures in the ± 30° plies would lie outside the fibre failures shown in Fig.13. They would pass through the equal-biaxial-strain points B and E with a slope of –1/3. These potential failures are therefore considered inoperative for this laminate, which is predicted to be totally fibre dominated under in-plane loads. The lamina strains-to-failure provided in Ref. 4, and necessary derived properties, are as follows:

TL = 0·02132, cL = –0·01065,

TL = 0·038,

vLT = 0·278,

= (1 + vLT), –vLTtL = –0·005927,

vTL = 0·092,

tL = 0·02725, and

 cL = 0·01660

It remains to convert these strain-based failure envelopes into stresses, as requested by the organizers.4 Doing so requires the generation of a stiffness matrix; evaluated here by using equations given by Jones.13 The resulting expressions are as follows:

Fig. 13. Failure envelope on the strain plane for (90°/ ± 30°) E-glass/LY556 epoxy laminate.

202 A11 1 = {[16E2] f90 + [9E1 + 6(v12E2 + 2G12) + E2] f ± 30}, t 16 A12 1 = {[16v12E2] f90 + [3(E1 + E2  4G12) + 10v12E2] f ± 30}, t 16 A22 1 = {[16E1] f90 + [E1 + 6(v12E1 + 2G12) + 9E2] f ± 30}, t 16

(23)

A66 1 = {[16G12] f90 + [3(E1 + E2  2v12E2  2G12) + 10(G12)] f ± 30} t 16

where  = 1  vLTvTL and f.. represents the fraction of fibres in the appropriate direction. Here, f90 = 0·1727 and f ± 30 = 0·8273. Since the failure envelope has been established on the basis of strains-to-failure, the appropriate stiffnesses to be used in the analysis are all secant moduli. These should all be evaluated for the particular strains in each ply at each point on the perimeter of the failure envelope, and not at some lesser or greater strain level. However, the complexity of doing so is far beyond normal practice. The most that has ever been done, of which the author is aware, is to establish two sets of properties, one for ultimate strain and the other for the lower strain levels due to limits imposed by bolt holes, cut-outs, and impact damage. (Even that is imperfect, because not all fibres become critical simultaneously at all points around the failure envelope. The problem is particularly easy to explain for this laminate. At the equal-biaxialtension point A in Fig. 10, all plies are equally strained in both directions. A single set of transverse properties would apply to both fibre directions. The same is true at point E, although the properties themselves would have changed. The real problems arise whenever the longitudinal and transverse strains are not equal in magnitude and of the same sign. In all of these cases, the ideal set of secant-moduli stiffness would change with every situation. No single representation of properties would be ideal along lines BAG and EDC in Fig. 10, for example. For better or worse, the author has elected to prepare the failure envelopes using totally tension-dominated properties. A justification for this philosophy is to be found by examination of the complete finite-element analyses for composite structures. While it would be only tedious, rather than impractical, to vary the lamina properties as a function of specified combinations of in-plane strains, another step of the analysis involves the inversion of stiffness matrices when the state of strain is not known. Practical analysis techniques favour the use of the most representative single set of lamina stiffnesses. Known matrix nonlinearities militate against the use of initial tangent values. Fortunately, for fibre-polymer composites, the greatest uncertainty is associated with the least dominant properties – provided that reasonable steps are taken to preclude the prediction of premature failures by matrix strainsto-failure which are too low when assessed in terms of the in-situ properties. The longitudinal modulus E1 (EL) is linear to failure in both tension and compression. It is given as 53·48 GPa in Ref. 4. Since the transverse strain-to-failure measurements were limited to the premature failure of the lamina by brittle fracture, which is not representative of its behaviour in the laminate, it will be necessary to assume the nonlinear form of the remainder of the curve. Given the constraint of not supplementing the data from other sources, the best available hypothesis is

203 that the unmeasured nonlinear behaviour (softening) will be proportional to that of the unreinforced matrix in shear. (Actually, it probably would not soften quite as much under transverse tension loads, because of the biaxial stresses induced by the fibres.) The factor relating direct and shear strains is 12 = (1 + v12)1 so, at an axial strain of 0·02132, the shear strain would be 0·02725. The ratio of this secant shear modulus to the initial tangent value follows from interpolation of the data in Ref. 4 as 0·4331, so that the effective transverse modulus E2 (ET) will be taken to be E2 = 0·433
17·7 = 7·67 GPa. The corresponding in-plane shear secant modulus GLT is given as G12 = 68·173/ 0·027 = 2·525 GPa. The major Poisson’s ratio vLT is given as v12 = 0·278, corresponding to which vTL = 0·0399. The associated value of  is 0·989. The laminate stiffnesses of this laminate then follow from eqns (23) as A11/t = 29·143 GPa,

A12/t = 9·512 GPa,

A22/t = 17·980 GPa,

and

A66/t = 9·881 GPa

The fibre pattern is significantly stiffer in the 0° (x) direction than in the 90° (y) direction, because there are proportionally fewer 90° plies than would have been needed to make a quasi-isotropic laminate. The results of the calculations are plotted in Fig. 14, including the identification of the critical fibre direction. The stress-based failure envelope is also essentially in the form of a parallelogram, with the two sloping sides defined by the strength of the 90° fibres, and the top and bottom by fibre failures in the ± 30° plies. The figure also includes the sloping top and bottom associated with the arbitrary cut-offs for matrix failures in the 90° plies according to the way the original maximum-strain failure model is used in much of the aerospace industry. These truncations, with respect to the limits set by the ± 30° fibres, are appreciable. Real or not, these design ultimate strain limits would leave the final failure envelope totally defined by the 90° plies, for both the original and truncated maximum-strain failure models. The lower right corner of the basic parallelogram in Fig. 14 has a local flat facet on which the 90° fibres are predicted to fail, according to the truncated maximum-strain failure model. As explained in Ref. 6, this cutoff is not credible for fibreglass-reinforced polymer composites as it is for carbon/epoxy reinforcements. The figure also shows predicted failures of the ± 30° plies by matrix shear, just outside the upper left and lower right corners of the envelope, which would apply for both models. The author has made no attempt to actually predict an initial (matrix) failure envelope for this laminate because to have done so would have endowed these two maximum-strain failure models with capabilities they do not possess. Customary aerospace use of transverse strain limits equal to those parallel to the fibre, for the original maximum-strain model, would limit the design failure envelope in the upper left and lower right corners as shown without actually attempting to predict whether or not the matrix would actually fail there. However, the author recommends disregarding these limits since they are contrary to the intent of the limits established for laminates in the 0°, ± 45°, 90° family. Since the 90° fibres are on the outside and could easily peel away whenever they failed first, leaving the ± 30° plies undelaminated, but with less stiffness than the original laminate, the question arises as to whether or not there could be a real ‘last-ply’ failure in this case, given that it would need higher strains to fail the ± 30 plies. Such calculations, however, predicted

204 that the residual strength would be less than at the ‘first-ply’ failures, for either failure model, assuming that the 90° plies would not fail until either version of the maximum-strain model predicted the 90° fibres to fail. Any secondary failure would then therefore be instantaneous in a load-controlled situation. Under conditions of displacement control, there would be no instantaneous failure, only a loss of stiffness, associated with having only ± 30° fibres remaining effective. These predictions could be very different if it were known that real matrix cracking occurred between the 90° fibres at far lower load levels. This issue is important in regard to the other section cut through the three dimensional failure envelope, along the vertical (x) axis in Fig. 14. Had this analysis retained the conventional empirical cut-offs for transverse loads on the 90° plies or, even worse, had they been set at the inappropriately low values given by tests on isolated 90° laminae, the effects of truncating the original maximum-strain model would have been barely apparent. Applying appropriate transverse-strength limits for the analysis is seen to have a far greater effect on the predicted laminate strengths. Figure 14 confirms that, as with many other laminates, uniaxial tests of this laminate would not be capable of clearly distinguishing between the two variations of the maximum-strain failure model. Biaxial tests, with stresses of opposite sign (–x, + y), could achieve this goal.

Fig. 14. Solution for test problem No. 4 (failure envelopes in stress space for (90°/ ± 30°) E-glass/ LY556 epoxy laminate).

205 8. Test problem No. 5: Biaxial (␴x ⴚ ␶xy) failure envelopes for (90°/ ± 30°)s glass/epoxy laminate This problem represents a second cut through the same three-dimensional failure envelope as contained in the immediately preceding solution. An examination of Fig. 14 for this section cut shows that, for compressive loads, the laminate strength is governed by the almost simultaneous failure of the ± 30° fibres in compression and the 90° fibres under shear, the latter according to the truncated maximum-strain failure model. The original maximum-strain failure model would predict that the 90° fibres were far from critical under compressive 0° loads. For tensile loads, however, both theories predict failure of the 90° plies at almost the same 0° stress. As noted above, despite the prediction that the ± 30° plies would then fail at a higher strain, it would be associated with a lower gross-section stress (load), so secondary failures can be disregarded. What Fig. 14 does not show is the interaction with the xy in-plane shear stresses. This is provided in Fig. 15, which represents quantified estimates of fibre failures without any estimate of matrix failures that might precede them. Again, the reason for this omission of initial failures is that neither of the maximum-strain failure models is capable of predicting them reliably. The effect of the truncation of the fibre failures in the tensioncompression quadrants in the characterization of the unidirectional lamina is evident only for tensile x stresses, being preceded by a prediction of compression failure in the ± 30° fibres for compressive x stresses applied to the laminate. Many different failure mechanisms are involved, or almost involved, depending on the state of combined stresses. Starting with the pure compression load, at the extreme left, there might be the slightest trace of failure of the 90° fibres, according to the truncated maximum-strain failure model. Then, as the applied in-plane shear load is increased, the laminate compressive

Fig. 15. Solution for test problem No. 5 (cross-section of failure envelopes in stress space for (90°/ ± 30°) E-glass/LY556 epoxy laminate).

206 strength diminishes while the ± 30° fibres fail. The plateau of shear strength, when the 90° plies are predicted to fail by matrix shear, lies barely above the strength limit set by the ± 30° fibres, according to both failure models. The in-plane shear strength is decreased as the ± 30° fibres begin to fail by tension, until the 90° fibres are predicted to fail by induced compression (according to the original maximum-strain failure model), or by shear (according to the truncated maximum-strain model, at the right side of the figure. The truncated maximumstrain failure model also predicts lower in-plane-shear strengths associated with failure of the ± 30° fibres under tensile laminate loads. As explained in the companion paper,6 these truncations could well be ignored for glass/epoxy laminates and the like, being necessary only for orthotropic fibres, like carbon. Nevertheless, the degree of any conservatism associated with retaining the cut-offs unnecessarily is seen to not be substantial. Although not requested by the editors, the other section cut through the failure envelope, in the y  xy plane, can also be characterized on the basis of the present analyses. The sides are vertical, for both failure models, defined by 90° fibre failures where shown in Fig. 14. The top is flat and almost horizonal, at a height evident from Fig. 15. The top would have been absolutely horizontal if the top and bottom lines in Fig. 14, for the fibre failures in the ± 30° plies, had been parallel to the y axis. 9. Test problem No. 6: Biaxial (␴y ⴚ ␴x) failure envelopes for ( ± 55°)s glass/epoxy laminate As for the preceding analyses, the first step in constructing the requested y  x failure envelope for this laminate is to derive it completely on the strain plane and then to convert the answers to the stress plane. The necessary lamina properties taken from Ref. 4 are:

tL = 0·02807, cL = –0·01754, vLT = 0·278, LT = 0·04 Using the strain transformation formulae given earlier in this paper, along with the form of lamina failure envelopes shown in Figs 2 and 4 for the original and truncated maximum-strain failure envelopes, respectively, the strain-based failure envelopes for the ± 55° laminate are as shown in Fig. 16. Actually, no distinction need be drawn between the two theories for this fibre pattern, because failure of the matrix under in-plane shear loads is predicted to precede that of the fibres under equal and opposite in-plane tension and compression by a large margin. The failure envelope is a parallelogram, with two opposite sides bounded by fibre failures, in tension or compression, and the other two set by matrix shear failures. Its construction is straightforward, involving only eqns (7), (10), (13), (15), and (22). Figure 16 also includes predictions of transverse tension and compression failures of the matrix, at strains equal to those of the fibres under the corresponding loads and to those corresponding with the specified even lower compressive strain-to-failure based on measured lamina properties. This mode of failure, if it occurred, would truncate the left corner of the failure envelope appreciably, as shown. However, as noted in the discussion about the 90°/ ± 30° laminate, these limits probably do not apply to cross-plied or even angle-plied laminates. They are included only to show what would happen with the blind use of a standard computer code without questioning the appropriateness of such limits. Both of these limits are disregarded here for a number of reasons. If the test laminate used to establish the 90°

207

Fig. 16. Failure envelope on the strain plane for ± 55° E-glass/MY750 epoxy laminate.

properties were only one ply thick, it would have buckled under test. If it were many plies thick, it would have suffered from the lack of stabilization afforded by interspersing the plies in the two fibre directions. (This phenomenon can be appreciated by thinking of stacks of logs, with and without periodic layers of cross beams, and how they collapse under their own dead weight when the pile is too high.) In either case, the measured strength would be too low for when the fibres were embedded in laminates. Also, the residual tensile stress within the resin matrix in each ply would be higher in the laminate than for an isolated ply, requiring the application of more compressive load to the laminate to overcome it before failure could commence. It should be noted, however, that standard coding of both maximum-strain failure models, as used in the US aerospace industry, would recognize at least the larger of these cutoffs for transverse matrix failures at strains corresponding to those at which the fibres would fail, in both compression and tension. Since both of these transverse cut-offs are arbitrary, and there is no obvious reason why compressive instability of the fibres should degrade the matrix compressive strength, the author would recommend setting equal transverse strain limits at the higher value. This would eliminate the cut-off at the left of the failure envelope, leaving only that on the right of Fig. 16 to be debated. (This situation would arise only rarely in the aerospace industry, however,

208 because of the normal design practice of requiring at least a minimum fibre content in each of the four standard directions of 0°, ± 45°, and 90°. For those industries in which this is not standard practice, one should question the empirical aerospace treatment of matrix failures under transverse loads, and seek a better model.) In converting these matrix strains-to-failure into a stress-based failure envelope, the procedure followed here, as in the previous examples, is based on using the lamina secant moduli for the strains at which the fibres are predicted to fail, using the data supplied by the organizers of the exercise.4 The stiffness coefficients equivalent to those in eqn (23) are simpler to evaluate for this laminate, because only one fibre direction is involved. They follow directly from the equations given by Jones,13 and these are shown in eqn (24). The longitudinal modulus E1 (EL) is linear, at 45·6 GPa, all the way from peak tension to peak compression loads, per the peak stresses and strains.4 The transverse modulus E2 (ET) is given to be 16·2 GPa initially, reducing to 12·1 GPa when the isolated lamina fails under transverse compression at a strain of 0·012.4 Since the fibres would not fail until a compressive strain of 0·0175 had been attained, a still lower secant modulus at failure is needed. The only nonlinear characterization given for the matrix at sufficiently high strains is that for in-plane shear, so it must be the basis for deducing the effective transverse tension and compression stiffnesses. A tensile strain of 0·02807 is equivalent to a shear strain (1 + vLT) times as high, i.e. 0·036. At this strain, the shear modulus has reduced to 34% of its initial value. The corresponding reduction in the transverse tension modulus would be to 5·51 GPa. (As a sanity check on this approach, it may be noted that, at the direct compressive strain of –0012 (for which E2 had been reduced to 75% of its initial value), the shear stiffness had been reduced to 66% of its original value. Clearly this approximation is plausible.) The major Poisson’s ratio vLT is given in Ref. 4 as 0·278 = v12. The corresponding minor Poisson’s ratio v12 follows as 0·034. Hence,  = 0·9907. The value of the shear modulus G12 (GLT) to be used is that which would give the most accurate estimate for in-plane shear failure of the matrix, i.e. 1·82 GPa taken at the final point of the stress strain curve provided in Ref. 4, when the matrix is about to fail. This would underestimate the stiffness where the laminate fails near the equal-biaxial-strain diagonal but, at that location, there are no shear stresses anyway. As a point of reference, this is far less than the initial tangent value cited of 5·83 GPa. The laminate stiffnesses can now be calculated per conventional lamination theory, as cited by Jones.13 Should a reader wish to verify the calculations, the following stiffness coefficients were used here: A11 ¯ 11 = 1 [0·1082E1 + 0·4415(v21E1 + 2G12) + 0·4503E2] = Q t  A12 ¯ 12 = 1 [0·2208(E1 + E2  4G12) + 0·5585v21E1] = Q t  A22 ¯ 22 = 1 [0·4503E1 + 0·4415(v21E1 + 2G12) + 0·1082E2] = Q t  A66 ¯ 66 = 1 [0·2208(1  v21)E1 + 0·2208(1  v12)E2 + 0·2403G12] = Q t 

(24)

209

Fig. 17. Solution for test problem No. 6 (failure envelope in stress space for ± 55° E-glass/MY750 epoxy laminate).

Hence, A11/t = 9·77 GPa, A22/t = 23·62 GPa,

A12/t = 10·65 GPa, and

A66/t = 10·92 GPa

The laminate failure envelope on the stress plane is depicted in Fig. 17. (Note that, per the organizers’ request, the x and y axes are transposed from their customary positions.) The failure envelope, omitting the questionable failures in the matrix in Fig. 16, has the shape of a skewed long narrow parallelogram, with the short ends defined by fibre failures in tension and compression and the long sides by matrix failures in shear. This particular fibre pattern is normally used for biaxial loads in pressurized circular cylinders, with y = 2x. This load vector is drawn on Fig. 17. It is clear that, if a liner were used to prevent leakage of contents through matrix cracks, the final catastrophic failure of such a laminate under this pressure load would be fibre dominated.

10. Test problems Nos 7 and 8: Stress–strain curves for ( ± 55°)s glass/epoxy laminate The author’s focus on predicting the strength of fibre/polymer composites has been on generating failure envelopes, particularly on the strain plane. He has little experience with generating the kind of nonlinear stress–strain curves which some of the organizers’ requests would involve. Any attempt to do so would therefore involve considerable speculation. In

210 addition, with no access to a suitable computer code, trying to solve these problems would simply delay the exercise unduly, with little likelihood of providing better information than what he hopes some of the micromechanicians participating in this exercise will provide. Therefore, no solution is provided here for these two test problems.

11. Test problem No. 9: Biaxial (␴x ⴚ ␴y) failure envelopes for (0°/ ± 45°/90°)s carbon/ epoxy laminate As for the preceding analyses, the first step in constructing the requested x  y failure envelope for this laminate is to derive it completely on the strain plane and then to convert the answers to the stress plane for presentation in the requested format. As before, solutions are provided for both the original and truncated versions of the maximum-strain failure model. The data necessary for establishing the strain-based laminate failure envelope are given in Ref. 4. These are that

tL = 0·0138, cL = –0·01175, vLT = 0·28, and

LT = 0·02

Now, since = (1 + vLT)tL = 0·01766 < LT, this laminate is found to be totally fibre-dominated for all points on the failure surface, according to the truncated maximum-strain failure model, but there are small local truncations at the top-left and lower-right corners of the failure envelope according to the original model. These are based on matrix shear failures and are far more likely to be real than predictions based on matrix cracking would be. However, since the primary loads in these regions are borne by the 0° and 90° fibres, the effect on the laminate strength and stiffness is likely to be slight. Traditional use of the original maximum-strain failure model in the US aerospace industry would recognize these matrix shear failures as real, and restrict the design failure envelope accordingly. Figure 18 shows the laminate failure envelopes on the strain plane for this quasi-isotropic laminate. Its construction, based on Fig. 9, is particularly simple, because those points in Fig. 18 not defined by the 0°-lamina failure envelope can be established by inspection through interchanging 0° and 90° properties. The critical fibre directions are marked on each facet. These, also, are self evident in this case. The only differences between the two models are the magnitudes and modes of failure of the horizontal plateaux crossing the in-plane shear axis and the vertical truncations on the base plane for strains of opposite sign. The envelopes are symmetric about the equal-biaxial-strain axis BE. It is most significant that the failure envelope is entirely flat faceted, and that the crosssection is rectangular at all points along the equal-biaxial-strain axis, as shown in Fig. 19. It MUST be rectangular for the cross-sections at the pure shear plane (equal and opposite tension and compression) because, there, all loads associated with the xy deformations alone are confined to the ± 45° fibres while those increments of load associated with only the y = –x strains load up only the 0° and 90° fibres. There is no interaction; the lengths of the other fibres are unaffected. Any lamina failure model which does not predict this rectangular crosssection for the laminate failure envelope is, therefore, in the author’s opinion, physically unrealistic.

211

Fig. 18. Strain-based failure envelope for (0°/ ± 45°/90°)s AS4–3501-6 carbon/epoxy laminate.

The necessary stiffness coefficients Aij for this laminate follow from the equations in Jones’s book.13 A11 A22 ¯ 11 = Q ¯ 22 = 1 [3E1 + 2(v12E2 + 2G12) + 3E2] = = Q t t 8 A12 ¯ 12 = 1 [(E1 + E2  4G12) + 6v12E2] = Q t 8

(25)

A66 ¯ 66 = 1 [(1  v21)E1 + (1  v12)E2 + 6G12] = Q t 8

As before, it is best to use secant values for the elastic constants, established for compatible deformations at the highest stresses. The laminate strengths will be overestimated if the higher transverse stiffnesses associated with linear strains are used instead. Incidentally, the use of strain-based assessments of failure avoids a very common problem with stress-based criteria, particularly for plies loaded transversely. For a given failure strain, a modification of the transverse ply stiffnesses has little effect on the larninate stiffness and does not discriminate between failure or not. On the other hand, if failure is based on stress (without using the appropriate fracture-mechanics procedures), changing the transverse ply stiffness has a big impact on whether or not the ply is predicted to fail. With progressive-failure analyses, it is not at all uncommon to have a ply be predicted to crack, and for its stiffness to be reduced so greatly that it would no longer be predicted to crack during the re-analysis.

212

Fig. 19. Flat-faceted failure envelopes for (0°/ ± 45°/90°)s AS4–3501-6 carbon/epoxy laminate in stram space.

The data supplied for this carbon/epoxy composite indicate significant strain hardening under tensile loads. Therefore, the most accurate estimate of the failure envelope would require that the higher modulus be used for all those fibres which actually became critically loaded around the perimeter of the failure envelope, while the initial modulus should be used for the less-than-critically-loaded fibres. This could be easily accomplished in the present context. However, standard analysis techniques involve other steps, for which the state of biaxial strain is not known a priori. Therefore, only the standard linear model is established here, since that is representative of standard procedures in the aerospace industry. (An exception is made for the equal-biaxial-strain point in tension, to indicate the magnitude of error thereby introduced which could not be avoided except by iterative analyses.) The longitudinal modulus E1 ( = EL) = 1950
103/0·0138 = 141 GPa for biaxial tension loads.4 Reference 4 gives 126 GPa as the initial tangent modulus, so this will be used for the basic failure envelope.

213 The transverse modulus E2 (ET) is given as 11 GPa, and the organizers’ data,4 shows that this would remain true for compressive strains up to 0·01175, at which time the fibres would fail. The nonlinear behaviour of the matrix tabulated beyond the ultimate fibre strains is not relevant for this laminate with fibres in all four directions. Even at the equivalent tensile strain of 0·0138, the matrix would have softened by less than 2%, assuming the same nonlinear behaviour in tension and compression. The value of 11 GPa will therefore be used throughout this exercise. The major Poisson’s ratio is given4 as v12 ( = vLT) = 0·28. The corresponding minor Poisson’s ratio would be v21 = 0·0244. Hence,  = 0·9932. For AS4/3501-6 material, Ref. 4 gives an in-plane shear modulus GLT of G12 = 4·2 GPa at a shear strain of 0·0177, significantly less than the initial tangent modulus of 6·6. The equalbiaxial-strain strengths are unaffected by the value adopted for in-plane shear stiffness, as was confirmed by checking the algebra. (The biaxial strength is given by 
[(1 + v21)E1 + (1 + vl2)E2]/2.) The 4·2 value for the shear modulus will be used throughout these calculations, even though a value of 4·6 would be more appropriate for the slightly lower strains associated with compression failure of the fibres. The laminate stiffnesses can now be calculated, using eqns (30). For the basic failure envelope, A11/t = A22/t = 54·60 GPa,

A12/t = 17·47 GPa,

and A66 = 18·57 GPa, so that Ex = Ey = 49·02 GPa while, for the equal-biaxial-tension point, A11/t = A22/t = 60·65 GPa, and A12/t = 19·48 GPa, so that Ex = Ey = 54·39 GPa. The stress-based failure envelopes are depicted in Fig. 20. Both models are included, to facilitate comparisons. The biaxial tensile and compressive strengths are 995 and 847 MPa in tension and compression, respectively, according to both failure models. (The strain hardening is predicted to increase the biaxial tension strength by some 10%, to 1098 MPa.) The predicted uniaxial tension and compression strengths are 676 and 576MPa, respectively, according to the original maximum-strain failure model, and 656 and 576 MPa, respectively, for the truncated version. In this case, according to the original maximum-strain failure model, the matrix shear failures would be real. They would occur between the ± 45° plies only, at in-plane shear stresses of + 372 MPa, suggesting that the laminate might not fail until the predicted fibredominated shear strength of ± 436 MPa was attained. The corresponding prediction by the

214 truncated maximum-strain failure model is ± 328 MPa for fibre failures, which would rule out possibly higher secondary strength. The most notable difference between this stress-based diagram and the strain-based formulation in Fig. 19 is that it is much more apparent now that the biaxial strengths for this laminate are some 50% greater than the corresponding uniaxial strengths. (This is a very real effect which has long plagued experimentalists seeking to measure the biaxial strengths of composite laminates, since there is no comparable phenomenon for homogeneous isotropic metals. In the case of a ± 45° laminate, the ratio of biaxial to unidirectional strengths is typically in excess of 5-to-1.) The zero-transverse-stress points lie slightly off the intersections of segments of the failure envelope because the Poisson’s ratio for this laminate is slightly higher than that for the lamina (0·30 vs 0·28). Fibres in all four directions are predicted to be equally critical at the pointed ends of the failure envelope, on the equal-biaxial-strain diagonal. The 0° and 90° fibres are predicted to be equally critical along the two 45° sloping lines while, for the other four lines reaching to the equal-biaxial-stress points, the more critical of the 0° and 90° fibres alternates around the perimeter and is identified by the stress axis which each segment crosses (or comes closest to). The ± 45° fibres, which carry the in-plane shear loads applied to this laminate, are most critical on the horizontal and sloping portions of the roof of this failure envelope, which is not shown in Fig. 20, but are depicted in Fig. 19. The differences between the original and truncated maximum-strain failure models are clearly confined to those regions of high stresses of opposite signs, as is to be expected. The

Fig. 20. Solution for test problem No. 9 (stress-based failure envelope for (0°/ ± 45°/90°) AS4–3501-6 carbon/epoxy laminate).

215 differences are significant, but not overwhelming, and are predicted to be further reduced by the limited shear-strain capacity of the matrix. If the matrix were sufficiently ductile not to fail by shear, the predicted difference between fibre failure strengths of as much as 50%, if the optimum ratio between x and y stresses were selected, could have been sufficient to distinguish between the two theories. Even the biaxial test rigs which have been doomed to fail while trying to measure unnotched strengths for biaxial loads of the same sign should succeed with biaxial loads of opposite sign. The comparison here shows that the choice between failure theories cannot be resolved by any other load case. Perhaps this exercise will spark a renewed interest in such tests.

12. Test problems No’s 10 and 11: Stress/strain curves for (0°/ ± 45°/90°)s carbon/epoxy laminate It is only with the greatest reluctance that the author refrains from attempting solutions for these two test problems. It is possible that the use of appropriate secant moduli alone, without postulating any matrix cracking at all, could explain any measured nonlinearities. What motivates this thought is the recognition that nonlinear matrix behaviour would affect the entire length of the test coupon simultaneously and that it would be instantly recognized in the laminate stiffness. Matrix cracking, on the other hand, would occur only locally at first, in the form of widely spaced narrow cracks. The initial effect of isolated cracks on the laminate stiffness would be almost imperceptible for this reason, until the laminate was so extensively cracked that the condition would be obvious. Matrix yielding could produce a kink in a laminate stress/strain curve, but matrix cracking would not. Cracking should be associated with a smooth transition in stiffness from the precracked to postcracked state. The other authors’ analyses of these two problems are therefore awaited with interest.

13. Test problems No’s 12–14: Stress/strain curves for ( ± 45°)s and (0°/90°)s e-glass/ epoxy composite laminate No solution is provided here for these three problems for the reasons stated in relation to test problems nos 7 and 8.

14. Concluding remarks The problems posed by the organizers of the composite-failure-theory-comparison exercise, Refs 4 and 12 have definitely substantiated the need for including ‘nonlinearities’ in parts of the analysis of composite laminates. The readers of the entire set of papers may now decide whether traditional progressive-failure or ply-discounting models were truly needed for carbon-epoxy laminates or if the use here of secant moduli, to reduce stiffnesses without predicting matrix failures, would not have been a better way to characterize matrix-dominated properties. As will be evident from an assessment of those problems which the author could solve and those which he could not complete, his earlier work has concentrated on failure criteria for

216 fibrous composites rather than nonlinearities in stress/strain curves. To a very great extent, this is a consequence of having worked in a field dominated by carbon rather than glass fibres. The need to expand his horizons is now clear. On the other hand, his close focus on mechanistic phenomena rather than arbitrary curve fits for failure criteria has been instrumental in validating some experimental data (unfortunately very little of the total) and in questioning (and in many cases rejecting) the rest of it. Part of his work has focused on developing better test specimens than the current standards and to publicize the need for doing so, if the full potential of composite structures is ever to be realized. The worked examples included in this document, all of which were solved without recourse to a computer code, make it quite clear that relying on macro-level lamina-to-laminate analyses for predicting laminate strengths does not make it necessary to homogenize the composite material properties and neglect consideration of what has historically been regarded by many as the exclusive domain of micromechanics. Nevertheless, the job is still incomplete, particularly in regard to intralaminar residual thermal stresses and the need for fracture-mechanics analyses of matrix cracking and ply-thickness effects, neither of which can be characterized by the traditional ply-by-ply decomposition of a laminate into a series of plies, the strength of which can be established independently of adjacent plies. The omission of these effects for the quasi-isotropic carbon/epoxy tubes made from thin layers and containing no free edges in the test section is not believed to be important, because the laminate is (and is predicted to be) fibre dominated for all states of in-plane stresses. However, for the other materials and fibre patterns, the fibre failures predicted should be regarded only as an indication of the validity or otherwise of conventional aerospace use of the maximumstrain failure models for matrix-dominated modes of failure, in keeping with the spirit of this exercise to compare the merits and limitations of various theories. (It is well known that the author’s personal belief in this regard is that, whenever possible, one should design fibredominated laminates to avoid matrix failures. Most US aircraft companies specify minimum permissible fibre contents of 10 to 15% in each of the standard 0°, + 45°, –45°, and 90° fibre directions, for this very reason. When this is not possible, one must include residual thermal stresses when predicting failures in the matrix, but equally, one must use a realistic rather than oversimplified model of the phenomena.) Solutions have been presented here for both the original and truncated versions of the maximum-strain failure criterion, as used in the aerospace industry. False predictions of matrix cracking equivalent to those of the first-ply failure analyses using tensor polynomial models for homogeneous solids are excluded from consideration by setting transverse strains high enough, for the original maximum-strain failure model, that such limits are not usually met. While some may question the justification for doing so, it should by now be clear that the consequences of not doing so are far worse – having to repeat all analyses using lamina properties which can neither be measured nor computed, failing to use fracture-mechanics analyses when they are needed, etc., etc. The traditional practice of setting a transverse strain equal to the longitudinal strain capability of any transverse plies is seen to be effective at preventing premature matrix failures for the 0°, ± 45°, 90° family of laminates. However, it is also seen to be too restrictive in the case of the other fibre patterns, 90°/ ± 30° and ± 55°. It is therefore suggested that this empirical limit be totally removed from analyses and replaced by limits imposed by other fibres or set by matrix shear failures – until such time as physically realistic analyses for real matrix cracking are developed in a sufficiently simple form for widespread use at the macro level of analysis. The issue should not be ignored in the

217 interim. Empirical cut-offs should be used for design, but they should be acknowledged as such. Better yet, the existing design philosophy of minimum fibre contents in a sufficient number of different directions to prevent matrix cracking should be used more widely, even when better analysis tools do become available. Both versions of the maximum-strain failure model give the same fibre-failure answers for all but shear (tension/compression) dominated loads, for which the differences can be appreciable. Consequently, the only experimental evidence favouring one over the other can come from the results of these specific tests. Nevertheless, on the basis of solving the problems for the glass/epoxy laminates, even before the experimental results are released, the author became convinced that the original maximum-strain failure model is more representative of the laminate strength than is the truncated version. The converse has long been true for carbon/epoxy laminates. (This issue is clarified in the companion paper,6 by what the author assumes to be a more encompassing failure model.) The author is very confident of his predictions for the carbon/epoxy laminate. However, those for the fibreglass/epoxy laminates lie beyond his prior experience. The predictions of fibre-dominated behaviour are likely to be correct. However, there is a possibility that such failures might be preceded by real matrix failures in the glass/epoxy laminates. Hopefully, other authors with more experience in this field will characterize such phenomena accurately. Otherwise, we must await the release of the test data and then, by parametric studies, identify the most likely explanations of any discrepancies.

References 1 Waddoups, M. E., Characterization and design of composite materials. In Composite Materials Workshop, ed. S. W. Tsai, J. C. Halpin and N. J. Pagano. Technomic, Connecticut, 1968, pp. 254–308. 2 Hart-Smith, L. J., The truncated maximum-strain composite failure model. Composites, 1993, 24(7), 587–591. 3 Hart-Smith, L. J., The role of biaxial stresses in discriminating between meaningful and illusory composite failure theories. McDonnell Douglas Paper MDC 91K0077, presented to 9th DoD/ NASA/FAA Conference on Fibrous Composites in Structural Design, Lake Tahoe, Nevada, 4–7 November 1991. In Proceedings, DOT/FAA/CT-92–25, Part 111, pp. 15071528; also published in Composite Structures, 25, 1993, 3–20. 4 Soden, P. D., Hinton, M. J. and Kaddour, A. S., Lamina properties, lay-up configurations and loading conditions for a wide range of fiber reinforced composite laminates. Compos. Sci. Technol., 1998, 58(7), 1011. 5 MIL-HDBK-17-lD, Polymer Matrix Composites, Vol. 3, Change Notice 1, September 1995. Department of Defense, Washington, DC. 6 Hart-Smith, L. J., Predictions of a generalized maximum-shear-stress failure criterion for certain fibre-polymer composites. McDonnell Douglas Paper MDC 97KOOII Compos. Sci. Technol., 1998, 58(7), 1179. 7 Hart-Smith, L. J., A re-examination of the analysis of in-plane matrix failures in fibrous composite laminates. McDonnell Douglas Paper MDC 95K0069. Compos. Sci. Technol., 1996, 56(2), 107–121. 8 Hadcock, R. N., Design and analysis of advanced composite structures. In Handbook of Composites, ed. G. Lubin. Van Nostrand Reinhold Company, New York, 1982, pp. 533–556. 9 Grimes, G. C., Ranger, K. W. and Brunner, M. D., Element and subcomponent tests. In Engineered Materials Handbook, Vol. I Composites, ed. C. A. Dostal. ASM Ohio, 1987, pp. 3;3–345.

218 10 Hart-Smith, L. J. Fibrous composite failure criteria – fact and fantasy, McDonnell Douglas Paper MDC 93K0047. Presented to Plenary Session of Seventh International Conference on Composite Structures, Paisley, UK, 5–7 July 1993. Abridgement to be published under the title, ‘A bona-fide fibrous composite failure criterion’ in Composites Part A: Applied Science and Manufacturing, 1997, 28. 11 Ewins, P. D. and Potter, R. T., Some observations on the nature of fibre reinforced plastics and the implications for structural design. Phil Trans. Roy. Soc.,1980, A294,507–517. 12 Hinton, M. J and Soden, P. D., Predicting failure in composite laminates: the background to the exercise. Compos. Sci. Technol., 1998, 58(67), 1001. 13 Jones, R. M., Mechanics of Composite Materials. Scripta, Washington, DC, 1975. 14 Puck, A., Festigkeitsberechnung an Glassfaser/KunstoffLaminaten bei zusammengesetzter Beanspruchung Kunststoffe, 59, November 1969, pp. 780–787. (English Summary: calculating the strength of glassfibre/plastics laminates under combined load. German Plastics, 1969, 59, 18–19). 15 Hart-Smith, L. J., Backing out equivalent unidirectional lamina strengths from tests on cross-plied laminates. McDonnell Douglas Paper MDC 91 K0078. In Proceedings of the 37th International SAMPE Symposium and Exhibition, Anaheim, California, 9–12 March 1992, pp. 977–990.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

219

CHAPTER 3.6

Predictions of a generalized maximum-shear-stress failure criterion for certain fibrous composite laminates L. J. Hart-Smith Douglas Products Division, Boeing Commercial Airplane Group, Long Beach, CA, USA Abstract The use of the author’s generalization of the maximum-shear-stress failure criterion for fibre/polymer composites is illustrated by sample solutions of specific problems provided by the organizers of the world-wide failure exercise. New refinements of the theory justify an earlier approximation of it for use with carbon/epoxy laminates and remove a degree of conservatism when the original theory was applied to glass-fibre-reinforced polymer composites. The intent of this exercise is to compare the independent predictions for these same problems made by several originators of composite failure models and, simultaneously, to compare the predictions with test data. Keywords: composite laminate strength, lamina failure criteria, fibre shear failures

1. Introduction The origin of the author’s generalization of the classical maximum-shear-stress yield or failure criterion for metals to fibre/polymer composites can be traced back to his recognition in 19831 that the highest measurements of the fibre-dominated in-plane shear strength of a ± 45° T300/N5208 carbon/epoxy laminate were almost precisely half of the uniaxial tension or compression strength of the corresponding 0°/90° laminates. No composite failure theory of the day predicted this. Indeed, no other one does so even now. Yet the shear strength of ductile metals has been known for centuries to be close to half the tension or compression strength. The author’s composite failure model is simply an attempt to develop an equivalent analysis method for fibrous composite laminates. The author’s fibre-dominated theory has already been described in several references (e.g. Refs 2 and 3). It is summarized here because significant improvements were made while solving the problems posed by the organizers of the failure exercise.4 This refinement, distinguishing between the transverse strain in each lamina and that in the fibres, has confirmed the validity of the original formulation (in which there was no distinction between these strains) for carbon/epoxy laminates and shown that the difference does need to be accounted for with glass-fibre laminates.

220 The theory did not evolve instantaneously. Indeed, it passed through a phase in which it was expressed on the stress plane, like so many other composite failure theories, before the benefits of expressing it on the strain plane instead became apparent. As the development progressed, it became clear that there were fundamental irrecoverable errors in the many published and coded interactive failure theories for composites, and an added goal has been to lay scientific foundations for future failure models of all inevitably heterogeneous composite materials by emphasizing mechanistic models and shunning the abstract mathematical models developed on the false assumption that composites of materials could be regarded as homogeneous anisotropic solids. This simplification is appropriate for computing stiffnesses, but not for strengths. In a fibre/polymer composite, for example, only a fibre, the matrix, or an interface can fail – and separate characterizations are required for each of these mechanisms, as indicated in Fig. 1. Indeed, multiple characterizations are sometimes required for each constituent of the composite, because more than one mechanism of failure can occur (depending on the state of stress) and a separate characterization is required for each of these, also. Fibres can fail by shear, as is indicated by the same longitudinal tensile or compressive strengths, by compressive instability, or by brittle fracture. The matrix can fail by ductile shear* under predominantly shear and transverse-compression loads) or by brittle fracture whenever the transverse-tension stress between the fibres is sufficient to cause microcracks in the resin matrix which are parallel to the fibres to fast fracture. Matrix failures are also influenced by residual thermal stresses within each lamina – the consideration of which is automatically precluded by the traditional false assumption of homogeneity within each lamina. Cracking of the matrix between the fibres, which is at the core of all progressive-failure and ply-discounting models, is particularly difficult to cope with analytically because, in contrast with the basic premise of laminated composite strength predictions – that each and every ply can be assessed independently of all others – matrix cracking is influenced by the adjacent plies and cannot be analyzed the same way. There must be a geometric factor in the analysis as well, just as is the case for all fracture-mechanics analyses of cracks in homogeneous materials. To assume otherwise is to imply that boron/epoxy crack-patching of metallic aircraft could not possibly extend the life of aircraft by retarding the growth of the cracks. Had the design of boron/epoxy crack patches relied on traditional composite stress analysis techniques for laminated structures, it is clear that the concept would never have been initiated since, without exception, these theories predict that no benefit could possibly be obtained. Unfortunately, the fact that such benefits have been demonstrated, many times (for example

* What is apparently ductile shear at the macroscopic level is actually better characterized at the microscopic level as linearly elastic behaviour at the lower stiffness remaining when many inclined microcracks have spread from fibre to fibre throughout the matrix. What is actually transmitting the shear load from fibre to fibre is a series of discrete ligaments of matrix. These cracks occur under the resolved tensile component of the applied shear load, at 45° to the fibre axes, and are stable once a saturation density has been established. A virtually full elastic recovery is made when the load is removed. The author is indebted to Professor Alfred Puck for explaining this to him. There is no permanent set of the kind associated with ductile yielding of metallic alloys. Regardless of the physics of the situation, all that needs to be noted for macro level analyses is that, after the first few load cycles, the in-plane shear stiffness is more accurately given by the secant modulus at failure than by the initial tangent modulus.

Fig. 1. Specification for fibre/polymer composite failure criteria.

221

222 by the pioneering work of Baker and his colleagues as in Ref. 5), does not seem to have caused the obviously necessary re-assessment of some of the best-known composite failure models. In any event, the author’s composite failure model, when first formulated on the strain plane several years ago, accounted for all three possible fibre-failure mechanisms cited above, but made no attempt to characterize possible failures of the matrix. The reason for this was that the author worked with carbon/epoxy composites in the aerospace industry and it was not difficult to establish simple design rules which would ensure that the strength of the fibres which carried most of the load would not be undercut by premature failures of the matrix which was there to stabilize the fibres, not to carry significant load itself. Consequently, the author’s forays into real matrix failures, as contrasted with those predicted to occur by so many interactive theories but which actually do not occur at either the stress levels or densities calculated, have lagged behind his efforts in regard to fibre failures and publicizing the need for mechanistic failure models. Other investigators, most notably Puck6 have worked more with glass-fibre-reinforced plastics and have been unable to avoid the need to characterize these added failure mechanisms. It was Puck who first formulated a mechanics-based composite strength-prediction theory in which failure of the fibres and matrix were covered by separate equations. It is not at all surprising that, given his focus on glass-fibre-reinforced plastics, he has developed a far more comprehensive model for matrix failures than the author’s and could rely on the simpler maximum-strain fibre-failure model for those elements of ‘his’ composites which failed last. In the models developed by both authors, the comparison between competing failure modes must necessarily be effected at a common strain reference – in each lamina. Additional fibre- or matrix-failure modes are added to either model by superposition, not by interaction. Each mechanism governs throughout a limited range of stresses – and none interacts with any other, even though individual stress components may interact within a single failure mechanism. Strength predictions by brittle fracture, from small and large flaws, and ductile failures in the same metals have co-existed for decades, the choice being dictated by the state of the applied stress and the degree of alloying and heat treatment of the metals. Why should carbon fibres be so unique as to be required not to behave similarly? And, given that glass fibres are even isotropic, why should this most common mechanism of failure, shear, have been excluded from fibrous composite failure analyses? The author has never wavered in his belief that it shouldn’t be. Progress in the development of this failure model over the years, coupled with objections, constructive criticism, and help from many other researchers around the world have strengthened the author’s belief that only mechanistic failure models are appropriate for predicting the strength of fibre/polymer composites – or any other material, for that matter. Before summarizing his theory and demonstrating how it can be used to solve at least some of the problems of the failure exercise described in Ref. 4, the author would like to take this opportunity to express his appreciation of the invitation to participate in the comparison and his hope that their goals will be achieved. The efforts made by the many participants certainly merit a successful outcome. The nature of this failure model is that most of its predictions must be bounded between those of the two theories covered in a companion paper7 involving the original and truncated maximum-strain failure models. (There are some minor exceptions, associated with changing from a constant-strain to constant-stress cut-off for compressive loads parallel to the fibre.)

223 Even if his best guesses at some of the matrix-failures prove to be wide of the mark, just trying to solve the problems has accelerated the author’s own learning of the subject and exposed just how fortunate he has been to have worked exclusively in a world which did not require such a focus on the more complicated portions of this discipline which have been encountered in other industries. Despite the risk of discrediting his fibre-based composite failure theory by making predictions about matrix-dominated failures under circumstances for which he has absolutely no prior experiences to guide him, the author has included his assessments of matrix failures in the belief that doing so would at least contribute to the technology by exposing those areas in which more work needs to be done. Particularly in the case of the ± 55° laminate, the failure envelope is defined by alternating regimes of fibre and matrix dominance. It cannot be defined by consideration of fibre failures alone.

2. The generalized maximum-shear-stress failure model for fibres Given that carbon fibres are transversely isotropic, and that glass fibres are essentially completely isotropic, any shear-failure mechanism would have the same critical conditions for both the longitudinal-transverse (L-T) and longitudinal-normal (L-N) planes within the fibres. It is possible that, since carbon fibres are orthotropic, the critical shear strain needed to cause failure in the transverse-normal (T-N) plane may not be the same as for the other two planes. For this reason, the T-N cut-offs shown in earlier presentations of the author’s theory have been relocated, to a parallel but possibly offset position beyond the original failure envelope. This is unlikely to have any effect on the in-plane strengths predicted for fibre/polymer composite laminates, and is done only because doing so simplifies the application of the analysis to the present problems and because it might be necessary for assessing the response of composites to transverse shear or other out-of-plane loads. The simplified failure envelope for the fibres is shown in Fig. 2, for glass fibres on the left and carbon (and other transversely isotropic) fibres on the right, drawn to scale, using data provided in Ref. 4. Since glass is isotropic, the failure envelope has the same form as for ductile isotropic metals. The corresponding corner points are labelled, to show equivalences and to identify the associated states of stress. The entire shear-failure envelope for glass fibres can be constructed from a single measured strength (or strain to failure) because the failure mechanism is prescribed to be constant around the entire perimeter. Other than this one reference strength, the only other quantities needed to construct the failure envelopes are the Poisson ratios, vLT ( = v12) and vLT ( = v21), to define the slopes of the constant-stress lines. If it is assumed that there is only one critical shear-strain-to-failure for transversely isotropic (carbon-type) fibres as well, the same can be said for all fibres. The diagram on the right of Fig. 2 shows additional cut-offs (line IJ and its mirror image) for the 2–3 plane transverse to the fibre axis in the event that the failure strains are unequal. The next step of the analysis has relied upon a standard simplifying assumption – that plane sections remain plane and that, therefore, the transverse strain developed in the fibre and the matrix is much the same as that developed in each lamina. This is a reasonably accurate approximation for carbon/epoxy laminates, because the fibres are so highly orthotropic, but is better regarded as a conservative design procedure for glass-fibre-reinforced plastics. Strictly, since the 45° sloping lines in Fig. 2 refer to the fibres, they cannot also refer to the composite laminae – unless the relevant moduli match. This is explained in Fig. 3 where, for the first time

224

Fig. 2. Strain-based failure envelopes for glass and carbon fibres, according to a generalization of the classical maximum-shear-stress criterion.

225 in the author’s works, the relationship between the transverse strains in the fibres and lamina is derived. The points (1), (2), (3), and (4) in Fig. 3 refer to the steps in creating the accurate cut-off 1–4, rather than the earlier cut-off passing through the measured point 1 at a slope of 45°. The first step in constructing the failure envelope for an embedded, rather than isolated, fibre is to draw radial lines from the origin at slopes defined by vfLT and vfTL for the fibres, and vLT for the lamina, as shown in Fig. 3. Point (1) on the lamina shear-stress cut-off line is at the uniaxial longitudinal strain point.

L = tL,

T =  vLTL

(1)

A vertical line is then drawn through the measured strain-to-failure of the fibres, tL, which is assumed to be the same for both the lamina and the embedded fibre. A 45°-sloping line, denoting constant shear strain, is then passed through the uniaxial-tension failure point (2) for the fibre, which can occur at a different transverse strain than that for the lamina reinforced by unidirectional fibres, because the two major Poisson ratios need not be the same. This sloping line will cross the purely transverse-stress line for the fibre close to the vertical axis at point (3), as shown. It remains only to compute the associated transverse strain in the

Fig. 3. Conversion of 45° slope for fibre failure on fibre strain plane to corresponding line on lamina strain plane.

226 lamina, at point (4). Strictly, this is a complicated micromechanical problem. However, with the same model as was employed by Chamis8 to derive an expression relating the transverse strain in the matrix between the fibres to the average lamina strain, the author has derived the following simple solution for the corresponding strain ratio between the lamina and the fibres. The formula results from an assessment of the compatibility of deformations along a transverse axis through the middle of a fibre.

冋

册

T EfT = 兹KVf + (1  兹KVf) (1  v2m) + vfTLvm = R fT Em

(2)

Here, Vf is the fibre volume fraction, vm the single Poisson ratio for the resin matrix, vfTL the minor Poisson ratio for an isolated fibre, Em is the modulus of the resin matrix, and EfT is the corresponding transverse modulus of the individual fibres. The transverse strain, T, is the strain in the lamina, not the matrix, and fT is the transverse strain in the fibre. The coefficient K is a function of the fibre array, being [(2兹3)/ ] for circular fibres in a hexagonal array, 4/ for circular fibres in a square array, and unity for square fibres in a square array. In other words, it has a value close to unity regardless of the stacking array. The effect of the value of K on R is not large, being greater for typical composites when K is least, particularly when R is much greater than unity. This equation satisfies obvious sanity checks that the strains are equal when the two stiffnesses match, regardless of the fibre content, and that the ratio is infinite for zero matrix stiffness. The derivation of eqn (2) is as follows, using the terminology in Fig. 4. It is not necessary to assume that the transverse stress is uniform throughout the thickness of each lamina, only that it is constant along the datum through the middle of the fibres. (Obviously, this stress will be less on other strata where the matrix makes up more of the total composite of materials, being lowest for any load path passing entirely through the matrix and totally missing the stiffer fibres.) The lamina strains along the transverse axis through the middle of each fibre, of ‘diameter’ d are

lam = mT(1  d) + fTd

(3)

where the subscripts lam, m, and f refer to the lamina, matrix, and fibre, respectively. The axial strain in the fibre for the particular state in which it has zero axial stress is, by definition, –vfTLfT, at point (3) in Fig. 3. The matrix is prescribed to undergo the same strain along the axis of the fibres. The transverse stress in each constituent of the composite would then follow from standard equations, of the type given in the standard text by Jones9 as 2 =

E2 (2 + v121) 

where  = 1  v12v21.

(4)

With 1 defined to match the state of zero axial stress in the fibres, as above, and 2 taken as constant throughout both fibre and matrix on this particular plane, it follows that 2 =

Em EfT [mT + vm1] = [fT + vfLT1], m f

(5)

227

Fig. 4. Fibre volumes for various arrays.

where 1 = –vfTLfT. Hence,

冋

Em E E E mT = fT fT  vfLT fT  vm m m f f m

册

(vfTLfT)

(6)

whence mT EfT = (1  v2m) + vmvfTL fT Em

(7)

Substitution of eqn (7) into eqn (3) then yields lam EfT = d + (1  d)[(1  v2m) + vmvfTL] fT Em

(8)

228 from which eqn (2) follows directly, once the fibre ‘diameter’ d is related to the fibre volume fraction Vf as a function of the fibre array. (Equation (8) also satisfies the obvious sanity checks for equal fibre and matrix properties and for zero matrix stiffness.) The effective fibre diameter is related to the form of the array, as is explained in Fig. 4. d = 兹Vf for a square fibre in a square array,

d=

冑

2兹3 Vf = 1·050兹Vf ,

(9)

(10)

for round fibres in a hexagonal array and

d=

冑

4 Vf = 1·128兹Vf

(11)

for round fibres in square arrays. (These expressions are derived from the solutions for the fibre volume fraction as a function of each array.) Setting the value of the array coefficient K at unity, eqn (2) would then predict the following strain-amplification factors for the composite materials used in failure exercise.4 (0°) T300/914C carbon/epoxy, R = 1·517 (0°) E-glass/LY556-epoxy, R = 5·257 (0°) E-glass/MY750-epoxy, R = 5·159 (0°) AS4/3501-6 carbon/epoxy, R = 1·488. Given that these amplification factors are effectively reduced in the ratio (vfLT/vLT), or roughly 0·2/0·3 for carbon/epoxy, in establishing the final slope of these shear-failure cut-offs, the validity of the earlier 45 deg; approximation for carbon/epoxy laminates is clearly confirmed. Conversely, the earlier approximation would be significantly conservative for glass fibres, as the author had suspected without actually being able to precisely quantify the effect until now. Point (4) in Fig. 3 follows from point (3), the zero axial stress point for the fibre, by retaining the same axial strain and multiplying the transverse strain by R from eqn (2). The line (1)–(4) in Fig. 3 then defines the locus of shear failures in the fibres in terms of strains in the lamina. It will be apparent that point (4) lies off the zero longitudinal stress line for the lamina, being associated with an effective transverse Poisson ratio of vTL = vfTL/R

(12)

instead of the unrelated vlamTL for the laminate as a whole. The reason for this is that, while the fibres have no axial stress at point (4), the matrix does.

229 The strains on the lamina strain plane at which the fibre would fail by shear under the application of purely transverse tension or compression are consequently given by

冉 冊 冉 冊

L = ± vfTL

T =  R

1 + vfLT t L, 1 + vfTL

(13)

1 + vfLT t L 1 + vfTL

(The fibre strain tL would be replaced by cL if the latter were numerically greater.) The line (1)–(4) in Fig. 3 thus defines the shear-failure locus of the fibre in terms of lamina strains. It is not exactly at a slope of 45°, something the author had previously adopted as what seemed to be a legitimate simplifying assumption, at least for carbon-epoxy composites.* The worked examples here will show that the correct solution is quite close to that slope for carbon-epoxy composites, but much closer to the 90° slope of the maximum-strain model for glass-fibrereinforced laminates. This same kind of modification to the effective longitudinal strain under a transverse stress must also be applied to the transition from a pure uniaxial load to the equal-biaxial-strain point. At the equal biaxial strain points, as defined above, the actual transverse strain in the fibre will now be less than in the lamina, per eqn (12) – and the equal-biaxial-strain point for the lamina will now be even closer to matching the uniaxial strain value than was the case for earlier presentations of this theory in which the transition from uniaxial to biaxial loads was assumed to be governed by the measured lamina transverse Poisson ratio vTL. The equalbiaxial-strain point B” in Fig. 3 has co-ordinates slightly less than the tL of the truncated maximum-strain failure model, at

L = T = tL

冉

冊

1  vTLvLT ≈ tL[1  (1 + vLT)vTL] 1 + vTL

(14)

where vTL, is defined in eqn (12). It is quite distinct from the measured (or computed) transverse Poisson ratio vTL. of the lamina. Points A, B and F in Fig. 3 refer to the corresponding points in Fig. 2. As regards the change in slope of the 45° line for fibre failures on the strain plane, the net effect of accounting for the differences in stiffness via eqns (1) and (13) is shown in Fig. 5, using the material properties supplied by the organizers.4 There would be a small increase in slope for carbon/epoxy laminae and a large increase in slope for glass-fibre-reinforced epoxies. The original maximum-strain model would be almost as good a representation of

* This improvement in the model is partly the consequence of responding to earlier unsubstantiated criticism in America that, even if it were conceded that the 45° slope were valid for isolated fibres (as it obviously can be for isotropic glass fibres), it would be replaced by nearly vertical lines at the lamina level even for carbon/epoxy laminates, because the matrix is so much softer than the fibres. The new analysis is also in response to the challenge presented by the organizers in evaluating glass-fibre laminates, beyond the author’s prior experience.

230

Fig. 5. Fibre shear failure cut-offs on lamina strain plane. (Dashed lines for E-glass/MY750 epoxy lamina and continuous lines for AS4/3501–6 carbon/epoxy lamina).

glass-fibre/epoxy laminates as the truncated maximum-strain model is for carbon/epoxy laminates, as will become evident from comparing the worked examples later in this paper with the corresponding solutions in Ref. 7. This similarity would be particularly strong if it were known that the actual failures under uniaxial tension were by brittle fracture rather than by shear because, then, the shear cut-off line should start from beyond the measured uniaxial strain to failure. If a matrix were so soft in comparison with the transverse stiffness of the fibres that it could exert no stress on them, the failure envelope would shrink to the points A and C in Fig. 2 at the ends of the two radial lines characterizing pure longitudinal loading on the fibre- and lamina-strain planes and be a simple square maximum-stress box on the laminate-strain plane. This is referred to as netting theory (see Refs 10 and 11). It should be noted that Puck’s maximum-strain characterization of glass-fibre failures on the lamina or laminate strain planes is not incompatible with the author’s 45° cut-offs for

231

Fig. 6. Superposition of additional fibre failure modes on basic maximum-shear-stress failure criterion.

carbon/epoxy laminates in the tension-compression quadrants. Both are close approximations, not precise answers. The 45° cut-off would still exist at the fibre level for both composite materials, but would simply not be evident at the lamina and laminate levels for glass-fibre/ epoxies. Figure 6 (failure of fibres on the fibre-strain plane) shows how possible cut-offs for fibre failures by brittle fracture, which is a constant-stress phenomenon because crack-tip stress intensities are unaffected by stresses parallel to the crack (transverse to the fibre), and compressive instability, which is also a constant-stress phenomenon,* are superimposed locally on the basic shear-failure envelopes. These three possible failure mechanisms are all that are considered for fibre failures in the author’s analyses of in-plane loads. * In all his earlier works on this subject, the author had described this cut-off as a constant-strain line, since the fibres would buckle once they had reached a critical shortening strain which would be unaffected by the simultaneous application of transverse stresses. The laminate stress at which this would happen would vary with the fibre pattern. The shortening strain would not. However, he had overlooked the Poisson-induced axial strains caused by those stresses, which changed the reference point for the buckling process. He is indebted to the editorial review for pointing this out. A constantstress cut-off for the lamina automatically accounts for this effect. Ironically, with the distinction derived above between transverse fibre and lamina strains, the new position of this cut-off, for both carbon and glass fibres, is almost coincident with the constant-longitudinal-strain line.

232 A difference between the longitudinal tensile and compressive strengths of unidirectional laminae should be interpreted as implying that at least one of the failures cannot be by shear. The 45°-sloping lines in Fig. 6 would then be passed through the numerically greater of the two measured strengths, on the assumption that the lower number denotes a premature failure by a different mechanism. (A more precise slope could be used on the lamina-strain plane when appropriate, as shown in Fig. 5). In the event that it is known by fractographic inspection of the broken fibres that neither of the failures is by shear, (as is quite likely for E-glass fibres), one could perform a shear test on a ± 45° laminate, to generate data near the middle of the sloping line, far away from any failures by other mechanisms. Unfortunately, based on past experience with carbon/epoxy laminates, at least, such a test is likely to result in a premature failure, giving a cut-off more severe than that based on the higher of the two measured axial strengths. This should be physically impossible if the test really represented the true material strength devoid of any influence of the geometry of the test specimen. The highest known test results have been obtained using the Douglas bonded tapered rail shear coupon described in Ref. 12. It is necessary to note that the formulation of the generalized maximum-shear-stress failure theory requires that laminates made from bi-directional woven fabric layers be treated as combinations of two equivalent unidirectional layers – at the same height within the laminate if it is a plain-weave fabric, or one above the other if it is a satin-weave fabric or the like. The transverse strains involved in Figs 2–6 are those associated with a unidirectional fibre, not a mixture of those acting on two orthogonal sets of fibres. A plain-weave cloth can be decomposed into its equivalent layers by using lamination theory in reverse. The combination of 0° and 90° fibres to produce a 0°/90° laminate results in a stiffness of something close to 55% of that of each individual layer. Therefore, once the stiffness and strengths of the fabric layer have been measured, they can be increased in the ratio 1/0·55 = 1·82 (or whatever more precise value is calculated for a specific material). When needed, the matrix-dominated properties can be adjusted accordingly; even the nonlinearities can be replicated. The process can either be performed using logic alone or by scaling (inversely) relevant details of the output from a complete analysis of a 0°/90° laminate for which it has been assumed that the in-plane-shear properties would not be altered by the separation of the constituents and that the transverse stiffness of each equivalent ply would be the same as for a real unidirectional lamina made from the same fibres and resin. The justification for this second assumption is that any crimping of the fibres in a real fabric would affect the longitudinal stiffness but would not affect the transverse stiffness within each tow of fibres. (There would be a minor effect because of the in-plane separation of the tows of fibres which would be filled with a different combination of resin and fibres than within the tows.) This cross-plying technique has already been used to generate more reliable measurements of unidirectional lamina strength than are usually obtained by direct measurement of the lamina strengths, as discussed in Refs 13 and 14. The process automatically accounts for the loss of stiffness by whatever degree of crimping was introduced by the weaving process and for the difference between tensile and compressive strengths which is exacerbated by this same crimping. Some readers of earlier articles on the generalization of the maximum-shear-stress failure criterion to nonisotropic homogeneous materials have expressed difficulty in accepting the concept of a 45°-sloping constant-shear-strain line representing a constant critical stress criterion for anything other than an isotropic solid, like a glass fibre. (The confusion seems to

233 arise from the obviously dissimilar differences between principal stresses in the L-T plane for fibres subjected to axial tension on the one hand and transverse compression on the other.) Reference 15 includes an attempt by the author to explain this apparent contradiction, in terms of the difference between isotropic and nonisotropic materials. Briefly, while isotropic homogeneous materials can undergo strains in the absence of stresses, as the result of uniform heating for example, or stresses in the absence of strains, as for hydrostatic compression of an incompressible material, there is a one-to-one relation between stress and strain for isotropic materials most of the time. This is the exception to the rule for homogeneous nonisotropic materials, however, as the equations in Jones’s work9 make clear. Consider, for example, uniform heating of a transversely isotropic solid. If the coefficients of thermal expansion in the principal axes differ, it is inevitable that shear strains will develop between axes inclined at ± 45° to the material axes, even though there are no stresses anywhere in the solid. However, only those components of stress and strain for which there is a matching strain or stress contribute to the distortional energy of deformation. Therefore, the criterion should not be applied to shear stresses deduced from Mohr circles, for example, but only to the shear stress associated with the shear strain. Even for transversely isotropic solids, there are three increments of stress for each strain, and vice versa. Obviously, if the shear strain is constant along some certain lines, the associated shear stress must also be constant. The remaining increments of shear stress, at the Mohr circle level, have been shown in Ref. 15 to have no matching shear strains. This is the explanation of the apparent inconsistency; isotropic behaviour can be inferred from that for nonisotropic materials, but not vice versa. 3. Cut-offs imposed by matrix shear failures The failure envelopes shown in Figs 2, 3, 5 and 6 lack a roof to define any limits imposed by the in-plane shear strength of the matrix between the fibres. Since being formulated on the strain plane, the author’s model has always included a non-interactive horizontal plateau, located by the shear-strain-to-failure, as shown by the lamina failure model in Fig. 7. This refers to shear with respect to fibres in the 0° and 90° directions. Most of any such load would

Fig. 7. Matrix shear failure cut-off for fibre/polymer laminae.

234 be reacted by fibres at ± 45°, if any were present. They would provide a far stiffer load path and impose different strains-to-failure, which are covered by the present analysis for fibres. In transverse compression, the failures of unidirectional tape laminae are akin to the collapse of too large a pile of stacked logs and little influenced by additional stress components other than transverse shear, which is not considered here. Naturally, in a welldesigned laminate with the layers of fibres in the different directions well interspersed, the fibres would be better stabilized to resist transverse compression loads – just as they are similarly able to withstand higher longitudinal compressive stresses – and this cut-off would be moved outward, possibly becoming totally ineffective. Like matrix cracking between the fibres under transverse-tension loads, this potential failure mechanism can be characterized properly only at the laminate level. Although he has developed a formula16 for ductile matrix failures which interact not only the stresses discussed above, but also the compatible matrix stress developed in the matrix parallel to the fibres, the author tends to assess the stresses discussed above separately (i.e. noninteractively) because his matrix-failure theory has yet to properly account for the residual thermal stresses in the matrix caused by curing at elevated temperatures. These stresses, which are customarily excluded from consideration by the standard assumption of homogenizing fibres and matrix to create one composite material, are typically very much greater than those stress components which are retained in most analyses – at least at the macro level. The author’s thoughts on what is needed to properly characterize matrix failures in fibre/polymer composites can be found in Ref. 16. The author’s empirical equation for ductile matrix failures (not cracking) under a combination of interactive stresses, extracted from Ref. 16, is

冋冉 冊冉 冊册 冉 冊 冉 冊 1 2S

Em EL

2

+

12 S

2

+

2 2S

2

≤1

(15)

in which Em is the modulus of the resin matrix, EL the longitudinal modulus of the unidirectional lamina, and S the measured in-plane shear strength of the lamina. The s and  represent the obvious in-plane stress components in the lamina, the direction 1 being along the length of the fibres. This is an entirely empirical expression; setting the direct reference strengths at twice the shear strength is based only on Mohr circles, not curve fits to data. A further reason for excluding the first and third terms from all but isolated unidirectional laminae is that the interactions between stresses often do not become significant below the strain limits imposed by the fibres. 4. Cut-offs imposed by transverse cracks in the matrix between the fibres The author’s assessment of cracking of the matrix under transverse-tension loads is explained in Ref. 16. This is a fracture-mechanics problem, with the failure stress varying from fibre pattern to fibre pattern. There is no universal characterization equivalent to that in Fig. 7 which can be formulated on the lamina strain plane, regardless of whether the assessment is in terms of stress or strain. The nominal failing stress in the lamina, when such cracking occurs, is a function of the orientation and thickness of adjacent plies, as well as the thickness of the ply under consideration. The transverse-tension strength measured on an isolated all-

235

Fig. 8. Characterization of intralaminar matrix cracking in fibre/polymer composites. (1) Transverse tension cracking of matrix in isolated unidirectional lamina, (2) arbitrary design limit imposed for cracking in brittle matrix, (3) inoperable cut-off for unattainable matrix failures in ductile matrices.

90° lamina applies only to that isolated lamina. It is neither an upper- nor a lower-bound estimate of strength for the very same lamina when it is embedded in a multidirectional structural laminate. This laminate strength cannot be predicted using the traditional ply-by-ply decomposition used for other failure modes in composite laminates. Nevertheless, once the laminate has been defined, and the operating environment specified, the influence of biaxial stresses on matrix cracking can be depicted as shown in Fig. 8 – in the form of a constanttransverse-tension cut-off. The only difficulty, at the macro level of analysis used here, is that the line can be located only by use of micromechanics or by empirical wisdom. This cut-off should not be interacted with any other failure mechanism. The slope of the line in Fig. 8 is defined by the Poisson ratio vLT for the lamina since, apart from small Poisson-induced internal stresses between fibres and resin, only a constantlongitudinal-stress locus makes no change to the transverse stress. This line is most conveniently defined, for purposes of relocating it for plies embedded in a multidirectional laminate, via the intercepts P and Q in Fig. 9 of the equal- and equal-and-opposite-strain lines shown. The factor k can be less than or greater than unity, depending on the fibres and resin from which the laminate is made, the difference between curing and operating temperatures, and the fibre pattern. It can even vary from ply to ply within a single laminate. It is far smaller for an all-0° lamina than for a 0°/90° laminate, for example, because of the absence and presence (respectively) of orthogonal fibres to act as crack arrestors. The co-ordinates of Points P and Q in Fig. 9 here are as follows, for a 0° lamina. Point P: 1 = 2 = ktL Point Q: 2 = –1 =

冉 冊

1 + vLT ktL 1  vLT

(16)

(17)

236

Fig. 9. Mathematical model for constant-transverse-tension stress cut-off to characterize in-plane cracking in the matrix between the fibres.

Point P in Fig. 9 will be unaffected by the strain transformation associated with the angles at which each ply is embedded in the laminate, while point Q will appear on the laminate strain plane somewhere else on this same –45° diagonal, with the following co-ordinates, per eqns (2) in Ref. 7.

Point Q: y = –x =

冉 冊 1 + vLT 1  vLT

ktL (cos2   sin2 )

(18)

(There can be no such intercept for a ± 45° laminate. However, the trace through point P on the laminate strain plane must then slope at –45°, so this cut-off can still be located.) The vertical line through point Q on the lamina strain plane will then intercept the xy axis at the following height.

Point Q: xy = ±

冉 冊 1 + vLT 1  vLT

ktL sin  cos 

(19)

according to eqn (3) in Ref. 7. This same constant-transverse-stress cut-off should also be used in conjunction with the truncated maximum-strain failure model, whenever it is known that matrix cracking is likely to occur. The difficulty remains, however, that the value of k can be established only by empirical wisdom or by micromechanical analyses. Rather than a definitive prediction, this empirical cut-off should be looked upon as a qualitative representation to show how laminate failure envelopes would be affected if suitable data were available. (This effect is known to be real for many fibreglass filament-wound pressure vessels, which need liners to prevent the contents from escaping through such cracks at stress levels far below what is needed to fail

237 the fibres. On the other hand, for carbon/epoxy laminates, this effect is ignored by the aerospace industry for laminates in which the fibres in the different directions are thoroughly interspersed in thin plies. This tradition is based on experience with the older intermediatestrain carbon fibres. The practice of ignoring the effect for the new high-strain fibres is probably still justifiable for laminates in which the design strain levels are limited by bolt holes and the like, but may need re-appraisal for those applications in which no such overriding stress concentrations abound. As indicated in the introduction, real matrix cracks are very difficult to handle analytically. It is better to design to avoid them whenever possible – but to acknowledge their existence whenever the choice of materials makes eliminating them impossible.) Raising the design transverse-tension strengths above what is measured on unidirectional laminae, to account for the benefit derived from the presence of adjacent orthogonal fibres in structural laminates, is relatively easy to justify, based on physical reasoning. Without such an interaction, boron-epoxy crack patching of metallic structures could not work, and neither could laminated bonded metal structures retard the rate of crack growth under fatigue loads with respect to that in monolithic structures. Raising the in-situ strength for transverse compressive loading, where there are no obvious cracks to prevent the strength-of-materials analyses from being applied, may seem to be a little more difficult to justify. However, if one looks at the problem in reverse, one would be forced to conclude that the strain-to-failure of a 0°/90° laminate under purely compressive loads could not exceed that for a 90° lamina in isolation unless the test in isolation were not representative of a complete structure. Since 0°/90° laminae fail in compression at far higher strains, at least as high as those for all-0° laminates, one is forced to concede that this lamina cut-off is also inoperative at the laminate level. (The author must conclude that, at the micro level in heterogeneous materials, the application of transverse compression loads induces internal transverse-tension stresses resulting in failures like a stack of logs collapsing whenever too many layers have been piled on top of each other. Transverse fibres in a laminate would suppress this mechanism, just as periodic transverse boards do in stacks of logs.)

5. Lamina failure envelope on the strain plane The failure envelope for an all-0° tape lamina, according to this generalization of the maximum-shear-stress failure criterion, is plotted in Fig. 10, based on the new corner points derived above. These are as follows: Point A: 1 = tL, 2 = –vLTtL

(20)

冋 冉 冊册

Point B: 1 = 1  vfTL

1 + vfLT 1 + vfTL

tL,

冋冉 冊 册

2 = R 

1 + vfLT  vLT tL 1 + vfTL

(21)

238

Fig. 10. Co-ordinates of corner points for lamina failure envelope according to generalization of maximum shear stress failure.

Point B: 1 = 2 =

冉

冊

1  vTLvLT t  L, 1 + vTL

(22)

where vTL = vfTL/R, per eqn (12).

冉 冊 冉 冊

Point C: 1 = –vfTL

2 = R

1 + vfLT t  L, 1 + vfTL

1 + vfLT t L 1 + vfTL

(23)

239 Point D = –Point A:

(24)

Point D: 1 = –cL, 2 = vLTcL

(25)

冋

册

c

(1  vTLvLT) | LLt | + (tan   vLT)vTL t L, Point D: 1 = (1 + vTL)

(26)

2 = vLTtL + (tL  | 1 |D) tan ,

where tan  =

冋

R(1 + vfLT)  vLT(1 + vfTL) (1  vTLvLT)

册

Point E: –Point B

Point E: 1 = 

(27)

冋冏 冏 冉 冊册 cL 1 + vfLT  vfTL tL 1 + vfTL

tL, (28)

冋 冉 冊 冏 冏册 1 + vfLT cL  vLT t 1 + vfTL L

2 =  R

Point E: 1 = 2 = 

冏 冏冉 cL tL

tL

冊

1  vTLvLT t L 1 + vTL

(29)

Point F: –Point C

Point G: 1 = –2 =

(30)

冋

Point G: 1 = –2 =

R(1 + vfLT)  vLT(1 + vfLT) + vLT(1  vfLTvfTL) R(1 + vfLT)  vLT(1 + vfTL) + (1  vfLTvfTL)

冉

冊

1 + vTLvLT t L 1  vTL

Point H: –Point G

Point H: 

冏冏 cL tL

册

(31)

(32)

(33)


Point

G

(34)

240 Not shown is the horizontal matrix shear failure cut-off at a height of LT on the 12 axis. The cut-offs which would be imposed by the additional presence of 90° fibres have been omitted here because their influence has already been addressed in Ref. 7, where their inclusion is normally part of the analysis process. Here, they can be treated just like fibres in any other direction. Figure 10 retains the constant-transverse-stress cut-offs for the fibres which the author included in earlier representations of this theory. Points B and E are the points of equal biaxial strains in the lamina. The fibre does not see equal biaxial strains until points B and E. The high magnification ratios R for the glass-fibre/epoxy laminates thus render the transverse-tension cut-off BC (and the corresponding compressive limit EF) inoperative for these laminates. In the case of carbon/epoxy laminates, these cut-offs should not be ignored, but they are undercut by the earlier failure of orthogonal fibres in the quasi-isotropic laminate and, again, play no part in the analyses presented here. With respect to the corresponding characteristic according to the truncated maximum-strain failure model, Fig. 10 in Ref. 7, the vertical lines denoting fibre failures are not quite vertical any more, and the 45° sloping lines for fibre failures are either close to 45° or to 90° (depending on the fibre), without precisely matching either. (Figure 10 here is deliberately not drawn to scale, to highlight these differences; its purpose is to summarize the co-ordinates of the corner points used in the subsequent analyses.) There are no changes for the modelling of matrix failures, confirming the contention which the author shares with many others that the failure mechanisms are truly independent and need to be characterized separately. The strain co-ordinates of Points Q, G, G, H, and H for a 0° lamina are particularly useful in generating failure envelopes because, no matter what the fibre orientation, they must remain on the same –45° diagonal, albeit possibly at a different distance from the origin. Other points on this same diagonal are equally useful, for the same reasons. The equal-biaxial-strain points B and E in Fig. 10 are even more useful; they never change. Neither does Point P on the same + 45° diagonal in Fig. 9.

6. Additional equations needed for analysis at the laminate level Except for the few new corner points in Fig. 10, which have been expressed in the preceding text, all equations needed to solve the problems posed by the organizers, Ref. 4, can be found either in the standard work by Jones9 or in the author’s companion paper.7 The solutions to these same problems in Ref. 7 have been solved using the original and truncated maximumstrain failure models which, between them, bound most of the solutions derived here. Once the strains for each lamina have been established on the x-y plane, the membrane ‘stresses’ follow directly from the following standard equations. x 1 y = t xy

A11 A12 0

A12 A22 0

0 0 A66

x y

xy

(35)

The stiffness coefficients Aij used here are those given by Jones in Ref. 9. Depending on the fibre pattern, these equations may need to be applied for each lamina, in turn. However, the Aij coefficients are those pertaining to the entire laminate in every case, not to the individual

241 plies. (The strain transformations, on the other hand, are applied to each fibre direction separately.) The approach used in every assigned problem is to solve it on the strain plane first, to identify which constituent of the composite is failing, and by what mechanism. These answers are then converted to the stress plane, using standard relations, to present the solutions in the form requested by the organizers.4

7. Test problem No. 1: Biaxial (␶xy ⴚ ␴y) failure envelope for unidirectional glass/epoxy lamina There should be evidence of two distinct failure modes for this failure envelope – matrix cracking between the plies under predominantly transverse-tension loads, and ductile shear failure under the influence of in-plane-shear and transverse-compression (or small transversetension) stresses. The latter is characterized by the interaction eqn (15), with 1 = 0, while the former should be covered by a fracture-mechanics analysis instead. However, since there is only one geometry to be considered here, the fracture-mechanics analysis for the lamina should degenerate into a constant-transverse-stress criterion unaffected by any shear stresses, at a stress level having precisely zero relevance to any other laminate. Also, because of the residual thermal stresses in the matrix, one should expect the maximum in-plane shear stress to be developed at that value of applied transverse-compression stress which neutralizes this residual thermal stress. Since the estimation of this stress by analysis would involve micromechanical analysis at a level way beyond what is used in the aerospace industry, the author will start his assessment with an assumed value of 30MPa. The justification for this choice is that the value cannot possibly exceed half the difference between the compressive and tensile transverse strengths, 114 and 35 MPa, respectively. Otherwise, the ductile shear failure envelope would not even reach the brittle-fracture line. Assuming that the ductile-stress interaction formula is an ellipse which is symmetric about the axis through this residual thermal stress line, one would predict a maximum shear strength of 77 MPa, some 7% higher than the cited measured strength. This would lead to a predicted intersection of the two failure mechanisms at an in-plane shear strength of 49 MPa, as shown in Fig. 11 (corresponding with* †Fig. 10 of the assignment package). (Smaller values of the residual thermal stress would be associated with peak shear strengths even closer to the nominal 72 MPa value; the greater sensitivity would be in relation to the intersection of the two failure mechanisms. The smaller the thermal stress, the higher the shear stress which would not cause failure on the constant-transverse-stress line in Fig. 11.) Repeating the calculations for an assumed thermal stress of 35 MPa would yield a predicted maximum shear strengths of 80 MPa and an intercept on the brittle-fracture line at 37 MPa, respectively. This result differs greatly from that given in Figs 6 and 7 of Ref. 7 because they referred to specific documented noninteractive theories, the original and truncated maximum-strain failure models. Figure 11, is the author’s best estimate, using undocumented techniques.

* The term †Fig. is used here in the text to refer to figures in the organizers’ task assignments. The term Fig. denotes a diagram in this paper or in the companion paper.7

242 8. Test problem No. 2: Biaxial (␶xy ⴚ ␴x) failure envelope for unidirectional carbon/ epoxy lamina The interaction between the compatible longitudinal stress in the matrix, as the fibres and matrix are strained together, with any transverse-shear stress in the matrix is one of the most misunderstood aspects of composite failure prediction – as indicated by the fact that only a handful of the theories even include terms (like the first in eqn (15)) permitting any account to be taken of this phenomenon. There are any number of theories which invoke a questionable interaction between the shear stress in the matrix and the longitudinal stress in the fibre (see, for example, Ref. 17), but the organizers have posed the interaction they requested in their †Fig. 11 at the lamina level. Actually, the traditional assumption of a homogeneous lamina masks the fact that there are two interactions needing to be plotted on this plane; one for the fibres and an inevitably totally different one for the matrix. None of the fibres in any of the four material systems cited is sufficiently weak in comparison with the soft matrixes to be affected by the relatively minuscule transverse shear stresses which the matrix could exert on them. To quantify this issue, consider a Mohr-stresscircle assessment of the influence of the maximum possible 80 MPa shear strength cited for this composite material on the in situ longitudinal fibre strength of (Ef/E1)
1500 = (230/ 138)
1500 = 2500 MPa. This interaction could reduce the longitudinal strength of the fibre by only 5 MPa, corresponding to which the lamina strength would be reduced by only 3 MPa, i.e. by 0·2%. Therefore, for all practical purposes, the fibre-failure envelope in this case is unaffected by matrix stresses. Since the carbon fibres are transversely isotropic, the author is unable to deduce the appropriate fibre transverse shear strength for Fig. 12, except to note that

Fig. 11. Interaction between in-plane shear and transverse tension and compression stresses on unidirectional E-glass/LY556 epoxy.

243

Fig. 12. Non-interaction between longitudinal and in-plane shear stresses acting on unidirectional T300/914C carbon/epoxy composite lamina, (a) showing the design envelope, (b) showing possible cutoffs.

it will be far greater than any matrix strength, but the form of the fibre failure envelope is presumed to be basically elliptical, with a cut-off for compressive strengths, as shown. On the other hand, the influence of the large longitudinal stresses on the shear strength of the matrix should not be ignored as it has so frequently in the past. Failing to account for this interaction is why the traditional analyses of off-axis laminae show shear strengths asymptoting to infinity as  approaches either 0° or 90°. Given that the strain to failure of the unreinforced resin is almost four times that of the fibres in this particular case, the interaction here will not be strong, even though it would be greater if the matrix-failure envelope were extended beyond the fibre failures until the matrix really did fail. However, this would require some way of accounting for the very great tensile residual stress in the matrix along the length of the fibre as the result of curing the composite at elevated temperatures and operating in far cooler environments. The data provided for the unreinforced matrix seem to exhibit a marked loss of strength in tension because of surface flaws on the test coupons. Otherwise, one should

244 have expected much the same tensile and compressive strengths. Indeed, the measured resin tensile strength is barely sufficient to tolerate the residual thermal stresses. Fortunately, the problem of surface-flaw sensitivity would be eliminated by the crack-stopping action of the very same fibres which would cause the residual thermal stresses to develop in the resin matrix as they resisted its shrinkage. For the design of structural laminates, the author would totally disregard any interaction between the two stress components in Fig. 12(b) – for the matrix as well as the fibres. A justification for ignoring any interaction in regard to matrix stresses, also, can be provided by using the first and second terms in eqn (15) with 2 = 0. In this case, the reference shear strength is 80 MPa. The tensile reference strength of twice this value in the matrix, corresponding with 2
80
(E1/Em) = 2
80
138/4 = 5520 MPa at the lamina level (scaled in proportion to the respective Young’s moduli), is deduced from the simple Mohr stress circle (max = 2
max), because there seem to be no data with which to assess whether or not there should be some scaling factor to reflect the influence of the fibres embedded in the matrix. While formulae of this type certainly eliminate the physically unreal asymptotes from the analysis of off-axis laminae, in this particular case the lesser strain to failure of the fibres limits the maximum possible correction to a purely shear-based failure criterion for the matrix to only 1%. 9. Test problem No. 3: Biaxial (␴y ⴚ ␴x) failure envelope for unidirectional glass/epoxy lamina The model shown in Fig. 13 in response to the organizers’ requests for their †Fig. 13 is that according to the author’s generalization of the maximum-shear-stress criterion, appropriately truncated locally to account for the additional failure mechanisms. It differs from the solution given in Fig. 9 of the companion paper7 in regard to predictions of both fibre and matrix failures. The constant-strain cut-offs in the companion paper are here replaced by constantstress cut-offs in both tension and compression. Since the glass fibres are isotropic, precisely the same cut-off, at 45° on the strain plane would apply for shear-failure of the fibres according to both the present theory and the truncated maximum-strain failure model. The corresponding transverse strain in the lamina, however, would be greater than that in the fibre, for the reasons explained earlier, so the slope of the shear cut-off for glass fibres is steeper in Fig. 13 than in Fig. 9 of Ref. 7. The location of this sloping shear cut-off is established as follows. One point is self-evident; the uniaxial longitudinal stress for the lamina, for which there is nominally zero transverse stress in the fibres. The other point, near the transverse stress axis, can be closely approximated by setting it right on that axis, at a transverse stress equal to the longitudinal stress needed to break the fibres because, in this case, they are isotropic. Assuming that the matrix is soft in comparison with the fibres, the longitudinal failing stress of the unidirectional lamina would be increased in the ratio 1/[Vf + (Em/EfL)(1  Vf)] = 1·62 to compensate for the fact that the axial fibre stress, when the lamina fails under axial load, is greater than that in the lamina. One could not, of course, actually reach this transverse-stress point because the matrix would fail first. This end point, at –2071 MPa, is merely an easily located extrapolation of the useful portion of the characteristic. (Strictly, this point should lie slightly off the transverse stress axis, because the lamina is stressed when there is zero axial stress in the fibres, and vice versa, but the error is

245 small provided that the fibres are stiff and the matrix soft.) An alternative derivation of this same end point, using the R factor to amplify the transverse strains, would have predicted –2240 MPa, instead. There is clearly consistency between these two calculations, lending credence to using eqn (2) to represent the strain-amplification between fibre and lamina. Figure 13 contains two sets of matrix-strength cut-offs. The first, using data supplied by the organizers, applies only to unidirectional laminae in isolation. The second, of greater strength, is indicative of the far higher in situ strengths developed in multidirectional laminates. Apart from this illustrative example of matrix cracking between the fibres at a constant transverse stress arbitrarily set at 300 MPa, the design strengths for matrix failures have been set high

Fig. 13. Failure envelope for unidirectional E-glass/ MY750 epoxy laminate on stress plane.

246 enough that any transverse fibres in a fibre-dominated laminate would fail first under axial loads in those fibres. The matrix-failure strengths measured on isolated laminae have absolutely no relevance to the performance of those same laminae when embedded in multidirectional laminates. They cannot even be used as upper or lower bounds. As noted in the companion paper,7 for the other failure criteria, it is necessary to distinguish between fibre and matrix analyses by using separate equations. Only a mechanistic failure model can suffice – and the governing phenomena for laminae when they are isolated and when they are embedded within a laminate are very different.

10. Test problem No. 4: Biaxial (␴y ⴚ ␴x) failure envelope for (90°/ ± 30°)s E-glass/epoxy composite laminate In responding to the organizers’ request in their †Fig. 13, starting with separate information on the strain plane for the 90° and ± 30° fibres clarifies both the differences and similarities between the present failure theory and those presented in the companion paper.7 While there is also a minor difference in the magnitude of the laminate strengths for equal biaxial strains, the major difference concerns the 45° cut-offs in the shear quadrants for the truncated maximum-strain failure model in Ref. 7 vs the more precisely located equivalent cut-offs here. Figure 14 shows the transformation of the 0°-lamina failure envelope in Fig. 10 into that for the 90° fibres, accomplished by merely rotating the original failure envelope through 180°

Fig. 14. Failure envelope on strain plane for 90° E-glass/LY556 epoxy lamina.

247

Fig. 15. Failure envelope on strain plane for ± 30° E-glass/LY556 epoxy lamina.

around the equal-biaxial-strain axis. Matrix shear-failure planes are offset from, and parallel to, this section cut, at heights of ± LT. This time, the fibre shear-failure cut-offs are almost horizontal, far from a 45° slope, favouring the original maximum-strain solution in Ref. 7 over that with truncations, for glass-fibre-reinforced laminates. Figure 15 presents the corresponding result for the ± 30° plies. The matrix shear-failure planes do intersect the x  y base plane in this case, parallel to the equal-biaxial strain diagonal, as shown, but are shown later to be just undercut by fibre failures in the 90° plies. The intercepts on the axes, at  = (2/兹3) LT, follow from eqn (4) in Ref. 7. Figure 16 shows the combination of cut-offs for both fibre directions on the x  y base plane. It is essentially a skewed parallelogram. The envelope is entirely fibre dominated, albeit only just in the upper left and lower right corners. The effect of the shear cut-off for the 90°

248

Fig. 16. Failure envelope on strain plane for (90°/ ± 30°) E-glass/LY556 epcxy laminate.

fibres is barely perceptible, in the upper left corner, so test data for this laminate cannot be expected to resolve the matter. The strains in Fig. 16 have been converted to stresses in Fig. 17, using standard formulae from Jones’s book.9 The result is basically another skewed parallelogram, but with a different orientation and a minor truncation in the lower right corner. A comparison of these predictions with those in Fig. 14 of Ref. 7 for the original and truncated maximum-strain failure models suggests that, for glass-fibre-reinforced polymer composites at least, the 45° shear cut-off in the truncated maximum-strain failure model is not an improvement on the original untruncated formulation. (Later in this paper, it will be shown that it is an improvement for carbon/epoxy laminates.) The error, however, is small and conservative. Ironically, given the proximity to the matrix shear failures which would destabilize the ± 30° fibres, it is likely that the predictions of the truncated maximum-strain model might show the better agreement with the test data – but not for the reasons predicted by these theories.

249

Fig. 17. Failure envelope on the stress plane for (90°/ ± 30°) E-glass/LY556 epoxy laminate.

The material properties used in preparing Figs 14–17 are those established in Ref. 7. Their justification is not repeated here, but the values actually used are as follows:

tL = 0·02132, cL = –0·01065,

LT = 0·038, vLT = 0·278, R = 5·257; E1 = 53·48 GPa, E2 = 7·67 GPa, G12 = 2·525 GPa.

250 The values of the stiffness coefficients used here are: A11/t = 29·143 GPa, A12/t = 9·512 GPa, A22/t = 17·980 GPa, and A66/t = 9·881 GPa. These lamina properties were established in Ref. 7 for fibre-dominated behaviour; the values of LT and G12 do not match, since the value of G12 was established for a shear strain LT of 0·027. This discrepancy has no effect on predicted fibre failures, but the issue is considered for the following exercise for those elements of the failure envelope defined by matrix failures. Selecting appropriate linearized matrix properties is complicated tremendously by an ability to insert only one such set of properties at a time into virtually all of the standard computer codes. One-off solutions, like these, are conducive to compensating for such discrepancies by hand. But large finite-element analyses are not, which is why no such effort was made in Ref. 7 for the two variants of the maximum-strain failure models used in production. Figure 17 includes speculative matrix-cracking cut-offs for transverse tension loads. These are for illustrative purposes only, to show the form of such truncations. The strain co-ordinates of these possible cut-offs are given by eqns (16) (18) for 0° laminae. The corresponding coordinates in the laminate strain plane are given by eqn (18). In this case, point Q is moved further from the origin to new co-ordinates of x = –y = –3·5402ktL for the ± 30° plies and to x = –y = 1·770ktL for the 90° plies, remaining on the same –45° diagonal in both cases. Since there are only half as many 90° fibres as would be needed to create a quasi-isotropic laminate, the ± 30° fibres are assumed to be more capable of retarding cracks between the 90° fibres than vice versa. This is why the two cut-offs are shown not crossing the equal-biaxial-strain line at the same point. Only the slopes of these two lines are definitive. Their locations could be anywhere inside or outside the fibre-failure envelopes, depending on the difference between operating and cure temperatures. (They would also vary with the fibre pattern and ply thicknesses.) The sample solution given (k = 1·1) would predict that there would be no cracking between the 90° fibres until after the ± 30° fibres had failed. Matrix cracking between the ± 30° fibres could precede fibre failures, for the sample solution shown (k = 0 8). The sample solution for matrix cracking between the ± 30° plies crosses the equal-biaxialstrain diagonal at x and y stresses of 659·3 and 468·9 MPa, respectively, and intercepts the y axis at 484·1 MPa, while the solution for cracking between the 90° fibres cuts the equalbiaxial-strain locus at x and y stresses of 906·5 and 644·7 MPa, respectively, crossing the x axis at 847·2 MPa. Precisely the same predictions would be made for matrix cracking according to the truncated maximum-strain failure model. Such analyses for matrix cracking are not yet in widespread use, even though the basic truncated maximum-strain failure model is, for fibre failures. This matrix-cracking feature is a relatively new addition, recommended in MIL-HDBK-17 for use with the truncated maximum-strain failure model because, despite ambiguity over the value of k, no alternative solution is available without recourse to micromechanics. 11. Test problem No. 5: Biaxial (␴x ⴚ ␶xy) failure envelope for (90°/ ± 30°)s glass/epoxy laminate In their †Fig. 14, the organizers have requested a further section-cut through the failure envelope for this laminate. While seeming to be a simple request, this is by far the most

251

Fig. 18. Elements of cross-section of failure envelope on strain plane for (90°/ ± 30°) E-glass/LY556 epoxy laminate.

difficult to quantify. An obvious possible failure mechanism possibly defining portions of this cut is matrix shear between the fibres within each layer. Most of the material properties should remain the same as for the preceding exercise. The only possible change is for G12 to be reduced to 1·895 GPa at the cited ultimate shear strain of 0·038. The corresponding change in A66, to 9·642 GPa, however, is not pronounced, because of the dominant contribution from the fibres in the ± 30° plies. The reduction, for this laminate with triangulated fibres, is less than 2·5% and is ignored. Figure 18 shows the diamond-shaped cylinder, which represents matrix shear failures in the ± 30° plies, running along the equal-biaxial-strain axis and intersecting the xy axis at a height of ± 2 LT. The width of this cylinder, on the x  y plane, is ± [兹(2/3)] LT, and the cylinder intersects the x and y axes at strains of ± (2兹3) LT. These intercepts are established by eqns (4) and (5) in Ref. 7. The figure also contains the plateau for matrix shear failures in the 90° plies, at a height of ± LT. Figure 18 is the section view A-A along the equal-biaxial-strain axis in Fig. 15. Even though the fibre-shear-failure line at the lower right in Fig. 15 is preceded by other failures on the xy = 0 section cut, the associated plane will cut the xy axis at some non-zero shear stress and, therefore, it cannot be omitted from this particular analysis; neither can the

252 line defining the right-hand side of the failure envelope in Fig. 15, for precisely the same reasons. (These two lines correspond with lines AF and AB, respectively, in Fig. 10.) Fig. 19 shows the view of Section B-B in Fig. 15, which has a slightly different cross section and a slightly different orientation, but is again dominated by sloping limits on the ply strengths. Section C-C in Fig. 15 would be very similar, except that the height would be slightly less and the width slightly greater. Actually sizing these two cross-sections requires establishing the intercepts of these two ± 30° fibre cut-offs and the tensile uniaxial load line (y = 0) for the laminate. The latter is defined on Fig. 15 by the laminate Poisson ratio vxy = 0·5290 for this laminate, evaluated as the ratio A12/A22. The intercept of line AB in Fig. 10 with the pure x stress line occurs at x and y strains of 0·03451 and –0·01826, respectively, while the corresponding strains on the line AF are 0·03473 and –0·01838. The corresponding x stresses are 837·5 and 832·1 MPa, respectively. The associated intercepts on the xy axis follow from eqn (3) in the companion paper,7 once the intercepts on the –45° diagonal for the 0° lamina in Fig. 10 are established. These are at strains of ± 0·021359 and ± 001913, respectively. The corresponding in-plane shear stresses are 487·4 and 436 ± 6 MPa, respectively.

Fig. 19. Further elements of cross-section of failure envelope on strain plane for (90°/ ± 30°) E-glass/ LY556 epoxy laminate.

253

Fig. 20. Still more elements of cross-section of failure envelope on strain plane for (90/ ~ 30) E-glass/ LY556 epoxy laminate.

Figure 14 shows the traces of the fibre-failure planes for the 90° plies. These planes are vertical and not influenced by any xy stresses acting on the laminate, as shown in Fig. 20 (Section A-A in Fig. 16). Not all of these possible individual failure modes in Figs 18–20 survive in the final complete failure envelope in Fig. 21, because some would be preceded by others. Except for the compression failures of the 90° fibres at the right-hand end of Fig. 21, the bulk of this

Fig. 21. Complete cross-section of failure envelopes in stress plane for (90°/ ± 30°) E-glass/LY556 epoxy laminate in stress space.

254 failure envelope is defined by fibre failures in the ± 30° plies. The various strains are related to stresses using the same elastic constants, as per Fig. 17. This section through the failure envelope is entirely fibre dominated even though, remarkably, at two points three failure modes are predicted to virtually coincide. This same dominance of the ± 30° fibres was also found in the companion solutions for the maximumstrain models in Fig. 15 of Ref. 7, with the right hand end of the failure envelope defined by the 90° fibres. Figure 21 does indicate that the matrix in the same ± 30° plies is almost ready to fail simultaneously when sufficient in-plane shear is added to the critical x load. Likewise, the matrix in the 90° plies almost becomes critical when a small x stress is combined with the critical xy stress. The matrix-shear cut-off for the 90° plies shown in Fig. 21 should not really be a flat plateau; it should droop, in accordance with eqn (15), as positive and negative 0° stresses are added. Whether or not this reduction would undercut the ± 30° fibre failures shown is unclear. Unfortunately, quantifying the issue would need a further formula, equivalent to eqn (8), but relating the higher-than-nominal transverse matrix strains to those in the lamina. If the test data show a need for further elaboration on this issue, it will be included in the subsequent discussion paper. As an indication of the magnitude of any such effect, eqns (3) and (6) can be combined to read as follows: mT = lamT

冉

1d

1 1 (1  v2m)EfTm + vmvfTL E

冊

(36) ≥1

The equation satisfies obvious sanity checks for infinitely stiff fibres and for fibres with the same properties as the matrix. Exploratory evaluation of this equation with typical fibre and polymer material properties indicates that the strain amplification would not be trivial. Although not requested by the editors, the other section cut through Fig. 17, the xy  y plane, is included here in Fig. 22, for completeness. The varying failure mechanisms for each facet are identified.

Fig. 22. Additional cross-section of failure envelope in stress space for (90°/ ± 30°) E-glass/LY556 epoxy laminate.

255 The illustrative matrix-cracking cut-offs in Fig. 17 have counterparts in the other section cuts through the failure envelope, in Figs 21 and 22. In addition to the points established in the preceding example, for the intercepts on the base plane, it is now necessary to also identify the intercepts on the xy axis. According to eqn (19), the intercept for the ± 30° plies will be at a height of 4·0878ktL 1291·8 MPa for the ± 30° plies, while the fracture surface for the 90° plies is parallel to this axis (i.e. vertical) and never cuts it. As before, it should be noted that only the form of these cut-offs is considered definitive; their locations are arbitrary, albeit not unrealistic. For the transverse matrix strengths assumed here, any matrix cracking between the 90° fibres would be preceded by other modes of failure, as shown in Fig. 17. The same is true for matrix cracking between the ± 30° fibres, but only just so, as indicated in Fig. 22.

12. Test problem No. 6: Biaxial (␴y ⴚ ␴x) failure envelope for ( ± 55°)s E-glass/epoxy laminate As for the preceding analyses, the first step in constructing the requested x  y failure envelope for this laminate is to derive it completely on the strain plane and then to convert the answers to the stress plane for presentation in the requested format. The strain-transformation matrix for this laminate is given by eqn (1) of Ref. 7, in the absence of shear strain at the laminate level, as

再冎

冋

x 0·3290 = –2·9238 y –0·6710

册再 冎

–0·6710 0·3290

1 2

(37)

These equations suffice to locate the fibre-failure cut-offs on the laminate strain plane. The matrix-failure cut-offs are 45°-sloping lines parallel to the equal-biaxial-strain diagonal given by

y = x ±

LT 0·939693

(38)

These formulae, in conjunction with the 0° lamina failure envelope defined in Fig. 10, enable the laminate failure envelope to be completed on the strain plane, as is done in Fig. 23. The specific strains used are

tL = 0·02807, cL = 0·01754, LT = 0·04, R = 5·159 The failure envelope is of a skewed diamond shape, with two opposite sides defined by fibre failures and the other two by predicted matrix shear failures. No other failure mechanisms come close to matching these, as is discussed in the companion paper.7 The only theoretical influence of the truncation for shear failures of the fibres is confined to the far upper left corner, the potential cut-off in the lower right corner being preceded by the fibre failure in compression. The corresponding failure envelope on the stress plane, per the request by the organizers in their Fig. 15, is a sloping rectangular box, as shown in Fig. 24. The material properties and stiffnesses used here to convert the strains into stresses are as follows, being essentially the same as in Ref. 7, which contains the justifications for these

256

Fig. 23. Failure envelope for ( ± 55°) E-glass/MY750 epoxy laminate on strain plane, according to generalization of maximum-shear-stress criterion.

precise values, particularly the establishment of secant values for the nonlinear matrixdominated stiffnesses. EL( = E1) = 45·6 GPa, ET( = E2) = 5·51 GPa, GLT( = G12) = 1·82 GPa, vLT( = v12) = 0·278 A11/t = 9·77 GPa, A12/t = 10·65 GPa, A22/t = 23·62 GPa, and  = 0·9907 These stiffnesses pertain to the fibre-dominated segments of the failure envelope. As in the previous problem, it is more appropriate to use the secant value for the in-plane shear behaviour of the matrix at the very end of the stress strain curve described in Ref 4. However, both the in-plane shear stiffness and the transverse stiffness have been estimated corresponding to an in-plane-shear strain of 0 036, which is close to the ultimate strain cited of 0·04. Thus, again, the distinction can be ignored, for this case, but not necessarily for all others.

257

Fig. 24. Failure envelope for ( ± 55°) E-glass/MY750 epoxy laminate on stress plane, according to generalization of maximum-shear-stress criterion.

The results, on both the strain and stress planes, are almost indistinguishable from those in Figs 16 and 17 of Ref. 7 for the original and truncated maximum-strain failure models. A possible matrix cracking cut-off has been included to show how this might affect the failure envelope, if the author were able to locate it with certainty. The cut-off shown corresponds with an assumed location for point P in Fig. 9 as being almost coincident with the equal-biaxial-tension point B in Fig. 10. The coefficient k in eqn (16) is set equal to 1; i.e. at point P, x = y = tL. According to eqn (18), point Q will remain on the same –45° diagonal on the laminate strain plane, but its distance from the origin will have changed (from that for the lamina) to x = –y = –5·1754tL. The stresses follow from these strains in exactly the same manner as for any other strains, per eqn (35). The predicted matrix-cracking characteristic, intercepting the x and y axes at stresses of 336·8 and –1884·3 MPa, respectively, would be scaled proportionally for other positive values of k. The effect of such matrix cracking would be to truncate the top corner of the failure envelope in Fig. 24, as shown. Since the fibres are more closely aligned with the y-, rather than the x-axis, this makes sense. Precisely the same cut-off would be applicable for the truncated maximum-strain failure model in Ref. 7. The author cannot assert that matrix cracking either would or would not prevail in this case; all he can predict is the form of such a cut-off. It would be a straight line, parallel to the one shown. The dominant emphasis of this paper is on characterizing the now more precisely located fibre-failure cut-offs derived from the adoption of a generalized maximum-shear-stress failure criterion.

258 13. Test problems No’s 7 and 8: Stress–strain curves for ( ± 55°)s E-glass/epoxy laminate No solutions are provided for the tasks assigned in the editors’ †Figs 16 and 17. The author’s contributions to this field have been in regard to physically realistic and approximate failure envelopes because he sees a very great need for improvement with respect to many of the traditional failure models. He has no special expertise to offer in regard to stress–strain curves. Of the two stress–strain plots which the organizers requested in their Figs 16 and 17, the uniaxial load case (x = 0) is expected to be highly nonlinear, since the matrix is predicted to fail by ductile shear, while that for biaxial loading (y = 2x) should be almost linear, since the fibres are predicted to fail first. 14. Test problem No. 9: Biaxial (␴x ⴚ ␴y) failure envelope for (0°/ ± 45°/90°)s carbon/ epoxy laminate As for the preceding analyses, the first step in constructing the requested x  y failure envelope for this laminate is to derive it completely on the strain plane and then to convert the answers to the stress plane for presentation in the requested format. For this laminate, however, the solution is simplified greatly by the absence of any influence of the fibres in the ± 45° plies, except for the equal-biaxial-strain points in tension and compression (for which the fibres in all directions are equally critical). The failure envelope needing to be quantified is thus defined by the fibres in only the 0° and 90° directions, and the matrix in the ± 45° plies. In this case, the author’s experience with this class of well-interspersed carbon/epoxy laminates enables him to state confidently that structurally significant matrix splitting within any lamina is most unlikely to precede failure of the fibres. Figure 25 shows the failure envelope on the strain plane for this quasi-isotropic laminate. The construction requires only the following properties.

tL = 0·0138, cL = –0·01175, LT = 0·020, vLT = 0·28, vfTL = 0‚0133333, R = 1·488 In comparison with the corresponding Fig. 18 of Ref. 7 for the solutions according to the original and truncated maximum-strain failure models, the present solution differs very little from that for the truncated failure model, but is appreciably different from the original maximum-strain failure model for strains of opposite sign. (All three models give almost identical answers for biaxial strains of the same sign.) Figure 26 shows the corresponding failure envelope plotted on the stress plane, in accordance with †Fig. 18 of the editor’s requests. Additional properties needed are as follows. E1 = 126 GPa for all stresses except equal biaxial tension (for which E1 = 141 GPa), E2 = 11 GPa, G12 = 4·2 GPa, A11/t = A22/t = 54·60 GPa, A12/t = 17·47 GPa, A66/t = 18·57 GPa. (The explanation of how these values are justified is given in the companion paper.7) In comparison with the truncated maximum-strain failure model evaluated in Fig. 20 of Ref. 7,

259

Fig. 25. Strain-based-failure envelope for (0°/ ± 45°/90°) quasi-isotropic AS4/3501–6 carbon/epoxy laminate, according to generalization of maximum-shear-stress criterion.

the equal-biaxial-stress points at the ends of the sharp failure envelope have been diminished by barely 1%, and the equal-and-opposite stress points on the sides bulged out by about 11%, confirming that the approximation of the constant-stress cut-off by a vertical constant-strain line, and the fibre shear-failure cut-off by a 45° sloping line, are quite reasonable for carbon fibres in polymer matrices. (The peak shear strain occurs on the –45° axis through the origin; the discrepancy is even less than this 10% conservatism off that axis, diminishing to zero for a pure uniaxial load.) The predicted uniaxial tension and compression strengths are 662 and 576 MPa, respectively. The predicted tensile strength differs only slightly from, and is bounded between, those estimates of 676 and 656 MPa given in Ref. 7 for the original and truncated maximumstrain failure models. All three predictions for the compression strength are identical. The equal-biaxial stress point is computed to be 983 MPa in tension, according to the basic set of elastic constants, versus 1086 MPa when the strain hardening of the fibres is accounted for. The equal-biaxial stress point in compression is predicted to be at –837 MPa. These strengths differ by only 1% from those predicted in Ref. 7. The predicted in-plane shear strength is 365MPa, indicating that the prediction of 328 MPa by the truncated maximum-strain failure

260

Fig. 26. Stress-based failure envelope for (0°/ ± 45°/90°) quasi-isotropic AS4/3501–6 carbon/epoxy laminate, according to generalization of maximum-shear-stress failure model.

model is conservative by only 10%. A comparison with the original maximum-strain model is complicated by the fact that the predicted failure for in-plane-shear loads is at 436 MPa, but for failure under compression of the fibres in only one of the four directions. Were it not for this limit on compression strength, the original maximum-strain failure model would have predicted an in-plane shear strength of 512 MPa for this laminate, confirming that the present model is more realistic for carbon/epoxy-type composites. The fact that the tension-compression (shear) corners of the failure envelope according to the original maximum-strain failure model are predicted to be truncated by both fibre and matrix shear failures seems to suggest that these data could not possibly prove the absence of such a cut-off as the author has long advocated for in-plane shear loads, at least for carbon/ epoxy laminates. 15. Test problems No’s 10 and 11: Stress–strain curves for (0°/ ± 45°/90°)s carbon/epoxy laminate No solutions are provided for tasks assigned by the editors in their †Figs 19 and 20. Comments on what the author would expect such solutions to reveal are given in Ref. 7. 16. Test problems No’s 12 to 14: Stress–strain curves for ( ± 45°)s and (0°/90°)s carbon/ epoxy laminate The author has not responded to the tasks in the editors’ †Figs 21–23, for the same reasons as he refrained from responding to the other similar requests. The absence of such responses in

261 no way inhibits the ability of the reader to evaluate the solutions to the problems for which an answer has been provided here (and in the companion paper for the two variants of the maximum-strain failure models).

17. Concluding remarks This paper, by means of the worked examples, has illustrated how the classical maximumshear-stress strength criterion for isotropic metals can be generalized to orthotropic materials like carbon fibres and to fibre-reinforced polymer composites. In contrast with the earlier formulations of this failure model for composites, in which it was assumed that the 45° cut-off on the lamina strain plane could be used for all fibre/polymer composites, is now seen to be very realistic only for carbon-type fibres. Therefore, the truncated maximum-strain failure model evaluated in the companion paper, Ref. 6, continues to be very suitable for use with carbon-epoxy type composite laminates. The original maximum-strain failure model, with no shear cut-offs at all, at the lamina level, is slightly unconservative for laminates reinforced by isotropic fibres, like glass. However, it is far closer to the ‘correct’ answer deduced by assuming that the 45° cut-off for shear loads applies precisely only to the fibres, and calculating the equivalent transverse strains in the laminae, than is the truncated maximum strain failure model with a 45° cut-off at the lamina level. This exercise has led to the realization that the compressive instability cut-offs, as for highstrain small-diameter fibres, are better characterized by constant strain lines at the lamina level, even though they originate from constant stress lines at the fibre level. This is true for both glass and carbon fibres, in the former case because of the high magnification factor between lamina and fibre transverse strains, and in the latter case because of the low transverse stiffness of the fibres. This difference is significant for glass-fibre-reinforced polymer composites but is inconsequential for carbon-fibre-reinforced laminates. Therefore, the truncated maximum-strain failure model evaluated in the companion paper,7 continues to be very suitable for use with carbon/epoxy type composite laminates. The present theory is presumed by the author to be the most suitable of the three he has evaluated for glass/epoxy type composites. It would seem, however, that the best classical approximation to this model would be the maximum-stress failure model for the fibres, with no truncations for fibre failures under shear loads. Ironically, at the lamina level, the maximum strain failure model more closely approximates the maximum stress failure model for fibres than does the lamina-level maximum stress failure model. (The maximum-stress and maximum-strain failure models will give essentially the same answers for carbon/epoxy type laminates because the transverse Poisson ratio is extremely small. But, again, the lamina-level maximum-strain model is slightly the better.) One of the advantages claimed for single-function interactive failure models was the associated mathematical simplicity of the ‘characterization’ of the lamina. This was lost, of course, once it became necessary to invoke ‘progressive-failure’ techniques to produce more acceptable strength predictions for laminates. It should now be clear that only mechanistic failure models can realistically characterize the strength of fibre-polymer composite materials. By a strange co-incidence, the precise characterizations of every fibre-failure mechanism considered here are simple flat surfaces, each of which is simple to transform from lamina to laminate and from strain to stress. Those matrix failure characterizations used here are also

262 presented as flat surfaces, but it is acknowledged that the ductile-shear failure mechanism really is more complicated than this whenever the fibres do not fail first and thereby limit the stresses to which the matrix is subjected before the laminate fails. The capability of assessing possible matrix failures by dedicated equations devoid of any inappropriate interactions with fibre failures is seen to be important, particularly in the case of the ( ± 55°)s laminate. This confirms the approach taken by Puck, as in Ref. 6, and casts doubt upon the reliability of those models in which the fibres and matrix have been homogenized into a single composite ‘material’. For these particular multidirectional composite laminates, the author does not see a need to have interacted other components of the matrix stress with the dominant in-plane shear component. However, the present assessment of such stresses applied in combination to a unidirectional lamina shows very great differences with respect to the noninteractive solutions according to the two variants of the maximum-strain failure model in Ref. 7. The present assessment of the effect of combined transverse and shear loads in such a lamina is strongly influenced by the customarily ignored intralaminar residual thermal stresses. However, the author has not addressed the far more difficult issue of possible intralaminar matrix cracking between the fibres in each ply – other than to explain why the traditional macromechanical approach to this has been scientifically unsound. The contributions of the micromechanicians to this exercise are awaited with interest. References 1 Hart-Smith, L. J., Approximate analysis methods for fibrous composite laminates under combined biaxial and shear loading. Douglas Aircraft Company, IRAD Report MDC-J9898, March 1984. 2 Hart-Smith, L. J. Predicting the strength of fibrous composites by an orthotropic generalization of the maximum-shear-stress (Tresca) condition, McDonnell Douglas Paper MDC 91K0068. Presented to Second SERCI.Mech.E. Expert Meeting, Failure Mechanism(s) and Criteria for Predicting the Performance of Polymer-Based Composite Structures, St. Albans, UK, 23–25 September 1991; published in Proc. Inst’n. Mech. Eng’rs, Part G, Journal of Aerospace Engineering, 1994, 208, 9–18. 3 Hart-Smith, L. J., How to calculate the in-plane strengths of fibre-polymer composites, McDonnell Douglas Paper MDC 92K0023. Presented to 25th MIL-HDBK-17 Meeting, Laughlin, Nevada, 31 March-2 April 1992; published in SAMPE Journal, 1992, 28(6), 25–35. 4 Soden, P. D., Hinton M. J. and Kaddour A. S., Lamina properties, lay-up configurations and loading conditions for a wide range of fibre reinforced composite laminates. Compos. Sci. Technol., 1998, 58(7), 1011. 5 Baker, A. A., Bonded composite repair of metallic aircraft components – overview of Australian activities. AGARD-CP-550, pp. 1.1–1.14. 6 Puck, A., Festigkeitsberechnung an Glassfaser/KunstoffLaminaten bei zusammengesetzter Beanspruchung. Kunststoffe, 1969, 59, 780–787. (English summary: calculating the strength of glassfibre/plastics laminates under combined load. German Plastics, 1969, 59, 18–19.) 7 Hart-Smith, L. J., Predictions of the original and truncated maximum-strain failure models for certain fibrous composite laminates, McDonnell Douglas Paper MDC 97KOO10. Compos. Sci. Technol., 1998, 58(7), 1151. 8 Chamis, C. C., Micromechanics strength theories in Composite Materials, Vol. 5, Fracture and Fatigue, ed. L. J. Broughtman, Academic Press, New York, 1974, pp. 126–131 9 Jones, R. M., Mechanics of Composite Materials. Scripta, Washington, DC, 1975. 10 Outwater, J. Q., Jr, The mechanics of plastics reinforcement in tension. Modern Plastics, 1956, 33, 156–162 and 245–258.

263 11 Brown, R. J., Jr, Simplified stress analysis of filament reinforced plastic pressure vessels. Soc. Plastic Engineers Journal, 1961, 17, 989–991. 12 Black, J. B., Jr and Hart-Smith, L. J., The douglas bonded tapered rail-shear test specimen for fibrous composite laminates, Douglas Paper 7764. In Proceedings of the 32nd International SAMPE Symposium and Exhibition, Anaheim, California, 6–9 April, 1987, pp. 360–372. 13 Hart-Smith, L. J. Generation of higher composite material allowables using improved test coupons. Douglas Paper 8557, in Proceedings of the 36th International SAMPE Symposium and Exhibition, San Diego, California, 15–18 April 1991, pp. 1029–1044; modified and republished in Aerospace Composites & Materials 1991 3(3), 26–30, 58, and 1991 3(4), 13–16, 18. 14 Hart-Smith, L. J., Backing out equivalent unidirectional lamina strengths from tests on cross-plied laminates, McDonnell Douglas Paper MDC 91K0078. In Proceedings of the 37th International SAMPE Symposium and Exhibition, Anaheim, California, 9–12 March 1992, pp. 977–990. 15 Hart-Smith, L. J., Fibrous composite failure criteria – fact and fantasy, McDonnell Douglas Paper MDC 93K 0047. Presented to Plenary Session of Seventh International Conference on Composite Structures, Paisley, Scotland, 5–7 July 1993, abridgement to be published under the title: A bonafide fibrous composite failure criterion in Composites. 16 Hart-Smith, L. J., A re-examination of the analysis of in-plane matrix failures in fibrous composite laminates, McDonnell Douglas Paper MDC 95K0069. Composites Science and Technology 1996, 56(2), 107–121. 17 Rosen, B. W., Fibre composite analysis and design, Material Sciences Corporation, Report DOT/ FAA/ CT88/18, 1988. 18 Hinton, M. J and Soden, P. D., Predicting failure in composite laminates: the background to the exercise. Compos. Sci. Technol., 1998, 58(7), 1001.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

264

CHAPTER 3.7

Failure analysis of FRP laminates by means of physically based phenomenological models* A. Puck and H. Schürmann† University of Technology Darmstadt, Fachgebiet Konstruktiver Leichtbau und Bauweisen, 30 Petersenstrasse, 64287 Darmstadt, Germany Abstract A realistic failure analysis of structural members made of FRP composites requires consideration of the non-linear stress/strain relationships. In laminate design and for determination of safety factors of FRP structural members, there is a strong need for fracture criteria and degradation models which are simple enough for application in common engineering problems while still being in good agreement with physical reality. It is essential to distinguish between fibre failure and inter-fibre failure by separate failure criteria. A recent success was the introduction of inter-fibre failure criteria which model the brittle failure behaviour of composites very realistically. These not only provide realistic stresses to failure, but also indicate the crack direction, which is an extremely important piece of information when evaluating the likely effect of fracture. After crack initiation, the stiffnesses of the affected plies degrade gradually with increasing load, until fibre fracture in one ply provokes ultimate failure of the laminate. Also, an inclined wedge-shaped inter-fibre crack can lead to ultimate failure. From now on, the risk of fracture caused by these phenomena can be evaluated. The determination of the fracture angle which is the key for this evaluation is derived in the present paper. Keywords: laminate design, non-linear stress/strain behaviour, failure criteria, fracture angle, degradation model

Notation Characteristics of the unidirectional layer x1, x2, x3 1, 2, 3

Coordinate system of a unidirectional layer (x1 = fibre direction, x3 = thickness direction) Normal stresses in a unidirectional layer

* This article represents the authors’ contributions to a worldwide exercise to confirm the state-of-the-art for predicting failure in composites, organised by Hinton and Soden.24 † To whom correspondence should be addressed.

265 12 = 21; 13 = 31; 23 = 32

Shear stresses of a unidirectional layer in the elastic symmetry directions. The first subscript indicates the direction normal to the plane on which the shear stress is acting; the second subscript indicates the direction of the shear force. Normal strains of a unidirectional layer.

1, 2, 3

12 = 21; 13 = 31;

23 = 32 Shear strains of a unidirectional layer 储; ⊥ Stressing parallel and transverse to the fibre direction ⊥储; ⊥⊥ Shear stressing transverse/parallel and transverse/transverse to the fibre direction E1, E2 = E3 Elastic moduli of a unidirectional layer in the directions x1, x2, x3 E1t, E2t, E3t; E1s, E2s, E3s Tangent moduli, secant moduli G21; G21s Shear modulus of a unidirectional layer in the x2, x1 direction; secant shear modulus v12 Major Poisson’s ratio XT; XC Tensile strength and compressive strength of the unidirectional layer parallel to the fibre direction YT; YC Tensile strength and compressive strength of the unidirectional layer transverse to the fibre direction S21 Shear strength of a unidirectional layer transverse and parallel to the fibre direction

Characteristics of the fibres f1; f2 Ef1 vf12

1T; 1C XfT; XfC mf

Fibre stress in x1 direction; fibre stress in x2 direction Fibre modulus in x1 direction Poisson’s ratio of the fibres (strain in x2 direction caused by a stress in the x1 direction) Tensile failure strain and compression failure strain of a unidirectional layer in x1 direction Tensile fibre strength and compressive fibre strength in fibre direction (in a UD composite) Mean stress magnification factor for the fibres in the x2 direction, due to the difference between the transverse modulus of the fibre and the modulus of the matrix

Characteristics of the potential fracture plane x1, xn, xt n nt n1

Coordinate system rotated with respect to the fibre direction by an angle  from the x2 direction to the xn direction Normal stress acting on the potential fracture plane Normal/transverse shear stress acting on the potential fracture plane Normal/longitudinal shear stress acting on the potential fracture plane

266 RA R(⊥+ )A RA⊥⊥ RA⊥储  fp p(⊥储 ) ) p(⊥⊥ fw 1d 1D fR() fE(IFF) 21c (1), (1), (r), (r)  =  fE(IFF)

Fracture resistance of a stress action plane against its own fracture caused by one single stress acting in it Fracture resistance of the action plane against its fracture due to transverse tensile stressing Fracture resistance of the action plane against its fracture due to transverse/transverse shear stressing Fracture resistance of the stress action plane against its fracture due to transverse/parallel shear stressing Angle between the x2 axis and the xn axis Angle of the fracture plane Slope of the (n, n1) fracture envelope for n 0 at n = 0 Slope of the (n, nt) fracture envelope for n 0 at n = 0 Weakening factor due to the degrading influence of high 1 stresses Empirical stress value, determining the degradation of the fracture resistance RA of the action plane due to 1 stresses (1d > 0 for 1 > 0; 1d < 0 for 1 < 0) Stress value for linear degradation (1D > 0 for 1 > 0; 1D < 0 for 1 < 0) Angle-dependent reserve factor, stretching factor for the applied stress state which is necessary to achieve IFF at  Angle-dependent effort of a unidirectional layer with respect to IFF (see Abbreviations) Shear stress at the ‘turning point’ of the (2, 21) fracture curve Load-dependent stress; residual stress Degradation factor representing reduction of transverse stiffness due to inter-fibre fracture

Abbreviations CLT FF FPF FRP IFF

Classical laminate theory Fibre failure First ply failure Fibre reinforced plastic Inter-fibre fracture

1. Introduction Strength analysis of laminates is still underdeveloped compared to the analysis of stresses and strains. Specifically, there is a lack of fracture criteria and degradation models which are both close to physical reality and simple enough for application in engineering design of laminates with optimum strength. Few of the present models meet both requirements. Significant characteristics of the fracture criteria and degradation models presented in this paper were already described more than 25 years ago in two publications.1,2 The most important statements of these papers were:

267 • Two basically independent fracture criteria have to be applied simultaneously: one for fibre failure (FF), and one for inter-fibre failure (IFF), because the effects of both fracture phenomena are totally different, as well as the design methods to avoid one or the other failure. • The (2, 21) failure envelope should not be described by a single equation, which includes both the transverse tensile strength, YT , and the transverse compression strength, YC. The reason for this is that the strength characteristics in the compression domain (2 < 0) are independent of the tensile strength, and the strength characteristics in the tension domain (2 > 0) are also independent of the compressive strength. • Stiffness degradation by smearing of the cracks after crack initiation should be continuous instead of sudden. Additionally, it has to be applied selectively for (2 < 0) different from that for (2 > 0). • When judging the significance of IFF, one has to be careful to distinguish between the fairly harmless crack formation caused by transverse tensile stresses (2 > 0), compared to the ‘explosive’ effect that oblique cracks, caused by transverse compressive stresses (2 < 0), have because of their wedge shape. On the basis of these principal findings, numerous details of the fracture criteria have been improved in the past; specifically, the extension of the fracture criteria for a three-dimensional state of stress has to be mentioned.3,4 IFF criteria, in particular, have been significantly improved recently by placing them on a solid physical basis5–10; an extensive coverage of these new developments has been given previously.11 Many efforts were made during recent years to describe the complex damage and fracture phenomena of composites physically in correct ways, i.e. in many cases by means of fracture mechanics, and the new research field of damage mechanics has rapidly grown up.12 The results of this approach provide important insights into micromechanical behaviour; however, for application in current engineering design they still seem to be inappropriate. Three different kinds of models for the description of damage and fracture behaviour have been used hitherto: (a) micromechanical models, (b) probabilistical models, (c) phenomenological models. In the present paper, phenomenological models are applied, but they are placed on a solid physical basis. They not only provide results for the stress level which leads to crack initiation and to fracture, but also indicate the direction of the cracks. Nevertheless, their application is surprisingly simple and therefore provide an interesting opportunity for use in everyday engineering design. In principle, damage and fracture analysis of laminates requires the following: 1. analysis of strains and stresses ply by ply; 2. fracture criteria for single plies; 3. degradation models to include the effects of partial fracture, which often does not lead to ultimate failure of the laminate;

268 4. a computer program which simulates the gradual fracture process by applying the above sequences iteratively. In the following, these four considerations are discussed in succession. According to the problems of the Failure Analysis Exercise,24,25 the discussion is limited to a state of plane stress (1, 2, 21).

2. Remarks on the analysis of strains and stresses For the analysis of strains and stresses of single plies the well-known classical laminate theory (CLT) is used. However, for application in fracture analysis of FRP laminates it has to be modified in order to include the non-linear relationships between stress and strain. These nonlinear effects can be observed, especially in (21, 21) diagrams, but also in (2, 2) diagrams if 2 is a compressive stress. Several opportunities to include non-linear effects exist, but not all of them are equally practical. Generally, it is assumed that the non-linear stress/strain relationships remain unchanged, no matter whether the load is 1 alone, 2 alone, 21 alone, or a superposition of all three stresses. This means, for example, that a (21, 21) diagram remains unchanged when, in addition to a 21 stress, a 2 stress is superimposed which is proportional to 21. This is not found in reality, as can be concluded from the stress/strain diagrams under several different loading combinations (2, 21)13 (Fig. 1). The additional microdamage which is caused by the 2 load leads to a shallower (21, 21) curve than with 21 load alone and vice versa. Our computer program makes it possible to take such interactions between the stresses 2 and 21 into account. However, in order to do this some stress/strain diagrams for combined (2, 21) loading must be available, which is not the case for the laminates which have to be analysed in the problems given in this exercise. Generally, it should be pointed out that neglecting the interaction in combined (2, 21) loading results in calculating a level of crack initiation which is too low. For calculation of the stresses 2 or 21, respectively, on the basis of experimentally determined (2, 2) or (21, 21) diagrams, it is possible to use either tangent moduli E2t = d2/d2 or secant moduli E2s = 2/2 for instance. Calculation with secant moduli is more practical, since the tangent modulus method normally requires many sufficiently small calculation steps, starting from zero load, before the final stress/strain state can be reached, whereas the secant modulus method allows one to reach this point with one large load step and only a few iterations. Figure 2 shows this for a 21 load. Furthermore, the secant modulus method offers the advantage of being self-correcting. With increasing load, the making of a mistake when computing the stress for a certain load level does not change the calculated stress state for a higher load level. In contrast, when the tangent modulus method is used, the current stress is always the sum of many , so that mistakes add up. For 21 shear stresses, it is irrelevant whether the secant modulus, G21s, is taken at the shear stress, 21, or at the shear strain, 21. However, when calculating with non-linear (2, 2) relationships and 1 and 2 are acting simultaneously, the secant modulus can only be deduced at the calculated stress 2, and not at the calculated strain 2, since the strain 2 is dependent

269

Fig. 1. Stress/strain diagrams of glass-fibre/epoxy UD specimens under combined (2, 21) loading.

270

Fig. 2. Calculating stress and strain of a UD layer with non-linear (21, 21) behaviour by iteration.

Fig. 3. Changing of the (2, 2) curve caused by the simultaneous action of a stress 1.

not only on the stress 2, but also on the stress 1 because of transverse contraction (Poisson’s effect) (Fig. 3). In our program, all stress/strain curves are approximated by spline functions of third order. Calculation above crack initiation level might require an extrapolation of the (2, 2) or (21, 21) diagrams beyond the fracture point. Nevertheless, the method of performing the stress analysis after crack initiation exactly can only be explained after having described the selected degradation model.

271 3. Failure conditions for fibre failure (FF) Up to now, a state of stress is regarded as being the limit for fibre failure in a unidirectional composite when, under combined loading, a stress 1 parallel to the fibres evolves which is equal to the longitudinal strength (tensile strength XT or compressive strength XC) determined by a uniaxial test. Correspondingly, a failure condition for the strain 1 which is parallel to the fibres of a unidirectional laminate is formulated: 1 must be equal to the longitudinal fracture strain, 1T or 1C.3,4 In reality, the truth lies between these two approaches, as shown in the following. Fundamentally, we assume that fibre failure in a UD composite under a combined state of stress (1, 2, 3, 23, 31, 21) will occur at the same fibre stress as that which is acting in the fibres at failure under a unaxial stress 1. Starting from this failure hypothesis we have to begin with a failure condition for the fibre instead of for the unidirectional composite: f1 = XfT for

f1 0

(1)

f1 = –XfC

f1 < 0

(2)

for

By this it is assumed that for the fibre, the failure condition of the maximum normal stress in the fibre direction holds. It should be pointed out that XfT and XfC are the tensile stress or the compressive stress in the fibre which are reached under uniaxial tensile or compressive load with 1 at fracture of the unidirectional composite material and not in single fibres or fibre bundles. Assuming linear-elastic material behaviour, these are given by: XfT =

XT XC Ef1 = 1TEf1 and XfC = Ef1 = 1CEf1 E1 E1

(3)

While XfT can be regarded as the ‘true’ tensile strength of the fibre (embedded in the composite), XfC is usually not the ‘true’ compressive strength of the fibre, because at f1 < 0, failure mostly occurs through elastic instability (so-called microbuckling or kinking) of the fibres embedded elastically in the matrix. However, it can be assumed that the buckling is not influenced by an additional stress 2, at most by an additional stress 21.14 Thus, XfC can be regarded as a constant. Already at a state of stress where 1 = 0, but 2 0, stresses of opposite sign occur in fibres and matrix in a direction parallel to the fibres because of their different elastic moduli and Poisson’s ratios (for 2 > 0 it is f1 < 0 and for 2 < 0 it is f1 > 0. This is taken into account by the equation:

f1 =

f1 vf12  mf 2 Ef1 Ef1

(4)

The factor mf accounts for a ‘stress magnification effect’ caused by the different moduli of fibres and matrix (in the 2 direction), which leads to an uneven distribution of the stress 2 from a micromechanical point of view: in the fibres, it is slightly higher than in the matrix. The factor mf is a mean magnification factor m of the transverse stress (index ) for the fibre (index f ) (for glass fibre mf 1.3; for carbon fibre mf 1.1).

272 Because of the tight bonding between fibres and matrix, the strain parallel to the fibres is the same for the fibres and the composite:

f1 = 1

(5)

Thus, the longitudinal stress in the fibre f1 under combined (f1, f2) loading is obtained from eqn (4) as: f1 = 1Ef1 + vf12mf 2

(6)

When this stress reaches the same stress level, 1, the fibre is subjected to at fracture of the unidirectional composite under uniaxial tension or compression, the fibre breaks: f1 = 1TEf1 or f1 = –1CEf1

(7)

Therefore, on the basis of the failure hypothesis mentioned, the failure condition for fibre failure under combined (f1, f2) loading is: 1 1T 1 1C

冉 冉

1 +

冊 冊

vf12 mf 2 = 1 for (. . .) 1 Ef1

(8)

vf12 mf 2 = –1 for (. . .) < 1 Ef1

(9)

1 +

when the fracture strain 1C is written as a positive value. In a previous publication15 it was emphasised that experimental results show that the compressive strength, XC, parallel to the fibres and, consequently, also IC are significantly reduced, when in addition to 1 < 0, a significant shear stress, 21, is superimposed. This can be explained by the observation that a 21 load promotes fibre kinking by compression parallel to the fibres. Edge noted an interaction equation,15 which yields XIC = 0 when 21 reaches the inter-fibre failure limit, i.e. 21 = S21. This is in contrast with our experimental experience: if a unidirectional ply in a laminate is already damaged by cracks caused by 21 loading, it can still transfer significant 1 compressive stresses. Therefore, we are using an empirical shear correction substantially lower than the one described by Edge: 1 1C

冏冉

1 +

vf12 mf 2 f1

冊冏

+ (10 21)2 = 1 for (. . .) < 0

(10)

where (10 21)2 is a pure empirical approach. It is preferable to use an expression with 21 instead of 21, because the value of 21 is uncertain after crack initiation while the value of 21 is not. It has been pointed out16 that XIC significantly increases when a hydrostatic pressure 2 = 3 acts at the same time. It is still not clear whether this effect also takes place when 2 0 and 3 = 0. Thus, we do not include any further interactions between 1 and 2 at fibre failure than the ones mentioned above. It should be mentioned that Hart-Smith17 contradicts the failure hypothesis of the constant longitudinal stress of the fibres. He assumes a distinct reduction of X1, when the fibres suffer

273 approximately equal strains but with opposite sign, in the longitudinal and transverse direction: this must be studied in more depth. 4. Failure conditions for inter-fibre failure (IFF) In this research field, the greatest improvements have been achieved recently.5–8 The knowledge gained by experiments with carbon-fibre/epoxy and glass-fibre/epoxy laminates (which are to be analysed in this paper) teaches that unidirectional layers behave in a very brittle manner at failure, particularly at inter-fibre failure. Without any previous macromechanically apparent plastic deformation, abrupt material separation occurs when the specimen reaches the point of failure.9 Considering this observation, it seems surprising that many scientists, when implementing failure criteria for brittle composites, decided to follow the yield criteria of von Mises or Hill which hold only for ductile materials. It seems much more appropriate to use the failure criteria of Mohr18 as guidelines, these having been developed for materials that exhibit brittle fracture characteristics. This idea was originally formulated, to the knowledge of the authors, by Hashin.19 He stated that transferring the fracture criteria of Mohr on unidirectional composites would put fibre failure criteria and inter-fibre failure criteria on a solid physical basis. Hashin never accomplished his idea, however, considering the computational effort to be far too great. However, the enormously increased computer capacities which are nowadays available for almost every design engineer make it desirable to develop and apply these ideas further.5 The current achievements are extensively discussed elsewhere.11 Surprisingly, it has recently been found that application of new fracture criteria which are based on physical foundations not only provides additional information about the angle of fracture, but also needs only little computer capacity when limited to a state of plane stress (1, 2, 21), compared to the capacity required by currently used failure criteria. Therefore, in this paper we already apply the new failure criteria, which are briefly explained in the following; a broad coverage can be found elsewhere.11 Like Mohr, we proceed on the hypothesis that fracture is exclusively created by the stresses which act on the fracture plane. In the case of inter-fibre fracture on an inclined plane parallel to the fibres these are a normal stress, n, and two shear stresses, nt and n1 (Fig. 4). The stress n represents a transverse stressing (⊥), the shear stress nt a transverse–transverse shear stressing ⊥⊥ and the shear stress n1 a transverse–longitudinal shear stressing (⊥储). The term ‘stressing’ has been introduced by Puck11 in order to distinguish between stress conditions of different character, e.g. 储, ⊥, ⊥⊥, ⊥储. When n > 0, i.e. transverse tensile stressing, it causes fracture in conjunction with nt and n1. Because of the existing symmetry, the direction of the shear stresses cannot influence fracture.11 The simplest approach one can think of for n > 0 would consequently be:5,19

冉 冊冉 冊冉冊 n R(⊥+ )A

2

+

nt RA⊥⊥

2

+

n1 RA⊥储

2

= 1 for n 0

(11)

In this equation, generally the common strengths cannot be used in the denominators. This can easily be seen when recalling the fundamental hypothesis: fracture of the plane in which the stresses act which cause fracture is exclusively created by the stresses which act in this plane.

274

Fig. 4. Three-dimensional stresses on a UD composite element. (x1, x2, x3) coordinate system is fixed to fibre direction (x1), laminate mid-surface (x2) and thickness direction (x3). The (x1, xn, xt) coordinate system is rotated by an angle fp from the x2 direction to the xn direction which is normal to the fracture plane. The inter-fibre fracture is influenced by the the three stresses n, nt, n1 only (according to Mohr’s strength theory).

Consequently, the three stresses n, nt, n1, which must necessarily have a common stress action plane, must be compared to the fracture resistances RA (A representing action plane) of their action plane, and not just to some strength which might belong to any other fracture plane. If a stress n > 0 is acting alone and fracture occurs in its stress action plane, the fracture resistance R(⊥+ )A equals the tensile strength YT. This is the case with the UD materials of laminates examined here, which does not mean that it is so in general. However, a ⊥储 shear stressing always causes fracture only in its action plane;11 thus, we can write RA⊥储 = S21. With RA⊥⊥, the situation is totally different for the materials to be analysed. A shear stressing ⊥⊥ which acts alone does not lead to shear fracture in its action plane, but to tensile fracture in a plane which is inclined by 45° to the action plane of the shear stress. Up to now, no experiment is known which would allow a direct measurement of the fracture resistance R⊥⊥A against fracture caused by a ⊥⊥ stressing. Hence, R⊥⊥A has to be derived from the transverse compressive strength, YC, assuming a mathematically formulated master fracture surface (see Table 1). The unusual part of the new fracture criteria can be seen in their definition in the (n, nt, n1) stress space. Still, stress analysis gives the existing stresses in the (x1, x2, x3) coordinate system, which is dependent on the fibre direction and the mid-plane of the unidirectional ply. From the (1, 2, 3, 23, 31, 21) stress space, one gets into the (n, nt, n1) stress space (Fig. 4) through the well-known transformation rules, which are for a state of plane (1, 2, 21) stress: 1 =  1 n = 2 cos2  nt = –2 sin  cos  n1 = 21 cos 

(12)

275 Generally, the calculation procedure is now to insert the stresses (n, nt, n1), which are in terms of 1, 2, 21 and , in a fracture condition like eqn (11). The angle fp (fp represents failure plane) then has to be determined for which the risk of fracture in the related stress action plane () has its global maximum. With fp being determined by this condition, the stresses (n, nt, n1) at fracture can be calculated, together with the reserve factor, fR, which is the factor all existing stresses have to be multiplied with in order to achieve fracture. The reserve factor, fR, is the same for the stresses (n, nt, n1) as well as for (1, 2, 3, 23, 31, 21) because of eqn (12). A tensile stress n > 0 promotes fracture, whereas a compressive stress n < 0 impedes shear fracture. For n < 0, the shear stresses nt and n1 (or just one of them) have to cause fracture against an additional fracture resistance, which increases with increasing | n | like an internal friction. A simple approach describing this effect is performed by the following equation,11,20 which resembles the fracture hypothesis by Mohr-Coulomb:21

冉

A ⊥⊥

R

nt )  p(⊥⊥ n

冊冉 2

+

冊

n1 R  p(⊥储 )n A ⊥储

2

= 1 for n < 0

(13)

Both fracture conditions, eqns (11) and (13), have been further modified to achieve a better agreement with experimental results:11

冉 冊 n R(⊥+ )A

c2

2

冉 冊冉冊

n nt + ( + )A R⊥ RA⊥⊥

2

n1 RA⊥储

2

= 1 for n 0

(14)

2nt 2 + A 2 n1 (  ) A = 1 for n < 0 () A )  2p⊥⊥R⊥⊥n (R⊥储)  2p⊥储 R⊥储n

(15)

+ c1

+

A 2 ⊥⊥

(R

These two equations are the basis for the inter-fibre fracture condition worked out in this paper. An essential simplification is performed in eqn (15) by coupling the thus far independent ) slopes p(⊥⊥ and p(⊥储 ). To calculate eqn (15) with reasonable computational effort, we assume in this paper that the following coupling exists:

冉冊

) p(⊥⊥ p(⊥储 ) p = = = const. A AA R⊥⊥ R⊥储 R

(16)

This simplification seems to be acceptable according to our experience. In general, p(⊥储 ) is fitted to the experimentally deduced slope of the (2, 21) fracture envelope for 2 < 0 at the ) point 2 = 0, so that p(⊥⊥ = p(⊥储 )RA⊥⊥/RA⊥储 is fixed as well. With the assumption made in eqn (16), eqn (15) reduces to a much simpler form:

冉 冊 冉 冊 冉冊 nt RA⊥⊥

2

+

n1 RA⊥储

2

+2

p n = 1 for n < 0 R

(17)

276 The fracture body described by eqns (14) and (17) in the (n, nt, n1) stress space is an ellipsoid on the tensile side n > 0 and a paraboloid (with parabolic contours and elliptical cross-sections) on the compression side n < 0. The surface of this fracture body will be referred to as the master fracture surface (Fig. 5). In accordance with experimental findings, applying eqn (14) for the special case of plane stress (2, 21) gives anywhere for 2 > 0 a fracture angle fp = 0°. Recalling the transformation rules (eqn (12)) with n = 2; n1 = 0; n1 = 21 the fracture condition eqn (14) reduces to

c2

冉 冊 2 R(⊥+ )A

2

+ c1

冉冊

2 21 + R(⊥+ )A RA⊥储

2

= 1 for 2 0

(18)

As described in full detail elsewhere11 we finally arrive at:

冑冉 冊 冉 21 RA⊥储

2

+ 1

p(⊥储+ ) ( + )A R⊥ RA⊥储

冊冉 冊 

2

2 ( + )AA ⊥

R

2

+

p(⊥储+ ) 2 = 1 for 2 0 RA⊥储

(19)

The fracture mechanism described by eqn (19), where 2 and/or 21 cause fracture, will be called Mode A (Fig. 6). Applying eqn (17) to the domain of 2 < 0, it can be found that also on the compression side there is a part of the fracture envelope where the fracture angle fp = 0°. Fracture is originated

Fig. 5. Master fracture surface (for 1 = 0) for UD material in the (n, nt, n1) stress space with fracture resistances R(⊥+ )A, RA⊥储, RA⊥⊥ of the stress action plane. If there is a degradation effect of 1, all dimensions have to be reduced by fw = 1  (1/1d)n = 1  | 1/1D |. The (2, 21) fracture curve follows a contour line (from a to c) and then a boundary line of a cross-section (from c to d). Compare with Fig. 6.

277

Fig. 6. (2, 21) fracture curve for 1 = 0, representing three different fracture modes A, B, C. The curve is generated by two ellipses and one parabola. If a degradation by 1 takes place, the fracture curve shrinks by a factor fw under the condition of geometric similarity.

by 21, whereas 2 impedes fracture. The 21 stress at fracture increases parabolically with increasing compressive stress. In eqn (17), nt = 0, n1 = 21 and n = 2 is assigned. Again, like eqn (19), a fracture condition can be found which is homogeneous and of first degree in terms of stresses:11

冑冉 冊 冉 冊 冉 冊 21 RA⊥储

2

+

p R

2

22 +

p R

2 = 1 for 2 < 0

(20)

This fracture mechanism is called Mode B (Fig. 6). In fracture tests under uniaxial transverse compressive load it can be observed that fracture angles fp ± 45° appear, with | fp | usually slightly higher than 45°.9 Consequently, it can be anticipated that the fracture angle, fp, in the course of the (2, 21) fracture envelope somewhere between the shear strength R⊥储 and the compressive strength R(⊥ ) changes from fp = 0° (in Mode B) to values fp 0°. For a closer examination, the fracture condition of eqn (17) has to be implemented for every possible fracture angle –90°  ≤ + 90° to detect the stress action plane with the highest risk of fracture. For a general three-dimensional stress state (1, 2, 3, 23, 31, 21) (Fig. 4), determination of the fracture angle is only possible by numerical procedures. For that purpose, the program FRACUAN (fracture curves analysis) was developed at the University of Kassel, Germany (Department of Mathematics, Working Group Engineering Mathematics, Mrs Prof. Dr R. Jeltsch-Fricker; MS Thesis D. Zeise, University of Kassel, 1993). Among other things, the program calculates the shape of the risk-of-fracture curve for –90°  + 90°. The risk of fracture is defined as the angle-dependent effort fE() for a given stress state and any angle  between –90° and + 90°. The risk of fracture or angle-dependent effort is the reciprocal value of the angle-dependent reserve factor fR(). fR() is the positive stretching factor which would

278 be necessary to increase a given state of stress (1, 2, 3, 23, 31, 21) so much that it leads to fracture in the sectional plane () of interest. The minimal stretching factor [ fR()]min = fR(fp) which occurs in the angle range –90°  + 90° at the angle fp is the conventional generally used reserve factor, fR. For the cutting angle fp which belongs to the minimal angle-dependent reserve factor fR(fp) the actual fracture can be expected, because at this angle the risk of fracture is highest. One of the computational results of FRACUAN deserves particular interest: applying the fracture models in eqn (13) or eqn (15), respectively, and assuming that eqn (16) is valid for (2, 21) stress states, the normal stress n on the fracture plane is constant on all inclined fracture planes (fp 0). This observation suggests a further analytical investigation. Therefore, we consider a (1, 2, 21) state of stress in the range 2 < 0 which leads to IFF.11 At the moment of fracture, two conditions are satisfied: 1. Fracture occurs in the one sectional plane where the angle-dependent effort fE() is a maximum, i.e. d ( fE()) = 0 d

(21)

2. The fracture condition is satisfied, i.e. fE(fp) = 1 

1 = fw 1D

(22)

In this case it is necessary already to introduce the weakening factor fw , which describes a certain degradation of the fracture resistances of the action plane by high 1 stresses (see Section 7.1). The fracture condition can be expressed as a function of cos . Therefore, expanding eqn (21) gives: d d d d ( fE()) = ( fE()) (cos ) = ( fE())(  sin ) = 0 d d(cos ) d d(cos )

The first solution for this equation is fp = 0°, which was already used when describing the fracture mechanism referred to as Mode B. A second solution can be expected from d ( fE()) = 0 d(cos )

To find the solution, we start with an alternative form of the fracture condition which follows from eqn (17), in which the value of the function on the left-hand side of the equation increases proportionally with the stress level.11 Thus, this alternative form cannot only be used to formulate the fracture condition, but also to calculate the effort:

fE(fp) =

冑冉 冊 冉 冊 冉 冊 冉 冊 nt RA⊥⊥

2

+

nl RA⊥储

2

+

p R

2n +

p R

n = fw for

n < 0

(23)

279 In accordance with the transformation rules in eqn (12), this equation may be written for an arbitrary angle  as:

冑冉 冊 冉冊 2 RA⊥⊥

fE() = cos()

+

p R

2

2 cos2  +

(1  cos2 ) +

1 1D

for

冉 冊 冉冊 21 RA⊥储

2

+

p R

2

22 cos2 

n < 0

Introducing the abbreviations

a=

冋冉 冊 冉 冊 册 冉 冊 冉 冊 冉 冊 p R

2



1 RA⊥⊥

2

21 RA⊥储

22; b =

2

+

2 RA⊥⊥

2

; c=

p R

2

the angle-dependent effort becomes a function of cos  of the form fE() = cos 兹a cos2  + b + c cos2  +

1 1D

It follows from eqn (21) and its first solution fp = 0° that d a cos2 fp ( fE()) = 兹a cos2 fp + b + + 2c cos fp = 0 d(cos ) 兹a cos2 fp + b

(24)

Since the fracture situation is considered, the fracture condition also holds: fE(fp) = cos fp兹a cos2 fp + b + c cos2 fp = fw Using this equation, the ‘unpleasant’ square root in the derivation can be eliminated, which finally gives: cos4  =

fp = arccos

f 2w c2  a

冑

(25)

fwRA⊥⊥ (  2)

In eqn (25) 2 is the stress at inter-fibre fracture. By this, it is possible to determine the fracture angle, fp, (with validity of eqns (15) and (16)) in a surprisingly simple way. It can be shown, moreover, that the normal stress n in the fracture plane is constant in the whole range where inclined fracture planes occur, because it is n = 2 cos2 fp = –fwRA⊥⊥. It is therefore not dependent on the stress ratio 21/2. Between 2 = 0 and 2 = –fwRA⊥⊥, the fracture angle is fp = 0°, and the normal stress is given by 2 = n. When the compressive stress exceeds the value of –fwRA⊥⊥, the fracture plane surprisingly always turns exactly so much that n = 2 cos2 fp remains constant.

280 On the master fracture surface (Fig. 5), the fracture curve for a (2, 21) load, starting from point (a) for pure ⊥ tension passing point (b) for pure ⊥储 shear up to the turning point (c) follows a longitudinal section contour line, and from (c) to (d) for pure ⊥ compression a cross-section contour line of the master fracture body. Knowing the fracture angle fp, eqn (17) or eqn (23) yield the fracture condition for inclined fracture through a (2, 21) load: 1 p 2[1 + (R)RA⊥⊥]

冋冉 冊 冉 冊 册 21 RA⊥储

2

+

2 RA⊥⊥

2

RA⊥⊥ 1 + =1 (–2) 1D

for 2 < 0

(26)

This equation may be written in many variations. One which is particularly simple results from the relation between RA⊥⊥ and the transverse compressive strength YC; this is given in Table 1. The associated fracture mechanism is called Mode C (Fig. 6).

5. Survey of fracture conditions and relationships between substantial parameters Table 1 summarises all fracture conditions used in this paper in a clearly arranged manner. The fracture resistances RA of the stress action plane are already replaced by the strength values YT, YC and S12 as far as possible. By an additional term fw , a possible influence of the stress 1 parallel to the fibres on the inter-fibre failure can be included. This is explained in detail in Section 7.1 which deals with degradation by 1 stresses. Regarding the physically-based inter-fibre fracture conditions given in Table 1, it is obvious that they do not require more computational effort than the present, widely used, fracture criteria. Thus, the only supplementary task is to decide which equation is valid for the stress state to be analysed. However, distinguishing between Mode A, B or C is easily possible, considering the criteria given in Table 1 (see also Fig. 6). As a result, it can be seen that with remarkably little extra work, one gets important additional information about the fracture mode and the anticipated fracture angle, which brings along a much better assessment of the consequences that inter-fibre failures may have for the laminate.

6. Reserve factor and effort in the presence of residual stresses When no residual stresses exist, the following definition is valid: (a) The reserve factor fR is the one (positive) factor all existing stresses would have to be multiplied with to originate failure. This means that the stress vector would have to be stretched in its original direction by this factor in order to cause fracture. In Fig. 7, such an increase of stresses can be seen in the stretching of the stress vector {}(1).

Table 1 Summary of failure conditions (for the calcuation of fE(FF) or fE(IFF) the terms with 21 or 1, respectively, have to be shifted to the left-hand side). Type of failure

Failure mode

Fibre failure (FF)

Failure condition (effort fE(FF) or fE(IFF)) 1 fT

Tensile

冏冉 冑冉 冊 冉 1 1C

Compression (kinking)

21 S21

Mode A, fp = 0°

Inter-fibre fracture (IFF)

Definitions

Parameter relationships

p(⊥储+ ) = 

RA⊥⊥ =

冉 冊 d21 d2

冑

1 + 2p(⊥储 )

T S21

2

(. . .) 0

= 1  (10 21)2

2 YT

冊 冉 冊册 冉 冊

21 () S21 2(1 + p⊥⊥

p(⊥储 ) = 

2 = 0

冉冑

冊冏 冊冉 冊

vf12 mf 2 Ef1

+ 1  p(⊥储+ )

冋冉

fwRA⊥⊥ (–2)

of (2, 21) curve, 2 0

YC S21 = () ) 2p(⊥储 ) 2(1 + p⊥⊥

1 +

冊

v 1 ` +f12 mf 2 = 1 Ef1

2

+ p(⊥储+ )

(. . .) < 0

冏 冏

2 1 =1 S21 1D

冏 冏 冏 冏

1 1 () (兹221 + (p(⊥储 )2)2 + p⊥储 2) = 1  S21 1D

Mode B, fp = 0°

Mode C, cos fp =

2

冉

Condition for validity

冊

YC 1 S21

d21 d2

2

+

2 YC

2

YC 1 =1 (–2) 1D

2 0

2 < 0 and 0

2 < 0 and 0

冏冏 冏冏

2 RA⊥⊥ ≤ 21 | 21c |

␶21 | 21c | ≤ A 2 R⊥⊥

of (2, 21) curve, 2 0

2 = 0

() () p⊥⊥ = p⊥储

RA⊥⊥ S21

() 21c = S21兹1 + 2p⊥⊥

281

282

Fig. 7. Stress vectors in the (2, 21) plane with and without a residual stress {}(r). Illustration of reserve factors fR, f (1) R and the inter-fibre failure effort fE(IFF). (1) Tip of the resultant stress vector {}; (2) stretching of {} until IFF occurs; (3) stretching of the load-dependent stress vector {}(1) until IFF occurs.

If residual stresses (index r) {}(r) exist, it is only possible to define a reserve factor of the load dependent stresses (index 1) {}(1): (a) The reserve factor f (l) R of the load-dependent stresses is the one (positive) factor all load dependent stresses would have to be multiplied with to cause fracture. If no residual stresses exist, the inverse value ( fR)–1 is the factor of effort or, abbreviated, the effort fE. –1 With acting residual stresses, the effort is not the reciprocal value ( f (l) R ) , but the reciprocal (r) value of that factor by which the resulting stress vector {} = {} + {}(l) would have to be multiplied with (maintaining its direction) to cause fracture (compare (1) → (2) and (1) → (3) in Fig. 7). Defining a reserve factor f (l) R for the load dependent stresses is reasonable (see Fig. 7), whereas introducing an ‘effort factor’ for the load dependent stresses {}(l) does not make sense. For a plane (1, 2, 21) state of stress, the existing stresses are: (l) (r) (l) (r) (l) 1 = (r) 1 + 1 ; 2 = 2 + 2 ; 21 = 21 + 21

(27)

To determine the fibre failure effort fE(FF) or the inter-fibre failure effort fE(IFF), respectively, which belong to a particular stress state, the stresses 1, 2, 21 calculated according to eqn (27) are inserted in the fracture conditions summarised in Table 1. These stresses either do not lead to fracture, or already exceeded the fracture limit. Inserting these stresses on the left-hand side of the fracture condition gives a number which is the value of the effort fE(FF) or fE(IFF), respectively. (The term (1/1D) has to be shifted to the left-hand side.) This comes from the fact that the functions characterising fracture on the left-hand side of the fracture condition are homogeneous of first degree in terms of stresses.11

283 Certainly an influence of loading path on the IFF strength can be expected, but we are not able to take this into consideration. We have to apply the same IFF curve for stress vectors with and without residual stresses.

7. Degradation models 7.1. Degradation of fracture resistances in the inter-fibre fracture conditions due to single fibre failure Since fibre strength follows a statistical distribution, single fibres already break under uniaxial 1 tensile stress long before fracture of many fibres leads to ultimate failure when XT is reached. These preliminary single fibre breaks cause local damage in the vicinity of the breaks in the form of debonding of fibre and matrix and microcracks in the matrix. By this damage the fracture resistances RA the composite offers to inter-fibre fracture are decreased. This is taken into account by equally decreasing all fracture resistances RA with a weakening factor fw .11 In order to keep the fracture conditions homogeneous and of first degree with respect to the stresses, then:11

fw = 1 

fw = 1 

冉冊 1 1d

n

is replaced by (28)

1 for iterative calculating 1D

where 1d ( 1.1XT or –1.1XC, respectively) and n are experimentally determined values, which express quantitatively the degradation d of the fracture resistances RA of the action plane coming along with 1. From our experimental experience, we assume that degradation of the fracture resistances by fw can only be recognised when 1 equals 70% of X or more. To describe this, one needs a large value of the exponent n. For matrices with a relatively high fracture strain, we select n = 8, whereas for matrices with a fairly low fracture strain (e.g. 3501-6 epoxy), we choose n = 6. For simplification, the calculation is performed with

1  (0, 9 fE(FF))n instead of 1 

冉冊 1 1d

n

Even though it is not entirely clear to what extent this is justified, we apply the same degradation for 1 > 0 and 1 < 0 (Table 1) (1d > 0 for 1 > 0 and 1d < 0 for 1 < 0). When the fracture conditions given in Table 1 are exclusively used to distinguish between such states of stress which can be sustained without fracture and states of stress which lead to fracture, the exponent may be n 1. If however they are used to calculate the effort fE(IFF), conditions must be kept homogeneous and of first degree in terms of stresses (i.e. the effort

284 value is proportional to the load level). Higher values of n can be taken into account by iterative adaption of the following: 1D from 1  (0, 9 fE(FF))n = 1 

冏 冏 1 1D

Apart from single fibre breaks, loading the composite with a 1 stress always also induces a stress in the matrix acting in the x1 direction because fibre and matrix are acting as parallel springs. Previous papers1–4 covered this by a supplementary term (1/1mf)2 in the fracture conditions (subscript mf represents matrix failure). However, since we now favour Mohr’s fracture hypothesis, we suspect that a matrix stress acting in the fibre direction does not have a noticeable effect on fracture, as long as the strain to failure of the matrix is considerably higher than that of the fibre. Since this is the case for the composites that are to be analysed in this work, we neglect matrix stresses in the x1 direction. 7.2. Degradation of elastic quantities after exceeding the crack initiation limits A degradation method has been described1 which covers realistically the progressive reduction of certain transverse stiffnesses of the unidirectional ply as a result of increasing crack density. Opening cracks are regarded as if they were ‘smeared’, and their global effect on the secant moduli E2s and G21s as well as the major Poisson’s ratio v12 are described by diminishing all three quantities with a reduction factor . When calculating (fictitious) stresses from stress/strain diagrams extrapolated to stresses above the level where uniaxial failure or pure shear failure occurs, and inserting these stresses into the inter-fibre failure conditions, one computes values for the inter-fibre failure effort fE(IFF) which are higher than 1. These fictitious effort values fE(IFF) > 1 have been used previously1 as a controlling parameter for the  reduction (see Fig. 8). Since cracks cannot open for 2 < 0 (or more correct n < 0), only a very small  reduction was designated for this case1 (see Fig. 8).

Fig. 8. Degradation factor  as a function of the calculated fE(IFF) > 1 above the fracture limit ( fE(IFF) = 1) of a UD layer.

285 The described degradation procedure should build on experimentally deduced curves of ( fE(IFF)). Because for the composites which are to be analysed here no such experimental values exist, in this paper a self-controlled  reduction is used, which will be explained below. Furthermore, we distinguish between the fracture modes A, B, and C. For inter-fibre fracture of Mode A, the secant moduli E2s and G21s evaluated from the stress/ strain diagrams (extrapolated if necessary) and the Poisson’s ratio v12 are diminished by the same factor  < 1, which is decreased gradually until the iterative calculation yields fE(IFF) = 1 again. For inter-fibre failure of Mode B and Mode C, the secant modulus E2s is still deduced from the (extrapolated) (2, 2) curve and not diminished, because the cracked surfaces can still transfer compressive stresses. The Poisson’s ratio v12 remains unchanged. Only the secant modulus G21s is reduced by the factor , until the calculation gives again fE(IFF) = 1. For the self-controlling procedure the calculated fE(IFF) of the affected layer is used by setting  = ( fE(IFF))–1 for each decreasing step.

8. Application of damage analysis and fracture analysis to FRP laminates 8.1. General procedure According to the given problem, only the procedure for a load increasing monotonically from zero load to fracture is discussed here. The described calculation models and procedures are utilised by our computer program NOLI FRAN COLAM (non-linear fracture analysis of composite laminates). The laminates analysed are described elsewhere.25 When manufacturing laminates, residual stresses develop both as a result of shrinkage when curing the matrix, and most of all as a consequence of cooling the laminate after curing at elevated temperatures. However, relaxation takes place with time. Through moisture absorption and the accompanying swelling, residual stresses develop which are of opposite sign, so that the residual stresses caused by cooling the laminate are partially compensated for. Since no further details are given about the laminates in this problem, we account for this effect by assuming that at the onset of loading, the residual stresses are already reduced to 50% compared to the level they had immediately after cooling. Whenever the calculation is performed using elasticity quantities which have been reduced by , the residual stresses are re-calculated with the reduced quantities for every load level (because of the stiffness loss which accompanies crack formation, the residual stresses are reduced11). In any case, after crack initiation in the first affected ply, increasing the load has to be carried out in small increments. However, the crack initiation threshold itself can in principle be reached with one large step by the help of iterations. Nevertheless, if the goal is to predict the stress/strain diagrams of the laminates by computation, increasing the load has to be executed in small load steps from the beginning. The deterioration of unreasonably designed laminates (which are – according to netting analysis – not able to transfer load, e.g. a ± 55° tube under axial tension), which progresses rapidly after crack initiation, cannot be represented correctly by means of calculation. In this case, we compute with large deformations (geometrically non-linear). When they exceed a limiting value ( 21 > 0·2), calculation is cancelled.

286 8.2. Interpretation of the computed results A reliable interpretation and evaluation of the computational result that an inter-fibre failure limit is exceeded is achieved not only by indicating the related stresses, but also the fracture mode (A, B, C). Least critical for a laminate is progressive crack formation after Mode A. Particularly when the crack formation is caused essentially by transverse tensile stresses, cracks open at an angle of fp = 0° which to some extent removes load from the affected ply. Such a load removal is only partly possible in Mode B. Only the shear modulus G21s is reduced, and not E2s. Thus, after exceeding the crack initiation limit, high 2 compressive stresses quickly build up when the load is increased further. The fracture point moves to the left on the (2, 21) fracture envelope, i.e. the fracture mode may change from B to C. Fracture in Mode C can be extremely dangerous for the laminate, especially when through a dominating compressive stress (2  21) a fracture angle fp between 30° and 55° results. The inclined fracture planes plus the effect of the high 2 compressive stress which is still acting constantly cause a wedge effect. When the frictional resistance on the originated inclined plane is overcome, one side of the broken layer tends to push its way over the other part gliding on the fracture plane, so that it ‘climbs up’ in the thickness direction (see Fig. 9). This has an explosive effect on the laminate, which may lead to local delaminations and subsequent buckling of neighbouring layers when further increasing the load. Of course, how critical these delaminations are is dependent on the local parameters (layer thickness, fibre direction, affected layer being inside or on the surface of the laminate, etc.). Thus, the results of the stress and fracture analysis can at first hand only point out the risks and lead to a more detailed examination of the local situation or, if possible, to a re-design of the laminate to avoid reaching this critical situation. Our newly developed fracture criteria with fracture angle determination allow a numerical prediction when a critical explosive effect can be anticipated for a laminate. As soon as inclined fracture occurs at an angle fp 0°, the shear stress nt which is needed to satisfy force equilibrium in the fracture plane is not transferred any more by the material. At the very most, it might be (entirely or partially) replaced by a ‘frictional stress’ n. A critical situation develops as soon as  | n | | nt |

(29)

because then both fracture planes start to slide on each other, if that is not prevented by vertical supporting forces of the neighbouring layers. How long the laminate withstands the explosive effect depends on its particular stacking sequence. We cancel calculation as soon as 1

| nt | = 2 | 2 sin 2fp | 3 | n | with

(30)

冉 冊

fp = arccos

fwRA⊥⊥ –2

1/2

because now, a severe delamination and buckling risk can be expected.

Fig. 9. Wedge effect of inclined mode C fracture planes caused by (1, 2, 21) stresses. (a) Stress state on an inclined cut under the angle : normal stress n. The shear stresses nt and n1 may be combined to n. (b) If nt* exceeds the maximum friction force nt*, the fractured pieces glide one against the other in the n, direction. (c) Equilibrium of forces on a fragmented piece 21 = n1 = 0; as long as | ntt* | remains lower than the maximum possible friction force  | nt* |, the fractured pieces do not glide. (d) Forces acting on fractured piece with large fracture angle fp; 21 = n1 = 0. The maximum friction force has been exceeded. Within a laminate the surplus force n2 will be introduced by an interlaminar shear stress 32 into the neighbouring layers. n3 produces, by interlaminar normal stresses, an explosive effect on the laminate.

287

288 When designing a heavily loaded GRP torsion tube spring for a passenger car,22,23 it was learned that only one single inclined fracture event of this kind may lead to ultimate failure of the structural member. The wall of the torsion spring consisted of two unidirectional layers with + 45° and –45° fibre orientation, respectively, and one intermediate crack arresting ply. The thicker outer layer experienced stresses 1 > 0 and 2 < 0. At load cycles of about N 2
106, the 2 compressive stress caused an inclined wedge-shaped fracture at an angle of fp ± 55°, which lead to the final collapse of the twisted GRP tube through the development of radial forces. This kind of failure is explained in more detail elsewhere.11

9. Theoretical results The theory described above was applied to predict the behaviour of laminates supplied by the organizers.25 In the analysis, the lamina properties of the four materials (T300/914C, AS4/3501-6, E-glass/MY750 epoxy and E-glass/LY556 epoxy) investigated were taken from the data provided. Figure 10 shows the failure envelope for the E-glass/LY556 unidirectional lamina subjected to combined transverse and shear loading (y, xy). Three modes of failure are predicted depending upon the state of stresses in the lamina. Under combined transverse tension and shear, the failure mode is IFF (Mode A); under moderate transverse compression it is IFF (Mode B) and at fairly large transverse compressive loading it is IFF (Mode C). Figure 11 shows the failure envelope for the T300/914C lamina subjected to combined longitudinal and shear loading (x, xy) Figure 12 shows the failure envelope for the E-glass/MY750 epoxy lamina subjected to combined direct loading (x, y). Four modes of failure are predicted, namely IFF (Mode A), IFF (Mode C), FFT and FFC. Each mode operates in different regions which are indicated. Figures 13 and 14 show the failure envelopes of the 90°/ ± 30°/90° E-glass/LY556 epoxy laminate under combined direct loading (y, x) and combined direct stress and shear loading (x, xy), respectively. Both initial and final failure envelopes are shown. Different modes of failure operate at different portions of the initial and final failure envelopes as indicated on the figures. Figure 15 shows the failure envelope for the ± 55° E-glass/MY750 epoxy laminate subjected to a variety of biaxial direct stresses (y, x). Initial and final failure stresses are shown. The predicted stress/strain curves for ± 55° E-glass/epoxy laminate under uniaxial loading (y /x = 1/0) and biaxial loading (y /x = 2/1) are shown in Figs 16 and 17. Figure 18 shows the predicted failure envelope for the 0°/ ± 45°/90° quasi-isotropic laminate under biaxial direct stresses (y, y). It is remarkable that in the compression quadrant only FF occurs. Figures 19 and 20 show the predicted stress/strain curves under uniaxial (y /x = 1/0) and biaxial (y /x = 2/1) loading for the 0°/ ± 45°/90° quasi-isotropic laminate made of AS4/3501-6 material. Biaxial stress strain curves for the ± 45° E-glass/MY750 epoxy laminate are shown in Figs 21 and 22 for two loading cases: (y /x = 1/1) and (y /x = 1/–1). Finally, the behaviour of 0°/90° E-glass/MY750 epoxy laminate under uniaxial tensile loading y /x = 1/0 is predicted in Fig. 23. The laminate fails in three stages. In the first stage

289

Fig. 10. Biaxial failure stresses of 0° lamina made of GRP material.

Fig. 11. Biaxial failure stresses of 0° lamina made of CFRP material.

Fig. 12. Biaxial failure stresses of 0° lamina made of GRP material.

290

Fig. 13. Biaxial failure stresses of (90°/ ± 30°/90°) laminate made of GRP material.

Fig. 14. Biaxial failure stresses of (90°/ ± 30°/90°) laminate made of GRP material.

Fig. 15. Biaxial failure stresses of ± 55° laminate made of GRP material.

291

292

Fig. 16. Stress/strain curves for ± 55° laminate made of GRP material (y/x = 1/0).

Fig. 17. Stress/strain curves for ± 55° laminate made of GRP material (y/x = 2/1).

293

Fig. 18. Biaxial failure stresses for (0°/ ± 45°/90°) laminate made of CFRP material.

Fig. 19. Stress/strain curves for (0°/ ± 45°/90°) laminate made of CFRP material (y/x = 1/0).

Fig. 20. Stress/strain curves for (0°/ ± 45°/90°) laminate made of CFRP material (ay/ax = 2/l).

294

Fig. 21. Stress/strain curves for ± 45° laminate made of GRP material (y/x = 1/1).

Fig. 22. Stress/strain curves for ± 45° laminate made of GRP material (y/x = 1/–1).

295

Fig. 23. Stress/strain curves for 0°/90° GRP laminate (y/x = 0/1).

the 90° plies fail in IFF (Mode A) as FPF. This is followed by failure of the 0° plies in IFF Mode A and finally total failure of the laminate. 10. Concluding remarks The introduction of new inter-fibre fracture criteria, which are based on the brittle failure behaviour of composites, makes fracture analysis even more realistic than it has been with the methods described previously.1 In addition, the new criteria make a distinction between different fracture modes (A, B and C) possible. Furthermore, application of separate fracture criteria for fibre failure and inter-fibre fracture provides a rapid overview of the fractures which can be expected for a laminate. This can be achieved by neglecting the  degradation of the elastic quantities when running the calculation program the first time (especially for CFRP, the effect of the  degradation is fairly small anyway). It is still not known for sure how much the thin layer effect and the in situ effect have to be taken into consideration, i.e. the fact that crack formation in very thin layers is delayed when they are embedded between stiff neighbouring plies.12 Moreover, the notch effect interfibre fractures have on the fibres of neighbouring layers should be covered by introducing a stress concentration factor. However, since no reliable findings concerning these questions are

296 available and the results of our work should be comparable to the results of other authors, we excluded correction factors accounting for these effects from this paper. We are strongly convinced that the Failure Analysis Exercise initiated in the UK will have a major impact on the development of this domain which has been abandoned for a long time, provided that the experience gained in different research fields can be taken into account.24

Acknowledgements The authors would like to thank Mrs Professor Dr R. Jeltsch-Fricker, Kassel, for discussion of mathematical aspects and Dr-Ing. D. Huybrechts, Aachen, for discussion of engineering aspects. The skilful typing of Mrs M. Streb is gratefully acknowledged. Thanks are due to Dipl.-Ing. S. Wenzel for computation and to Dipl.-Ing. A. Knickrehm for translation of the manuscript.

References 1 Puck, A., Calculating the strength of glass fibre/plastic laminates under combined load. Kunststoffe, Gennan Plastics, 1969, 55, 18–19 (German text pp. 780–787). 2 Puck, A. and Schneider, W., On failure mechanisms and failure criteria of filament-wound glassfibre/resin composites. Plast. Polym., 1969, Feb., 33 43. 3 Puck, A., Fracture criteria for highly stressed fibre plastic composites which meet requirements of design practice. Kunststoffe, German Plastics, 1992, 82, 34–38 (German text pp. 149–155). 4 Puck, A., Should fibre-plastics composites be designed with strain or stress criteria? Kunststoffe, German Plastics, 1992, 82, 34–36 (German text pp. 431–434). 5 Puck, A., A failure criterion shows the direction. Kunststoffe, German Plastics, 1992, 82, 29–32 (German text pp. 607–610). 6 Michaeli, W. and Huybrechts, D., A new approach for the dimensioning of thick laminates using physically based strength criteria. In Proc. 39th Conf. of the Society for the Advancement of Material and Process Engineering (SAMPE), Vol. 2, 11–14 April 1994, Anaheim, CA, pp. 2829–2840. 7 Michaeli, W. and Huybrechts, D., A new approach for the dimensioning of thick laminates. In International Mechanical Engineering Congress (IMECE) of the American Society of Mechanical Engineers (ASME), Vol. 49, 6–11 Nov. 1994, Chicago, IL, pp. 307–319. 8 Cuntze, R. G., Evaluation and application of a new, physically based 2D/3D inter-fibre-fracture (IFF) strength criterion. In Proc. Int. Symp. on Advanced Materials for Lightweight Structures, ESTEC, March 1994, Noordwijk (ESA-WPP-070), pp. 133–139. 9 Huybrechts, D. and Michaeli, W., Dimensioning of thick laminates: New IFF strength criteria based on the determination of the fracture plane, experiments for verification. In 16th Conf. of the European Chapter of the Society for the Advancement of Material and Process Engineering (SAMPE), 30 May–1 June 1995, Salzburg, Austria, pp. 211–222. 10 Kroll, L. and Hufenbach, W., New proof of laminate design by a physically based failure criterion. In Proc. ICCM-10, Vol. 1, ed. A. Poursatrip and K. Street. August 1995, Vancouver, Canada, pp. 715–720. 11 Puck, A., Festigkeitsanalyse von Faser-Matrix-Laminaten, Modelle für die Praxis (Strength Analysis of Fibre-Matrix/Laminates, Models for Design Practice). Carl Hanser Verlag, Munich, Germany, 1996.

297 12 Talreja, R. (ed.), Damage Mechanics of Cornposite Materials. Elsevier, Amsterdam, 1994. 13 Schürmann, H., Zur Erhöhung der Belastbarkeit von Bauteilen aus Faser-Kunststoff- Verbunden durch gezielt eingebrachte Eigenspannungen (How to Increase the Load Bearing Capacity of Fibre-Plastic Composite Components by the Introduction of Internal Stresses). Fortschritt-Berichte VDI, Reihe I, Nr. 170, VDI-Verlag Düsseldorf, Germany, 1989. 14 Rosen, B. W., Mechanics of composite strengthening. In Fibre Composite Materials. American Society for Metals 1995, pp. 37–76. 15 Edge, E. C., Does transverse and shear loading affect the compression strength of unidirectional CFC? A reply to Dr Hart-Smith. Composites, 1994, 25, 159–164. 16 Parry, T. V. and Wronski, A. S., Kinking and compressive failure in unaxially aligned carbon fibre tested under superimposed hydrostatic pressure. J. Mater. Sci., 1982, 17, 893–900. 17 Hart-Smith, L. J., Fibrous composite failure criteria – Fact and fantasy. In 7th Int. Conf. on Composite Structures, Paisley, UK, July 1993. McDonnell Douglas MDC 93 K0047. 18 Mohr, O., Welche Umstände bedingen die Elastizitatsgrenze und den Bruch eines Materials? (Which circumstances are causing yield limit and fracture of a material?). Civilingenieur, 1900, 44, 1524–1530 and 1572–1577. 19 Hashin, Z., Failure criteria for unidirectional fiber composites. J. Appl. Mech., 1980, 47, 329–334. 20 Puck, A., Physically based 3D fracture criteria for unidirectional composites. Presentation at the Working Group ‘New Fracture Criteria’, Kassel University, AG Ing. Math., June 1993. 21 Paul, B., A modification of the Coulomb-Mohr theory of fracture. J. Appl. Mech., 1961, 2, 259–268. 22 Puck, A., Torsion tube springs to replace highly stressed steel springs. Kunststoffe, German Plastics, 1990, 80, 27–30 (German text pp. 1380–1384). 23 Garbe, J. and Puck, A., Experience with criteria for fracture of FRP torsion springs in cyclic loading. Kunststoffe, German Plastics, 1993, 83, 28–31 (German text pp. 406–411). 24 Hinton, M. J. and Soden, P. D., Predicting failure in composite laminates: the background to the exercise. Compos. Sci. Technol., 1998, 58 (7), 1001. 25 Soden, P. D., Hinton, M. J. and Kaddour, A. S., Lamina properties, lay-up configurations and loading conditions for a range of fibre reinforced composite laminates. Compos. Sci. Technol., 1998, 58 (7), 1011.

Appendix Discussion of difficulties with nomenclature and symbols for “Reserve Factor” and “Effort” In the Anglo-American aerospace literature, the symbol RF is used for the Reserve Factor. Since that can also mean the product between R and F, we substituted with fRes or the short form (in recent publications) fR. In the German literature, the term “Anstrengung” is preferred. All possible translations (effort, exertion, stress exposure factor, exploitation factor) are not completely satifactory. However, we decided to use “effort” as the shortest term, symbolised by the letter E in calligraphic style (Ᏹ). Unfortunately, this is often mixed up with epsilon (). To avoid that, we use the symbol fE instead in our most recent publications, which also emphasises the close relation to the Reserve Factor fR; both are dimensionless factors. In the absence of internal stresses, it is fE = ( fR)–1.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

298

CHAPTER 3.8

Prediction of laminate failure with the Rotem failure criterion* A. Rotem Faculty of Mechanical Engineering, Technion-lsrael Institute of Technology, Haifa 32000, Israel Abstract The mechanical behaviour of four types of composite material in various laminate lay-ups is examined by using the Rotem failure criterion. The criterion distinguishes between fibre failure and matrix failure. Only in-plane loads are considered, neglecting the possibility of interlaminar failure. Mainly the effect of biaxial loading is examined, including tension, compression and shear. Most of the cases of biaxial loading were predicted to fail by matrix failure. Only those cases where the load was unidirectional or almost unidirectional in the fibre direction causes fibre failure. The results of the predictions are given in graphs of failure envelopes. For some cases, stress/strain curves are shown. Keywords: laminate failure, Rotem failure criterion, biaxial loading

Notation Aij Ei Go h Ni Qij Si i i 

Stiffness component ij of the laminate Young’s modulus of i Initial shear modulus Lamina thickness Force component i per unit length in the laminate plane Stiffness component ij of the lamina Strength component i Strain component i Stress component i A constant

Subscripts A

Axial direction (fibre direction) of the lamina

* This article represents the author’s contribution to a worldwide exercise to confirm the state-or-art for predicting in composites, organised by Hinton and Soden.10

299 AT m s T x, y + 

Shear in the lamina plane Matrix material Shear in the coordinate system Transverse to the fibre direction (in the lamina plane) Coordinate system directions of the laminate Tensile Compressive

1. The Rotem failure criterion The first version of the failure criterion was suggested in 19731 and has been modified later in 19752 and 1981.3 The criterion was postulated particularly for fibre-composite materials and is not suitable for other types of anisotropic materials. The criterion is based on three basic assumptions: 1. The failure of a fibre-composite material laminate will occur either in the fibres or in the matrix. The onset of the failure is a localized phenomenon. 2. The laminate has no free edges, i.e. the laminate is wide enough and clamped on its outer contour and has no holes. Therefore only in-plane stresses are effective. There are no interlaminar stresses which may cause failure. 3. The matrix material is weaker and softer than the fibres. On the basis of these assumptions the failure criterion actually combines two separate criteria, namely a fibre failure criterion and a matrix failure criterion. The fibres, being stiffer and stronger than the matrix, can only fail by loads acting in their axial direction. With the notation  for stress and S for strength, the fibre failure criterion is expressed mathematically in the following fashion: A SA + (a) A SA  (b)

(1)

where S is the composite lamina strength, the index A denotes the axial direction, + indicates tensile strength and  indicates compressive strength. The matrix failure is controlled by all the stress components. For polymeric matrix materials a quadratic function is a good approximation for failure criterion. In laminates, a plane-stress situation exists and therefore only plane stresses enter the criterion. Therefore the failure criterion takes the form

冉

冊冉 冊冉冊

Em(AA) ± Sm

2

+

T ± ST

2

+

AT SAT

2

≥1

(2)

where the subscript m index indicates the matrix material, T is for directions transverse to the fibres and AT is for shear in a plane axial and transverse to the fibre direction. The first right expression on the left-hand side of eqn (2) is the relative shear strength (stress/strength). The sign has no effect on this expression. The second to the right expression is the relative normal stress transverse to the fibre direction. Usually the transverse strength is different for tension

300 and compression. Therefore if the stress is positive, the tensile strength should be used and if the stress is negative, the compressive strength should be used. The third to the right expression is for the axial stress in the matrix. Although this stress is smaller than the others and usually may be neglected, it may sometimes become large enough to be considered. Since the matrix is polymeric, the stress/strain relationships are not linear and the axial stress is calculated as a function of the axial strain, A, with the relevant Young’s modulus. The pure matrix normal strength Sm is also sign dependent and must be chosen by the sign of the axial strain. For every state of stress, both criteria must be checked by using the strengths appropriate to the sign of the normal stresses (and axial strain). This procedure is easily carried out with the use of modern computers. The first criterion to be reached will determine the failure mode (fibres or matrix) and the load level which would cause failure. Fibre failure in tension is caused by the accumulation of fibre fractures until a critical amount is reached4,5 and the lamina fails. The failure is not a progressive process in this sense, and therefore, the failure of a lamina within a laminate is on the same stress level as a single lamina. Fibre failure in compression is caused by localized micro-buckling,6 which causes catastrophic failure after a suffficient amount is accumulated. Again the failure of single lamina is of the same stress level as the failure of a lamina within a laminate. Matrix failure is in the form of a crack parallel to the fibres. The matrix failure criterion predicts the first crack in the lamina. However, for a lamina within a laminate, this is not a final failure. Another crack will be initiated at a distance7 with a load increase. Therefore, total failure of the lamina will be at a higher load level, after the crack density is high enough so that load cannot be transferred through the lamina. 2. Calculation of laminate failure 2.1. Single-lamina laminate We assume that the lamina is very thin and it is subjected only to in-plane forces. Therefore the loads are homogeneously distributed in the lamina plane on the laminate boundary. Using the classical plate theory, and assuming homogeneous strain distribution, the load/strain relationships may be formulated by8 Nx Axx Ny = Ayx Nxy Axx

Axy Ayy Asy

x y xy

Axs Ays Ass

(3)

where N is a force per unit length and A is the stiffness of the laminate. The x–y directions are the coordinate system where the loads are defined. Since the stiffness matrix depends on the orthotropic moduli and geometry, and the moduli are non-linear, the stiffness matrix is also a function of the strain. Therefore for every strain state the stiffness must be recalculated to match the load. The calculation is iterative and it is done by looping. First the reciprocal of the matrix expression is obtained Axx x y = Ayx xy Axx

Axy Ayy Asy

Axs Ays Ass

–1

Nx Ny Nxy

(4)

301 Aij are then calculated from the material moduli at ij = 0. The A matrix is symmetric (Aij = Aji) and for a single lamina with x direction parallel to the fibres direction, Axs = Ays = 0. For example: Ass = Qss
h = (G0   log(AT))
h

(5)

where h is the lamina thickness, G0 is the initial shear modulus and  is a constant that is to be evaluated from the shear-stress/shear-strain relationships (assuming a logarithmic fit). After calculating the strains for the given loads, Nij , they are substituted in the Aij expressions, by using the secant-modulus approach, and the strains are recalculated. This procedure is repeated until the strains match to a desired accuracy. If the x, y coordinate system differs from the principal A–T coordinate system, the strains are rotated to the A–T coordinate system. Once the strains are computed, the stresses are calculated with QA A T = QAT AT

QAT QT QS

A T 2AT

(6)

where Qij are the lamina moduli. With the stresses calculated, they are substituted in the failure criteria eqns (1) and (2) to check for failure. When a load pattern is given, the loads Nij are increased monotonically, the matching strains are computed and the resulting stresses are substituted in the failure criteria until they are satisfied. These are the loads which will cause failure.

2.2. Balanced symmetric laminates In this exercise all the laminates were of this kind. Such laminates do not have tensionbending coupling and therefore eqn (3) still holds for the load/strain relationships. However, the stiffness matrix, Aij, must include all the layers: Aij =

冘

Qij
hk

(7)

The behaviour of a lamina within a laminate differs from the behaviour of a single isolated lamina in two characteristics: strength and stiffness. In a single lamina, when a crack starts in the matrix material, it opens in a plane-stress situation and propagates through the material until it is separated, and therefore the lamina fails. However, for a lamina within the laminate, as a crack is initiated, it is bridged by the neighbouring laminae to form a 3-D stress field.9 The crack may propagate to the whole length of the laminate but yet the lamina does not fail. Since the first crack starts at the weakest point of the lamina, another crack will initiate on load increase, and so on, until enough cracks are accumulated to cause lamina failure. It was found,2 that for laminae inclined in the range 15–90° within a laminate, the strength increase over the strength of the single lamina is on average 20%. In order to calculate the stress/strain relationships in shear and transverse to the fibre direction over the whole range of loading, the ultimate shear and transverse strengths of the laminae were increased by 20% and the momentary shear and transverse moduli were also increased by 20% (except for the cases where the moduli were constant over the whole range of loading) to match the ultimate

302 strengths. An example is given in the Appendix. The lamina cracking decreases the apparent stiffness of the lamina. The decrease is proportional to crack accumulation or crack density. Since the matrix strength is statistically distributed, the rate of crack initiation after the first one will increase exponentially or logarithmically with load increase. For a Weibull distribution of the matrix strength, the stiffness of the cracked lamina will decrease proportionally with the logarithm of the load increase on the laminate. After the lamina has failed by the matrix criterion, the Q moduli of this lamina decrease to zero except in the fibre direction, QA. This modulus has an effect on adjacent laminae because of some residual matrix material along the fibres. However, this effect vanishes rapidly with strain increase. Therefore an exponentially decaying QA is used for strains after the failure of the lamina. Given the load pattern and the coordinate system, the principal Qij (strain dependent) moduli of each lamina are rotated to that direction by the respective angle and the Aij matrix is calculated by eqn (7) with the initial strain. Again the strains are calculated using eqn (4), the Aij are corrected with these strains and so on as described above until the strains match to the specified accuracy. Once these strains are determined, they are rotated from the load coordinate system to the principal coordinate system of each lamina. Then the principal stresses in each lamina are computed by eqn (6) and substituted in to the failure criteria eqns (1) and (2). If these criteria are not satisfied, then the load is incremented and a new strain level is computed. When the first lamina satisfies the failure criterion, this load level indicates the initiation of the first crack in this lamina. If the failure criterion eqn (1) is satisfied, this lamina is completely failed and all its Qij are set to zero. If eqn (2) is satisfied, the load is further increased until it is satisfied with 1·2ST and 1·2SAT. This load level marks the final failure of the lamina and the Qij are set to zero, except the QA which is decreased exponentially. The load continues to increase to the next lamina failure until all the laminae fail.

3. Prediction of laminate failure Two types of prediction were performed to obtain stress/strain curves to failure for specific load pattern and first and final failure envelopes for given biaxial loadings. Computer programs were written to compute these predictions. The results are shown in Figs 1–14. The material types are designated as follows: Mat1 Mat2 Mat3 Mat4

E-glass/LY556/HT907/DY063 T300/BSL914C E-glass/MY750/HY917/DY063 AS4/3501-6

Details of the lamina properties and lay-up configurations and loading of the laminates analyzed are provided in Ref. 11. Shear behaviour and some transverse behaviour were given as data points. The data have been processed and the results which were used in the programs are given in the Appendix. Figures 1–3 are failure envelopes for single lamina. The single-lamina strengths were used in the failure criterion eqn (2). The first failure is also the final failure because one crack causes failure of the laminate. Since there is no interaction between the criteria, Figs 2 and 3 have these polygon shapes. On the two sides of the figures the laminates fail by the fibre

303

Fig. 1. Biaxial failure stresses for 0° lamina of Mat1.

Fig. 2. Biaxial failure stresses for 0° lamina of Mat2.

criterion, 1500 MPa and –800 MPa in Fig. 2 and 1280 MPa and –800 MPa in Fig. 3. Then, on top and bottom the laminates fail by the matrix failure criterion. Figure 4(a) shows failure envelopes for (90°/ ± 30°)s laminate. The + 30° and the –30° laminae have the same failure envelopes, therefore only two solid lines are seen. The dashed lines are for the first initiation of cracks in each lamina and the solid lines represent the final failure computed with a factor of 1.2 in eqn (2). The horizontal extension to the right is fibre failure due to eqn (1) and initiation and final failure coincide here. Each lamina has its own failure envelope and therefore the inner space bounded by the dashed lines is the first-cracking envelope of the laminate. The outermost solid line all around is the final-failure envelope of the laminate. The combined lines are shown in Fig. 4(b). Figure 5(a) shows the failure

304

Fig. 3. Biaxial failure stresses for 0° lamina of Mat3.

envelopes for the same material but for different loading. The biaxial loading includes shear and transverse stresses. The failure envelope of each lamina is different here, but the rule for first and final failure of the laminate is the same. The inner space bounded by the dashed lines is the first-crack envelope and the outer solid line makes the final failure envelope. Here again the combined lines are shown in Fig. 5(b). Figure 6 shows the failure envelopes of ( ± 55°)s laminate made of material Mat3. The two laminae have the same envelopes and therefore there is only one envelope for crack initiation and one envelope for final failure. Notice that the failure criterion eqn (2) is for matrix failure all around because biaxial loading always causes high shear and/or transverse stress in the matrix. Figure 7 shows the stress/strain curve that would result upon loading a ( ± 55°)s laminate made of material Mat3 loaded in the y direction. This is equivalent to loading an angle ply laminate of ± 35°. The figure shows the extension in the loading direction (Ep-y) and the contraction in the perpendicular direction (Ep-x). The symbol marks the beginning of the cracking. After the first crack, the Q moduli (except QA) were decreased by the following expression until final failure. Qij = Qij
(1  0·35 ln(mff ))

(8)

where mff is the matrix failure factor calculated by the left hand side of eqn (2). Figure 8 shows the stress/strain curves of the same laminate under biaxial loading, Ny/Nx = 2. Again the symbols mark the beginning of cracking and the end of the solid lines mark the stress level at failure in the y direction and the extension in the x and y directions. The biaxial failure envelopes of a quasi-isotropic laminate, (0°/ ± 45°/90°)s, made of Mat4, are shown in Fig. 9(a). Again the inner space bounded by the dash lines of all the laminae is the first crack envelope, and the outer solid line is the final failure. The biaxial loading causes a matrix failure even when the resultant load is in the fibre direction, like 45°. When the load is uniaxial, e.g. for 0° and 90°, there is a fibre failure of the lamina which has its fibres in this

Fig. 4. Biaxial failure stresses for 90°/ ± 30°/90° of Mat1.

305

306

Fig. 5. Biaxial failure stresses for 90°/ ± 30°/90° of Mat1.

direction. When the load is compressive and equal in both directions, the lamina fails by eqn (1), that is fibre failure in compression. Therefore, there is no initiation of lamina cracking prior to the final failure. The combined envelopes are shown in Fig. 9(b). Figure 10 shows the stress/strain curve of such loading, that is in the y direction (in the 90° lamina direction). The curves shows the extension in the y direction and the contraction in the x direction. The 0° lamina (perpendicular to the load direction) is the first to fail, marked by the first two symbols. It causes sudden extension and contraction. Then, the ± 45° laminae crack and fail causing another sudden extension in the load direction. However, as these laminae fail they cease to contract the laminate and it remains as a unidirectional lamina that fails by fibre fracture. Biaxial loading of Ny/Nx = 2/1 of this laminate causes the weakest situation for tension– tension loading. The stress/strain curve of this loading is shown in Fig. 11, for Sy versus the

307

Fig. 6. Biaxial failure stresses for ± 55° of Mat3.

Fig. 7. Stress/strain curves for ± 55° of Mat3 (Ny/Nx = 1/0).

Fig. 8. Stress/strain curves for ± 55° of Mat3 (Ny/Nx = 2/1).

308

Fig. 9. Biaxial failure stresses for 0°/ ± 45°/90° of Mat4.

309

Fig. 10. Stress/strain curves for 0°/ ± 45°/90° of Mat4 (Ny/Nx = 1/0).

Fig. 11. Stress/strain curve for 0°/ ± 45°/90° of Mat4 (Ny/Nx = 2/1).

310

Fig. 12. Stress/strain curve for ± 45° of Mat3 (Ny/Nx = 1/1).

Fig. 13. Stress/strain curves for + 45° of Mat3 (Ny/Nx = 1/–1).

311

Fig. 14. Stress/strain curves for 0°/90° of Mat3 (Ny/Nx = 0/1).

strain in the y direction Ep-y. The ± 45° laminae fail first, followed by the 0° and then finally the 90° laminae. Note that after the ± 45° laminae have failed, the laminate contracts by a small amount. This behaviour would only be observed in a testing machine which is operating under load control. It would not be observed in a machine operating under displacement control. Next, an angle-ply laminate of ( ± 45°)s made of material Mat3 was biaxially loaded by Ny/Nx = 1. The stress/strain curves of the stress Sy versus the strains Ep-y and Ep-x are shown in Fig. 12. Both curves are the same, that is they coincide to one curve. The curve is linear, unlike loading in the bisector direction. The failure mode is of the matrix, even though the resultant load is in a fibre direction. However, when the biaxial load is Ny/Nx = –1 the stress/ strain curves become non-linear as shown in Fig. 13. This load is, in fact, a shear load and therefore causes the non-linear behaviour. The beginning of cracking is marked by the symbols. The last graph demonstrates the behaviour of a crossply (0°/90°)s laminate made of material Mat3. Figure 14 shows the stress/strain curves for uniaxial loading in the x direction. At a low stress level the 90° lamina fails and the 0° lamina continues to carry the load until fracture by failure of the fibres (eqn (1)). The interesting part is the behaviour of the perpendicular strain. In the initial stages of loading, the 90° lamina resists the contraction of the laminate and the perpendicular strain is minimal. After the lamina fails, the stiffness in the fibre direction of this

312 lamina degrades exponentially and then the laminate acts as a unidirectional laminate. This transition is shown by the curved line in the graph. That completes the task of prediction of the laminates behaviour with the Rotem criterion.

4. Summary The prediction of laminate failure was carried out by using the Rotem criterion. This criterion separates the failure into fibre and matrix failure. Only in-plane loads were considered, neglecting the possibility of interlaminar failure. The polymeric matrix material induced nonlinear behaviour on the laminate stiffness where the matrix is dominant. Biaxial loading induces matrix failure and only unidirectional or almost unidirectional loading in a fibre direction can cause fibre failure. The criterion distinguishes between tensile and compressive stresses to account for the different strengths. By using a computer program it is easy to take all these restrictions into account. The different laminates of glass and carbon fibres, ductile and stiff matrices, were analyzed and their strengths were predicted. It is summarized in failure envelopes, and for certain cases, the stress/strain curves are given.

References 1 Hashin, Z. and Rotem, A., A fatigue failure criterion for fiber reinforced materials. J. Compos. Mater., 1973, 7, 448–464. 2 Rotem, A. and Hashin, Z., Failure modes of angle ply laminates. J. Compos. Mater., 1975, 9, 191–206. 3 Rotem, A. and Nelson, H. G., Fatigue behavior of graphite-epoxy laminate at elevated temperatures. ASTM STP 723, 1981, pp. 152–173. 4 Rosen, B. W., Tensile failure of fibrous composites. AIAA J., 1964, 2, 1985. 5 Lifshitz, J. M. and Rotem, A., Longitudinal tensile failure of unidirectional fibrous composites. J. Mater. Sci., 1972, 7, 861–869. 6 Rosen, B. W., Mechanics of composite strengthening. In Fiber Composite Materials. American Society for Metals, Metals Park, OH, 1965. 7 Highsmith, A. L. and Reifsnider, K. L., Stiffness reduction mechanisms in composite laminates. In Damage in Composite Materials. ASTM STP 775, 1982, pp. 103–117. 8 Ashton, J. E. and Whitney, J. M., Theory of Laminated Plates. Technomic, Stamford, CT, 1970. 9 Altus, E. and Rotem, A., A 3-D fracture mechanics approach to the strength of composite materials. J. Engng Fract. Mech., 1981, 14, 637–644. 10 Hinton, M. J. and Soden, P. D., Predicting failure in composite laminates: the background to the exercise. Compos. Sci. Technol., 1998, 58(7), 1001. 11 Soden, P. D., Hinton, M. J. and Kaddour, A. S., Lamina properties, lay-up configurations and loading conditions for a range of fibre reinforced composite laminates. Compos. Sci. Technol., 1998, 58(7), 1011.

Appendix The materials moduli were calculated by the following method. As an example, the calculated stress/strain curve of a lamina embedded in a laminate and the given stress/strain curve of a free lamina (of Mat2), are shown in Fig. 15.

313

Fig. 15. Stress/strain curves of Mat2.

Comment. The initial values of the matrix moduli Em( j ) for z = 0, were taken from TABLE (3). The final values of the moduli were calculated from the data on the tensile strength and the tensile failure strain. Exponential behaviour was assumed between these two points. Since longitudinal tensile and compressive failure strains of the composite are lower than the tensile failure strain of the matrix, it was assumed that these values are valid for tension and compression over the whole range of loading. Moduli calculation program No epl(1, j) epl(2, j) epl(3, j) Em( j ) Rd( j )

Number of layers in laminate Axial strain in lamina j (mm/mm) Transverse strain in lamina j (mm/mm) Half the shear strain in lamina j (mm/mm) Matrix extension and compression modulus of lamina j (MPa) Decay function of failed lamina j + ST = ftt  ST = ftc + SA = fat –SA = fac (MPa) SAT = fs + Sm = fmt –Sm = fmc Mat1: fmt = 80 fmc = –120 fac = –570 fat = 1140 ftt = 35 ftc = –114 fs = 72 z = ABS(epl(1, j)) Em(j) = 3350 * EXP(–14·8 * z)

314 Rd(j) = EXP(–600 * z) x = ABS(2 * epl(3, j)) GA(j) = 5830 IF x > 0·00628 THEN GA(j) = –5266  5039 * LOG(x) IF x > 0·037 THEN GA(j) = (72·1 + 350 * (x  0·037))/x IF No > = 2 THEN GA(j) = 1·2 * GA(j) ET(j) = 17700 EA = 53480 NU = 0·278 Mat2: fmt = 75 fmc = –150 fat = 1500 fac = –900 ftt = 27 ftc = –200 fs = 80 EA = 138000 NU = 0·28 z = ABS(epl(1, j)) Em(j) = 4000 * EXP(–18·9 * z) Rd(j) = EXP(–600 * z) x = ABS(2 * epl(3, j)) ga1 = 5743  273251 *
+ 102174268 *
ˆ 2  16114670368 *
ˆ 3 + 1065425000000 *
ˆ 4 ga2 = –27999800000000 *
ˆ 5  224851000000000 *
ˆ 6 + 3·056337D + 16 *
ˆ 7  6·606309D + 17 *
ˆ 8 + 4·797875D + 18 *
ˆ 9 GA(j) = ga1 + ga2 IF
> 0·034 THEN GA(j) = (77·29 + 452·08 * (x  0·034))/x IF No > = 2 THEN GA(j) = 1·2 * GA(j) ET(j) = 11000 Mat3: fmt = 80 fmc = –120 fat = 1280 fac = –800 ftt = 40 ftc = – 145 fs = 73 EA = 45600 NU = 0·278 z = ABS(epl(1, j)) Em(j) = 3350 * EXP(–14·8 * z) Rd(j) = EXP(–600 * z)

315 x = ABS(2 * epl(3, j)) GA(j) = 5830 ga = 24789  9·2575e6 *
+ 1·9379e9 *
ˆ 2  2·3214e11 *
ˆ 3 + 17161e13 *
ˆ 4  8·1248e14 *
ˆ 5 + 2·4714e16 *
ˆ 6  4·6719e17 *
ˆ 7 + 4·9949e18 *
ˆ 8  2·3074e19 *
ˆ 9 IF x > 0·005979 THEN GA(j) = ga IF x > 0·04 THEN GA(j) = (72·99 + 263 * (x  0·04))/x IF No > = 2 THEN GA(j) = 1·2 * GA(j) ET(j) = 16200 y = epl(2, j) IF y < – · 005 THEN ET(j) = 19140·7 + 588142·85 * y IF y < – · 012 THEN ET(j) = (–145 + 5154·6 * (y + 0·012))/y IF No > = 2 THEN ET(j) = 1·2 * ET(j) Mat4: fmt = 69 fmc = –250 fat = 1950 fac = –1480 ftt = 48 ftc = –200 fs = 79 EA = 141300 NU = 0·28 z = ABS(epl(1, j)) Em(j) = 4200 * EXP(–2·01 * z) Rd(j) = EXP(–1200 * z) x = ABS(2 * epl(3, j)) ga1 = 6587  27890 *
+ 26352988 *
ˆ 2  8251908844 *
ˆ 3 + 501907700000 *
ˆ 4 ga2 = 26836510000000 *
ˆ 5  4717776000000000 *
ˆ 6 + 2·298241D + 17 *
ˆ 7  4·766197D + 18 *
ˆ 8 + 3·3182285D + 19 *
ˆ 9 GA(j) = ga1 + ga2 IF x > 0·02 THEN GA(j) = (79 + 1754·4 * (x  0·02))/x IF No > = 2 THEN GA(j) = 1·2 * GA(j) y = epl(2, j) ET(j) = 11000 IF y < – · 011 THEN ET(j) = 12222 + 111111 * y IF y < – · 02 THEN ET(j) = (–200 + 6849 * (y + 0·02))/y IF No > = 2 THEN ET(j) = 1·2 * ET(j)

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

316

CHAPTER 3.9

Prediction of failure envelopes and stress/strain behavior of composite laminates* C. T. Sun and Jianxin Tao School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907-1282, USA Abstract Failure envelopes for unidirectional composites and multilayered composite laminates have been predicted by linear laminate theory. The ply-by-ply discount method with parallel spring stiffness reduction model was used in the prediction. The stress/strain behavior of four laminates under uniaxial or biaxial loading were predicted by considering both material non-linearity and progressive matrix cracking. Material non-linearity was considered by using a one-parameter plasticity model for fiberreinforced composites. Progressive matrix cracking and the corresponding stiffness reduction were predicted by use of a simplified shear-lag analysis and finite element analysis Keywords: laminate analysis, failure prediction, ply-by-ply discount method, stress/strain behavior, progressive matrix cracking

1. Introduction Over the last three decades, efforts have continued in the development of failure analysis methods for unidirectional composites and their laminates. Currently, there are a large number of lamina failure criteria and laminate failure analysis methods.1,2 There are two major elements in the failure analysis of laminated composites, i.e. the lamina failure criterion and the laminate stress analysis with lamina stiffness reduction. A lamina failure criterion predicts failure at the lamina level. The laminate stress analysis combined with lamina failure criteria predicts failure in the laminate. All existing lamina failure criteria are basically phenomenological, and detailed failure processes are not described. There are three major failure modes in composites, namely fiber failure, matrix failure and fiber/matrix interfacial failure. Since, in general, the stresses in the fiber and matrix are different, their respective failures are determined by different stresses and have different failure mechanisms. Thus, a lamina failure criterion that separates matrix

* This article represents the authors’ contribution to a worldwide exercise to confirm the state-of-the-art for predicting failure in composites, organised by Hinton and Soden.20

317 failure from fiber failure, such as the Hashin–Rotem criterion,3,4 is mechanistically more sound. Laminate ply-by-ply failure analysis is performed on the basis of the 2-D stress field in the laminate. In reality, however, the failure mechanisms in laminates are a great deal more complicated than the failure mechanisms assumed in the 2-D analysis. Matrix cracking occurs in the laminate before ultimate failure. The effect of matrix failure on the laminate failure is not completely understood. Non-linear behavior and the 3-D failure modes, such as delamination and failure induced by free edge singular stresses, add to the problem. A failure analysis including all the aforementioned effects is prohibitive, if not impossible. For practical purposes, the 3-D effects are usually neglected. Classical laminate strength analysis with appropriate lamina failure criteria and stiffness reduction models is still more popular among composites engineers. The non-linearity of the stress/strain response of a laminate results primarily from three sources: (1) material non-linearity, (2) progressive failure in the laminate, and (3) geometrical non-linearity. For fiber-reinforced composites, significant material non-linearity is displayed in shear and transverse deformations. This non-linearity is governed by the matrix material. The progressive failure in the laminate also leads to non-linear stress/strain response. The geometrical non-linearity comes from the change of fiber orientation. For certain laminates, relatively large applied loads result in interlaminar and/or intralaminar yielding. This causes fibers to move and change the orientation, and, thus, the increases or decreases in the stiffness of the laminate. In this study, predictions of the laminate stress/strain response and failure envelopes are performed separately. Failure envelopes are predicted by linear laminate analysis, in which the ply-by-ply discount method in conjunction with the parallel spring model is used. Laminate stress/strain responses are predicted by considering material non-linearity and matrix cracking. 2. Predictions of failure envelopes 2.1. Failure criterion The Hashin–Rotem criterion is given by the following equations: 11 =1 X

(1)

冉冊冉冊 22 Y

2

+

12 S

2

=1

(2)

in which 11, 22 and 12 are longitudinal stress, transverse stress and shear stress, respectively, and X, Y and S are longitudinal strength, transverse strength and shear strength, respectively. This criterion is generalized for either tensile or compressive stresses; the corresponding (tensile or compressive) strengths must be chosen based on the sign of the applied stresses. Failure is assumed when one of the two equations is satisfied. If eqn (1) is satisfied, then we have fiber breakage. Equation (2) represents the condition for matrix failure. The mode of matrix failure is determined by comparing the ratios 22/Y and 12/S.

318 The Hashin–Rotem criterion includes well the concept of separating the fiber and matrix failure modes while maintaining some stress interactions. Therefore, in this exercise, the Hashin–Rotem criterion is chosen as the lamina failure criterion.

2.2. Laminate failure analysis method The ply-by-ply discount method is used in the laminate failure analysis. Linear elastic laminate plate theory is used to initially calculate stresses and strains in each ply of the laminate. The lamina failure criterion (Hashin–Rotem criterion) is then used to determine the ply that would fail first and the corresponding failure mode. A stiffness reduction model is then used to reduce the stiffness of the failed ply. The laminate with reduced stiffnesses is subsequently analyzed for stresses and strains. The lamina failure criterion predicts the next ply failure, and laminate stiffness is accordingly reduced again. This cycle continues until ultimate failure is reached. For fiber dominated laminates, fiber failure is the ultimate failure. However, matrix dominated laminates, such as [ ± 45]S laminates, may fail without fiber breakage under uniaxial loading. Consequently, in this study, ultimate failure is defined either when fiber breakage occurs in a ply, or when excessive laminate strains take place. The stiffness reduction is accomplished by using the parallel spring model. Each lamina is modeled as a pair of springs representing the fiber (longitudinal) and matrix (shear and transverse) deformation modes. When fiber breakage occurs, the longitudinal modulus, E1, is reduced to zero; and when matrix cracks occurs, the shear, G12, and transverse E2, moduli are reduced to zero. For simplicity, Poisson’s ratios v12 and v21 are kept constant, and transverse modulus reduction is treated the same way for both tensile failure and compressive failure. The model is also capable of differentiating between types of matrix failure if desired; i.e. the transverse and shear moduli can be reduced separately depending on the specific type of matrix failure mode. The model which reduces E1 for fiber failure and E2 and G12 for either transverse or shear matrix failure is denoted as PSM. The model which reduces E1 for fiber failure, E2 for transverse matrix failure and E2 and G12 for shear matrix failure is denoted as PSMS. The idea behind the PSMS model is that a transverse matrix failure does not necessarily inhibit the ability of the lamina to carry significant shear loads. For fiber dominated laminates, these two models may predict almost the same laminate strength. For matrix dominated laminates, however, the difference in the ultimate failure stresses predicted by these two models may be significant. In this exercise, the PSMS is used for the prediction of failure envelopes of the laminates described in Ref. 21.

2.3. Thermal residual stresses The mismatch of thermal expansion coefficients of the fiber and matrix can cause thermal residual stresses (curing stresses) to form in the laminate. Typically these stresses place the matrix in tension. In certain cases these curing stresses can be of significant magnitude compared to the transverse and shear strengths of the unidirectional composite. For example, for the [0/ ± 45/90]S laminate of AS4/3501-6 cured at 177°C, the transverse thermal residual stress (in the direction perpendicular to fibers) is as high as 40 MPa, which is almost the same

319 as the ply transverse strength. Thus, thermal stresses should not be neglected if prediction of matrix failure is of importance. In this study, thermal residual stresses are taken into account in the laminate stress analysis. Temperature drop is the difference between room temperature (20°C) and the stress free temperature.

2.4. In situ strengths It has been found that transverse matrix cracking initiation stress (or strain) of a lamina in a laminate is usually larger than the transverse strength of the unidirectional laminate.5,6 This socalled ‘in situ transverse strength’ in some cases could be as high as 2·5 times the unidirectional transverse strength as shown by Flaggs and Kural.6 The in situ strength is dependent on the lamina thickness and the constraints from the adjacent layers. It is believed that the same phenomenon exists for shear strength, although it has not been reported in the literature. In our previous investigation,7 shear strength of an AS4/3501-6 lamina measured from [ ± 45]S specimens is 110·3 MPa, which is 40% higher than the value (79 MPa) specified in this exercise.21 Clearly, to accurately predict strength of the laminates, the in situ strength must be taken into consideration. By examining the results presented previously,5–7 we take the in situ tensile transverse and shear strengths to be 1·5 times those measured from unidirectional laminates for all the laminates discussed. This factor reflects overall considerations of layer thickness, lay-up and the location of the individual layer. This treatment is approximate but necessary.

2.5. Initial failure In this exercise, initial failure is defined as first-ply failure. In most cases, first-ply failure is matrix failure. Thus, the prediction of initial failure is very sensitive to the matrix dominated lamina strengths, Y and S.

2.6. Predictions A FORTRAN program developed in the Composite Materials Laboratory of Purdue University is used to perform the laminate analysis and failure prediction. The initial elastic constants provided in Ref. 21 are used in the program. Figures 1–3 plot failure envelopes of unidirectional composites. For unidirectional laminates subjected to transverse and shear combined loading (22–12), failure is governed by eqn (2). Hence, the failure envelope is quadratic (Fig. 1); but for 1–22 and 1–12 combined loading cases, failure envelopes are determined by two sets of straight lines (Figs 2 and 3), since there is no interaction between the stress components. Laminate failure is the eventual result of progressive failure processes taking place in the constituent laminae under loading. To predict the laminate failure, it is necessary to describe the failure sequence. The following notations are used to facilitate the discussion of laminate failure. The failure mode of a ply is indicated in the parenthesis following the name of the failed ply, with ‘1’ representing fiber failure, ‘2’ representing transverse matrix failure, and ‘3’

320

Fig. 1. Biaxial failure stresses (yy–xy) for 0° lamina made of E-glass/LYS56/HT907/DY063.

shear matrix failure. A plus ( + ) superscript indicates tensile failure and a minus (  ) superscript denotes compressive failure. Arrows indicate the failure sequence. In some cases, laminate failure is dominated by fiber failure. In some other cases, matrix failure will result in an immediate fiber failure of some plies without any increase of load. In other words, the matrix failure leads to a catastrophic failure of the laminate. In those cases, the ultimate failure

Fig. 2. Biaxial failure stresses (xx–xy) for 0° lamina made of T300/BSL914C.

321

Fig. 3. Biaxial failure stresses (xx–yy) for 0° lamina made of E-glass/MY750/HY917/DY063.

stress and strain of the laminates are the stress and strain corresponding to the matrix failure. Therefore, we use an underline ( ) to signify the ultimate failure. Figures 4 and 5 show failure envelopes of the [90/ ± 30/90] laminate of E-glass/LY556/ HT907/DY063 subjected to xx–yy and xx–xy biaxial loading. First-ply failure envelopes are relatively simple. For xx–yy loading (Fig. 4), there is no shear stress in the 90° plies, and, thus, no stress interaction exists among stresses. Therefore, the failure envelope is linear in region PR and SO due to the tensile and compressive transverse matrix failure, respectively, of the 90° plies. However, owing to the existence of shear stress in the ± 30° plies, and thus the existence of the interaction between 22–12, the failure envelope is quadratic in regions RS and OP, which correspond to the tensile and compressive transverse matrix failure, respectively, in the ± 30° plies. Under xx–xy loading (Fig. 5), due to the existence of the stress interaction between 22–12 in all the plies, the first-ply failure envelope consists of three quadratic curves in the upper half plane. Regions JK, KL, and LM correspond to tensile transverse matrix failure in the 90° plies, tensile transverse matrix failure in the –30° plies, and compressive transverse matrix failure in the 90° plies, respectively. The complexity of the ultimate failure envelopes in Figs 4 and 5 indicates the complexity of the failure sequences under these two loading conditions. Ultimate failure envelopes are generally determined by the fiber failure of plies. However, intermediate failures also play very important roles in determining the ultimate failure envelopes. Take Fig. 4 as an example. In regions EF and FG, the tensile transverse matrix failure of the 90° and ± 30° plies are followed by the tensile fiber failure of the 30° and 90° plies, respectively. In most parts of regions AE, GH, JK, and QA, the shear matrix failure of the ± 30° plies leads to catastrophic failure of the laminate. The failure sequence of the laminate in region HO is more complicated, and the ultimate failure is either compressive fiber failure or matrix failure which leads to an immediate catastrophic failure of the laminate without any increase of loading. In

322

Fig. 4. Biaxial failure stresses (xx–yy) of [90/ ± 30/90] laminate made of E-glass/LY556/HT907/DY063.

region NA, the compressive fiber failure of the 90° ply dominates the failure of the laminates. Figure 6 shows failure envelopes of the [ ± 55]S larninate of E-glass/MY750/HY917/DY063 under xx–yy biaxial loading. In regions AB and CD, the failure process is tensile transverse matrix failure in both the + 55° and –55° plies followed by shear matrix failure. Region BC represents tensile fiber failure in the ± 55° plies after initial tensile transverse failure. In regions EF and IA, initial shear matrix failure is catastrophic, and the failure envelope is quadratic since both transverse stress and shear stress are active up to failure. In region FG, the initial failure is compressive transverse matrix failure, followed by the shear matrix failure of ± 55° plies. Regions DE and HI are actually transitional points from one failure sequence to another failure sequence. Failure envelopes of [0/ ± 45/90]S laminate under xx–yy biaxial loading are plotted in Fig. 7. The failure envelopes are completely symmetrical about 45° (xx–yy), due to the

323

Fig. 5. Biaxial failure stresses (xx–xy) of [90/ ± 30/90] laminate made of E-glass/LY556/HT907/ DY063.

Fig. 6. Biaxial failure stresses (xx–yy) of [ ± 55]S laminate made of E-glass/MY750/HY917/DY063.

324

Fig. 7. Biaxial failure stresses (xx–yy) of [0/ ± 45/90]S laminate made of AS4/3501-6.

nature of this quasi-isotropic laminate. Ultimate failure is governed by fiber failure for most loading ratios: except in regions BC and EF, where the shear matrix failure in the ± 45° is catastrophic. Consequently, the ultimate failure envelope is piece-wise linear except in BC and EF. In region FG, the first-ply failure is also the ultimate failure of the laminate, i.e. the two failure envelopes coincide in this region. Caution should be exercised in interpreting results presented in Figs 4–7 for compressive loading. Under compression, premature buckling failure may occur in reality, whereas the laminate analysis performed here does not include the buckling failure mode. If buckling occurs, the actual failure stresses could be much lower than those shown in these figures.

3. Prediction of stress/strain responses In this exercise, we include material non-linearity and effects of matrix cracking in our predictions of stress/strain responses of laminates.

325 3.1. Material non-linearity A number of models have been proposed for predicting laminate response resulting from material non-linearity. Some existing models were reviewed by Nahas.8 In most of these models, material non-linearity is taken into account by considering a non-linear stress/strain relationship of the lamina either in shear, or both in shear and transverse loading. High order polynomials or spline interpolation functions are usually used to represent the stress/strain relationships. The material non-linearity in fiber composites can also be considered a plastic behavior and can be modeled by well developed classical plasticity theories. In this approach, an appropriate plastic potential function for the composite material must be defined. A plastic potential function for a general 3-D fiber-reinforced composite was introduced by Sun and Chen:9,10 2f (ij) = a11211 + a22222 + a33233 + 2a121122 + 2a131133 + 2a232233 + 2a44223 + 2a55213 + 2a66212

(3)

where stresses ij refer to the principal material directions. The coefficients aij describe the amount of anisotropy in the initial plasticity. This expression for f (ij) satisfies the orthotropy condition. The values of ij can be determined from experimental data. If the plastic deformation in the fiber (x1) direction is neglected, a one-parameter plastic potential can be obtained for plane stress as: 2f = 222 + 2a66212

(4)

where a66 describes the initial plastic anisotropy between in-plane transverse and shear directions. A smaller a66 corresponds to smaller shear plasticity, whereas a larger a66 indicates more significant shear plasticity than transverse plasticity. It was discussed previously10 that parameter a66 can be determined by testing a series of off-axis unidirectional specimens (theoretically only two off-axis specimens with different off-axis angles are required to determine a66). If only the non-linearity in shear loading is considered, a66 can be chosen to be a sufficiently large value, say a66 = 100. In this investigation, due to the insufficient information on transverse plasticity of E-glass/MY750/HY917/DY063 material, we consider shear plasticity only and neglect the transverse plasticity. In this case, a66 is taken to be 100. For AS4/3501-6, a66 = 2·5 is taken, which includes both shear and transverse plasticity.11 Commercial finite element code ABAQUS (Version 5·4) is used to perform the elasticplastic analysis of the composite laminates. In ABAQUS, Hill’s yield function for anisotropic solids is used to provide the plastic potential given by eqn (4) by properly defining the coefficients. Because of the uniform loading in one or two directions, a simple four-element by four-element mesh of eight-node plate element is used. The one-parameter plastic potential function together with the associated flow rule is input in *POTENTIAL command, and the plasticity is input in the *PLASTIC command. 3.2. Transverse matrix cracking It is well understood that transverse matrix cracking may occur long before final failure. Matrix cracking causes stiffness degradation and induces other failure modes. Extensive

326 research has been conducted to understand laminate behavior after first matrix cracking. A number of approaches have been used to predict stress distribution in a cracked laminate and stiffness reduction caused by matrix cracking.12–17 Among those approaches, shear-lag methods13 and finite element methods17 appear to be relatively simple and accurate. To predict stress/strain behavior of a laminate caused by matrix cracking, two relationships are required. One is the relation between crack density and applied stress; the other is the relation between stiffness reduction and crack density. We have performed a finite element analysis to predict stiffness reduction of laminates containing matrix cracks.18 The effective stiffnesses of a cracked layer and several parameters which affect the effective stiffness of that layer were evaluated. Some conclusions were obtained: (1) normalized crack density (the number of cracks over the length equal to the thickness of the cracked layer) is a more suitable parameter for characterizing cracking damage, and effective stiffnesses of a cracked layer depend on normalized crack density rather than crack density; (2) the location of a cracked layer has a negligible effect on the effective stiffnesses of that layer; (3) the constraining layers have some effect on the effective stiffnesses. By using the finite element model, the effective transverse modulus and shear modulus of a cracked layer can be determined.18 It is found that they can be approximated by exponential functions of normalized crack density in the form: E2 = exp(–E*) E 02

(5)

G12 = exp(–G*) G 012

(6)

where E2 and G12 are effective transverse modulus and shear modulus of a cracked layer, respectively; E 02 and G 012 are initial transverse modulus and shear modulus, respectively; * is normalized crack density; and E and G are coefficients to be determined by the finite element analysis. In this exercise, as an approximation, the effective transverse modulus and shear modulus of the cracked layer in a cross-ply [0/90/0] laminate of AS4/3501-6 and E-glass/MY750/HY917/DY063 are determined by the finite element analysis and used for prediction. For AS4/3501-6, E = 1·45, G = 0·89; and for E-glass/MY750/HY917/DY063, E = 1·64, G = 0·92. Equations (5) and (6) only relate the effective stiffnesses of a cracked layer with the normalized cracked density *. The crack density for a given loading condition needs to be determined first. Lee and Daniel proposed a simplified shear-lag analysis to model progressive transverse cracking of cross-ply composite laminates.13 Closed form solutions for stress distributions and transverse crack density were obtained as functions of applied load and geometry of the constituent layers. The transverse stress, x2, in the cracked layer of a crossply laminate which is subjected to a longitudinal loading a is expressed as: x2 =

冉

冊冉

E2 a + r2 E0

L

1

cosh( 21  1x) L cosh( 21 )

in which 21 =

(h1 + h2)E0 3G12G23 h1h2E1E2 (h1G23 + h2G12)

冊

(7)

327 where E1 and E2 are, respectively, the longitudinal and transverse Young’s moduli of the lamina; E0 is the undamaged laminate stiffness; G12 and G23 are the shear moduli of the lamina; L is the crack spacing; h1, and h2 are the thicknesses of the outer 0° layer and the (transverse matrix) cracked layer, respectively; and r2 is the transverse thermal residual stress in the cracked layer. Following Lee and Daniel,13 we obtain the in-plane shear stress, xy2, in the cracked layer of a cross-ply laminate under shear loading a:

冉

xy2 = a

L

cosh( 22  2x) 1 L cosh( 22 )

冊

(8)

in which 22 =

(h1 + h2) 3G23 h1h2 (h1G12 + h2G23)

Note that in eqn (8), there is no shear thermal residual stress term since it is absent in the cross-ply laminate. The stress state in a cross-ply laminate under general in-plane loading can be regarded as the superposition of normal loading and shear loading. We assume that, when the matrix failure equation of Hashin–Rotem criterion is satisfied, a new matrix crack will develop in the middle of two existing adjacent cracks. Substitution of eqns (7) and (8) into eqn (2) and replacement of x by L/2 give:

冉

E2 a r2 + E0 Y Y

冊冉 2

1

冊 冉 冊冉

1 cosh(1L/2)

2

+

a S

2

1

冊

1 cosh(2L/2)

2

=1

(9)

Normalized crack density (2h2 /L) corresponding to the combined applied stresses (a and a) can be determined by numerically solving the above equation. Equations (7)–(9) can be easily extended to a symmetric balanced laminate, such as [ ± /90]S, provided the effective moduli of sub-laminates are used in the derivation. To apply eqn (9) in conjunction with the finite element model to predict the stress/strain behavior of a general laminate, one additional assumption is needed. For general laminates, the constraining sub-laminates of a cracked layer are not necessarily orthotropic and symmetric; the equation and the finite element model cannot be used directly without modifications. To simplify the analysis, effective stiffnesses of constraining sub-laminates are used in the finite element analysis and in eqn (9). The effective stiffnesses can be calculated as described elsewhere.19

3.3. Predictions For simplicity, the overall stress/strain response of the laminate is taken approximately as the ‘superposition’ of those predicted separately by considering material non-linearity and matrix

328 cracking. Specifically, under a certain stress (uniaxial loading) or stress combination (biaxial loading), the longitudinal and transverse strains determined by the elastic-plastic analysis are simply superposed with the additional strains caused by the stiffness reduction due to matrix cracking. Figures 8–14 are plots of overall stress/strain curves for different laminates under various loading conditions. For comparison, stress/strain curves predicted by the linear laminate theory in conjunction with the ply-by-ply discount method are also plotted in the figures. Solid squares in the figures mark the onset of matrix cracking. For the [0/90]S laminate under uniaxial loading and the [ ± 45]S laminate under equal-biaxial loading (y /x = 1), the nonlinearity in the stress/strain curve is solely caused by matrix cracking, because there is no shear stress in these laminates and we neglect the transverse plasticity for E-glass/MY750/ HY917/DY063. On the other hand, for the [ ± 45]S laminate subjected to biaxial loading with y /x = 1/–1, the non-linearity comes primarily from material non-linearity since the shear stress is very prominent in this case. For the quasi-isotropic graphite/epoxy laminate subjected to uniaxial loading, the material non-linearity and matrix cracking contribute nearly the same amount to the non-linearity of the stress/strain curve. When it is subjected to the biaxial loading with y /x = 2/1, matrix cracking has a greater effect on the stress/strain nonlinearity.

Fig. 8. Stress/strain curves for [0/90]S laminate made of E-glass/MY750/HY917/DY063 (xx /yy = 1/0).

329

Fig. 9. Stress/strain curves for [ ± 45]S laminate made of E-glass/MY750/HY917/DY063 (yy /xx = 1/1).

Fig. 10. Stress/strain curves for [ ± 45]S laminate made of E-glass/MY750/HY917/DY063 (yy /xx = 1/–1).

330

Fig. 11. Stress/strain curves for [ ± 55]S laminate made of E-glass/MY750/HY917/DY063 (yy /xx = 1/0).

Fig. 12. Stress/strain curves for [ ± 55]S laminate made of E-glass/MY750/HY917/DY063 (yy /xx = 2/1).

331

Fig. 13. Stress/strain curves for [0/ ± 45/90]S laminate made of AS4/3501-6 (yy /xx = 1/0).

Fig. 14. Stress/strain curves for [0/ ± 45/90]S laminate made of AS4/3501-6 (yy /xx = 2/1).

332 4. Summary In this exercise, failure envelopes and stress/strain responses of a number of composite laminates were predicted. Failure envelopes of unidirectional composites and multi-layered composite laminates were predicted by linear laminated plate theory together with Hashin– Rotem lamina failure criterion and a ply-by-ply discount method with parallel spring stiffness reduction model. The stress/strain curves of four laminates were predicted by considering both material non-linearity and progressive matrix cracking. Material non-linearity was modeled by the one-parameter plasticity model for fiber composites. Progressive matrix cracking and the resulting stiffness reduction were predicted by use of a simplified shear-lag analysis and the finite element analysis. References 1 Nahas, M. N., Survey of failure and post-failure theories of laminated fiber-reinforced composites. J. Compos. Technol. Res., 1986, 8, 138–153. 2 Labssiere, P. and Neale, K. W., Macroscopic failure criteria for fiber-reinforced composite materials. Solid Mech. Archives, 1987, 12, 65–95. 3 Hashin, Z. and Rotem, A., A fatigue failure criterion for fiber reinforced materials. J. Compos. Mater., 1973, 7, 7448–7464. 4 Rotem, A. and Hashin, Z., Fatigue of angle-ply laminates. AIAA J., 1976, 14, 868–872. 5 Bailey, J. E., Curtis, P. T. and Parvizi, A., On the transverse cracking and longitudinal splitting behavior of glass and carbon fiber reinforced epoxy cross ply laminates and the effect of Poisson and thermally generated strain. Proc. R. Soc. Lond., 1979, A366, 599–623. 6 Flaggs, D. L. and Kural, M. H., Experimental determination of the in situ transverse laminate strength in graphite/epoxy laminates. J. Compos. Mater., 1982, 16, 103–116. 7 Sun, C. T. and Zhou, S. G., Failure of quasi-isotropic composite laminates with free edges. J. Reinf. Plast. Compos., 1988, 7, 515–557. 8 Nahas, M. N., Analysis of non-linear stress–strain response of laminated fibre-reinforced composites. Fibre Sci. Technol., 1984, 20, 297–313. 9 Sun, C. T. and Chen, J. K., Effect of plasticity on free edge stresses in boron-aluminum composite laminates. J. Compos. Mater., 1987, 21, 969–985. 10 Sun, C. T. and Chen, J. L., A simple flow rule for characterizing nonlinear behavior of fiber composites. J. Compos. Mater., 1989, 23, 1009–1020. 11 Sun, C. T. and Wu, P. S., Pin-contact failure in composite laminates. Proc. ICCM-10, Whistler, BC, Canada, August 1995, pp. III-637–644. 12 Highsmith, A. L. and Reifsnider, K. L., Stiffness-reduction mechanism in composite laminates. In Damage in Composite Materials, ASTM STP 775, ed. K. L. Reifsnider. American Society for Testing and Materials, Philadelphia, PA, 1982, pp. 103–117. 13 Lee, J.-W. and Daniel, I. M., Progressive transverse cracking of crossply composite laminates. J. Compos. Mater., 1990, 24, 1225–1243. 14 Hashin, Z., Analysis of cracked laminates: a variational approach. Mech. Mater., 1985, 4, 121–136. 15 McCarty, L. N., Theory of stress transfer in a 0–90–0 cross-ply laminate containing a parallel array of transverse cracks. J. Mech. Phys. Solids, 1992, 40, 27–68. 16 Laws, N., Dvorak, G. J. and Hejazi, M., Stiffness changes in unidirectional composites caused by crack systems. Mech. Mater., 1983, 2, 123–137. 17 Hahn, H. T., Kiusalaas, J., Burns, B. P and Bogetti, T. A., Constitutive modeling of composite laminates with progressive ply cracking. In Proc. 1992 ASME Int. Computers in Engineering Conf. and Exposition, ed. G. A. Gabriele, 2–6 August 1992, San Francisco, CA, pp. 185–193.

333 18 Tao, J. X. and Sun, C. T., Effect of matrix cracking on stiffness of composite laminates. Mech. Comp. Mater. Struct., 1996, 3, 225–239. 19 Sun, C. T. and Li, S., Three-dimensional effective elastic constants for thick laminates. J. Compos. Mater., 1988, 22, 633–639. 20 Hinton, M. J. and Soden, P. D., Predicting failure in composite laminates: the background to the exercise. Compos. Sci. Technol., 1998, 58(7), 1001. 21 Soden, P. D., Hinton M. J. and Kaddour, A. S., Lamina properties, lay-up configurations and loading conditions for a range of fibre reinforced composite laminates. Compos. Sci. Technol., 1998, 58(7), 1011.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

334

CHAPTER 3.10

A progressive quadratic failure criterion for a laminate* Kuo-Shih Liu and Stephen W. Tsai† Department of Aeronautics and Astronautics, Stanford University, CA 94305-3045, USA Abstract The application of failure criteria to laminated composites is a critical step in the determination of the strength and safety of a point in a structure under combined loading conditions. Non-homogeneous stresses within a structure may induce a complicated failure scenario whereby one ply at a point can initiate failure and affect other plies at the same point or the same ply at a neighboring point. After the first-ply failure, the stiffness of the ply is reduced by either matrix or fiber failures. The strength of the laminate at the same point is evaluated again to see if the laminate can carry an additional load. This ply-by-ply analysis progresses until the ultimate strength of the laminate is reached. The process can be extended to structures subjected to non-homogeneous stresses where both ply-by-ply and point-to-point strength analyses are progressively applied. Keywords: failure, progressive, plotting envelopes

Notation Ei E 0i Ef E *f Ex, Ey Em E *m Es Fij, Fi F *xy FPF

Young’s modulus in the ith direction; i = x, y, s In-plane Young’s moduli of a laminate; i = 1, 2, 6 Fiber longitudinal Young’s modulus Fiber degradation factor (to model fiber collapse) Longitudinal and transverse Young’s modulus of a ply Matrix Young’s modulus Matrix degradation factor (to model a ply with micro-cracks) Shear modulus of a unidirectional ply Strength parameters in stress space of a quadratic failure criterion; i = x, y, s Normalized interaction term of a quadratic failure criterion First Ply Failure

* This article represents the authors’ contribution to a worldwide exercise to confirrn the state-of-the-art for predicting failure in composites, organized by Hinton and Soden.1 † To whom correspondence should be addressed.

335 Gij, Gi h0 k LPF {N}, Ni n R S s X, X  x, x Y, Y  y, y {}, i {}, i i

Strength parameters in strain space of a quadratic failure criterion; i = x, y, s Unit ply thickness Failure index, equal to 1/R Last Ply Failure Absolute in-plane stress resultant components; i = 1, 2, 6 Exponent for longitudinal compressive strength degradation Strength/stress ratio or strength ratio, equal to 1/k Shear strength in the xy or 1–2 plane of a ply A subscript for the shear component in the xy or the 1–2 plane; or ultimate shear strain Longitudinal tensile and compressive strengths Longitudinal tensile and compressive ultimate strains Transverse tensile and compressive strengths Transverse tensile and compressive ultimate strains Strain components; i = x, y, s Stress components; i = x, y, s Ply orientation

1. Quadratic failure criterion It is important to realize that failure criteria are purely empirical. Their purpose is to define a failure envelope by using a minimum number of test data. These data are obtained from relatively simple uniaxial and pure shear tests. Combined stress tests are difficult to perform and are therefore not included in the determination of the failure envelope. There are, however, geometric and material considerations which will limit the mathematical form of a failure criterion and the shape of the envelope. For example, a failure envelope must be closed in order to prevent infinite strength. Another example is that the envelope must be convex so that unloading from a state of stress will not lead to additional failures. A two-dimensional representation of a general quadratic criterion in stress space is shown in the equation below:23

冋 册 冋 册

2x 2F *xyxy 2y 2s 1 1 1 1 + + + +  x +  y = 1 XX  兹XX YY  YY  S 2 X X Y Y

(1a)

F *xy = Fxy/兹Fxy Fyy

(1b)

–1 F *xy ≤ 1

(1c)

where

and, for closed envelopes

In eqn (1a) X and X  are the longitudinal tensile and compressive strengths; Y and Y  are the transverse tensile and compressive strengths; S is the longitudinal shear strength; and the normalized interaction term is bounded to ensure that the failure envelope is closed.

336

Fig. 1. Range of values for the normalized interaction term for this exercise and for two other common materials, see also Ref. 4.

The value of this interaction term can best be determined by combined stress tests. In place of performing such difficult tests, it is possible to narrow the bounds between –1 and + 1 cited in eqn (1a) by considering a more restrictive shape for the envelope. A restriction, for example, can be invoked by the admissible tangents to the envelope at the axis intercepts. This is shown in Figs 8.10 to 8.13 of Ref. 3, where the admissible tangents at X, X , Y and Y  are defined by shaded areas. These constraints are rationalized by virtue of the facts that: (1) there is only one interaction term for the entire stress space, and (2) the envelope must be convex. More restrictive bounds for each material can thus be calculated. The results are also shown as bars in Fig. 1. In this figure, the first four materials are those proposed for this exercise.4 The next two are common composite materials, one carbon-fiber-reinforced plastic (CFRP) and one glass-fiberreinforced plastic (GRP). Note that the values are all zero or negative. In fact, an average value of F *xy = –1/2, referred to as the generalized von Mises model, is reasonable for all materials shown in the Fig. 1. A comparison with the values of F *xy in other quadratic failure criteria is given in Table 1. We shall use the generalized von Mises model as the basis of our failure criterion in this exercise. The interaction term for the first two criteria is essentially zero for highly anisotropic materials.

2. Strength ratios In order to facilitate the application of a failure criterion, it is convenient to use either the strength ratio, R, or its reciprocal failure index, k. The strength/stress ratio, or strength ratio for short, is based on proportional loading applied from any state of stress. If loading is from the origin of stress space, the ratio R is easily defined and shown in Fig. 2. By substituting the maximum stress components into the quadratic failure criterion, we have a quadratic equation in strength ratio R and can easily determine its value. Letting i reach maximum values when Fij i | maxj | max + Fi i | max = 1

(2a)

337 Table 1 Interaction terms for various quadratic criteria. Criteria

Uniaxial strengths

Fxy

Tsai–Hill

X = X , Y = Y 

Hoffman

X ≠ X , Y ≠ Y 

1 2X 2 1  2XX 

Generalized von Mises

X ≠ X , Y ≠ Y 

Tsai–Wu

X ≠ X , Y ≠ Y 

F *xy (all materials)



Y ≤ –0·008 2X

–0·014

–0·041

冑

1 2

F *xy 兹XX YY  F *xy 兹XX YY 

YY  ≤ –0·022 XX 



1 2

–1 F *xy ≤ 1

we substitute Ri | applied for i | max: [Fij i j ]R 2 + [Fi i]R  1 = 0

(2b)

aR 2 + bR  1 = 0; a = Fij i j , b = Fi i

(2c)

Solving the quadratic equation:

gives the strength ratio R equal to the positive quadratic root R =  (b/2a) + [(b/2a)2 + 1/a]1/2

(2d)

This approach is easy to use because the resulting strength ratio provides a linear scaling factor; i.e. If R = 1, failure occurs. If R = 2, the factor of safety is 2. Load can be doubled or laminate thickness reduced by 1/2 before failure occurs. The same strength ratio can be determined from the equivalent quadratic criterion in strain space. This may be preferred in a laminate because ply strains are either uniform or vary

Fig. 2. Strength ratio R as the scaling factor of a loading vector.

338 linearly across the thickness. Similarly, the failure index can be determined in the strain space. R = strength ratio: [Gij i j ]R 2 + [Gi i]R = aR 2 + bR = 1 (3)

k = failure index: [Gij i j ][k]2 + [Gi i][k]2 + k = 1 1

1

b

3. Hygrothermal stresses Laminates are normally cured at an elevated temperature, and after curing moisture absorption normally occurs. The effects of thermal and moisture stresses from these sources can be readily estimated by using a linear theory of thermoelasticity. These stresses and strains can significantly affect the first-ply-failure condition. Since many designs are based on this approach, these hygrothermal stresses should not be ignored. It is therefore recommended that the effects of both elevated-temperature curing and moisture absorption be included in the strength prediction of a laminate. The expansion coefficients used for our calculation are taken from Ref. 3, Table 4.4, and Ref. 4 where • Swanson material is based on T300/5208 in Table 4.4; • Schelling material is based on AS/3501 in Table 4.4; • Krauss and Hinton materials are based on E-glass/epoxy in Table 4.4. We have assumed that the temperature difference is –100°C and that the moisture content is 0·005 or 0·5%. These two factors tend to cancel each other out if both occur. If a laminate is cured at room temperature, then moisture absorption can have a significant effect on ply failures. Conversely, if a laminate is cured at elevated temperature but used in a dry environment, then curing stresses will not be cancelled and can be significant.

4. Micro-cracking A ply in a laminate may fail by micro-cracking when the transverse strain on the ply axis is positive (tensile). A saturation level of periodically dispersed cracks is reached. This phenomenon will be explained below. When the transverse strain component is zero or negative the failure mechanism will not be micro-cracking. It would fail by crushing or buckling. Since transverse compressive failure strain, y, shown in Table 2, is many times higher than the transverse tensile strain, y, or even higher than the longitudinal compressive failure strain, x, we assume that the ply failure will be a combined matrix and fiber failure. The failure will occur in one location, totally different from the widely dispersed microcracks. If micro-cracking takes place in a ply, we assume that it happens instantaneously within a limited region of high stress in a ply. Having other plies at the same point in a laminate, the

339 Table 2 Ultimate stress and strain of the exercise and other materials.4

Fiber Matrix

Swanson

Schelling

Krauss

Hinton

CFRP

GRP

AS4 3501-6

T300 BSL914C

E-glass LY556

E-glass MY750

T300 F934

E-glass epoxy

1140 570 35 114 72

1280 800 40 145 73

Max stress (MPa) X 1950 X 1480 Y 48 Y 200 S 79 Max strain, eps* (
10–3) x 151·37 x 11·67 y 4·33 y 18·06 s 11·97

1500 900 27 200 80 10·80 6·48 2·44 18·07 14·55

20·77 10·39 1·93 6·28 12·35

27·30 17·06 2·40 8·70 12·52

1314 1220 43 168 48 8·88 8·24 4·46 17·41 10·55

1062 610 31 118 72 27·51 15·80 3·75 14·27 17·39

laminate as a whole may be capable of continuing to carry the prevailing load. It is therefore useful to model the presence of micro-cracking in a ply by reducing its transverse and shear moduli while maintaining the longitudinal stiffness. The extent of this reduction is a function of the number of cracks at saturation as measured by their density or aspect ratio. A shear lag model for a [0/90] laminate of T300/5208 CFRP is shown in Fig. 3, where the reduction in stiffness of the laminate approaches 5% as micro-cracks reach a saturation level.5 Although reductions in transverse and shear moduli at the crack saturation level can be calculated for each laminate subjected to a uniaxial tensile load, a simpler method of estimating this reduction can be obtained in terms of a matrix reduction factor by using micromechanics.35 For the purpose of this exercise, we only list the modulus reduction in both

Fig. 3. Asymptotic loss in laminate stiffness based on shear lag analysis as micro-cracking reaches a saturation level.

340 Table 3 Reduced and transverse shear moduli of plies due to micro-cracks. Degraded moduli GPa/normalized Matrix Transverse Shear Fiber volume fraction

Em E *m Ey E *y Es E *s

Swanson AS4/35

Schelling T300/9

Krauss E-gl/ep

Hinton E-gl/ep

CFRP T300/F9

GRP E-gl/ep

→ 0·51 → 0·15 1·93 0·18 1·07 0·16 0·60

→ 0·51 → 0·15 1·93 0·18 1·04 0·19 0·60

→ 0·51 → 0·15 2·19 0·12 1·11 0·19 0·62

→ 0·51 → 0·15 1·71 0·11 0·86 0·15 0·60

→ 0·51 → 0·15 1·93 0·20 1·00 0·22 0·60

→ 0·51 → 0·15 1·32 0·16 0·66 0·16 0·45

absolute and normalized forms (with respect to those of the intact plies and shown with asterisks) in Table 3. The estimated fiber volume fractions are also listed.

5. Longitudinal compressive strength Another factor that is affected by the presence of micro-cracking is the longitudinal compressive strength of a ply within a laminate. Using the concept of a beam on an elastic foundation, we can show the loss in the compressive strength by the reduction in the longitudinal shear modulus. This approach was used by Rosen6 for the loss in compressive strength by in-phase fiber buckling (symmetric). The predicted loss is linearly proportional to the reduction in shear modulus. The loss, however, seems more severe than necessary. We therefore introduce an exponent n to vary the loss. When n = 1 we have Rosen’s linear model. When n < 1 we can dampen the effect of the loss in compressive strength. We recommend the use of n = 0·1 which lies between the values n = 0·0 and n = 0·2 shown in Fig. 4.

Fig. 4. An estimated reduction in compressive strength due to the reduction in shear modulus caused by micro-cracking.

341 Table 4 A summary of sffffness and strength of virgin and degraded AS4/3501-6 ply material (Swanson). Intact

E* Em Ex Ey vx Es X X Y Y S F *xy

Baseline 210 3·40 126·00 11·00 0·28 6·60 1950 1480 48 200 79 –0·50

Matrix degradation Modified 210 0·51 126·00 1·930 0·042 1·070 1950 1258 48 200 79 –0·08

Mod/B 1·000 0·150 ← E *m 1·000 0·175 0·150 ← E *m 0·162 1·00 0·85 ← n = 0.1 1·00 1·00 1·00 0·15 ← E *m

Fiber degradation Modified 2·100 0·034 1·260 0·110 0·003 0·066 1950 976 48 200 79 –0·01

Mod/B 0·010 ← E *f 0·010 ← E *f 0·010 ← E *f 0·010 ← E *f 0·010 ← E *f 0·010 ← E *f 1·00 0·66 ← n = 0.1 1·00 1·00 1·00 0·01 ← E *f

6. Summary of degradation factors Subjective derived factors of degradation as absolute and normalized (modified/baseline) values are tabulated in Table 4 where: • matrix degradation factor, E *m = 0·15 (Table 3); • fiber degradation, E *f = 0·01, signifying a catastrophic collapse; • compressive strength degradation exponent, n = 0·1. The composite ply material used by Swanson in this exercise will sustain the degradation indicated in Table 4. We have made many studies of the sensitivities of various degradation factors to the resulting final (LPF) failure envelopes and found that there are so many uncertainties that the exact numerical values will not significantly affect the qualitative nature of the resulting predictions. For example, the selective degradation (as determined by the sign of the transverse ply strain) can have a greater influence than the values of the degradation factors.

7. Progressive failure scenario With the degradation factors, we can now outline a progressive failure scenario of a laminate subjected to applied stresses. First of all, the traditional application of a failure criterion is shown in the box on the left-hand side of Fig. 5. Ply material and ply orientation are selected to form a laminate. After a load {N} is applied, the first ply failure (FPF) is determined based on the ply having the minimum or lowest strength ratio among all plies. Secondly, the transverse strain of the failed ply will dictate whether micro-cracking has taken place. Only when this strain component is positive will microcracking occur. The ply

342

Fig. 5. A flowchart of the progressive failure modeling that extends the traditional FPF to include matrix and fiber failure modes.

will be degraded by a reduced matrix modulus, say to 0·15 of the virgin modulus or the values of transverse and shear moduli listed in Table 3. If the transverse strain is zero or compressive, the ply is assumed to remain intact (no micro-cracking) and the only possible failure mode would be that of the fiber. This failure mode is catastrophic as shown by the application of an arbitrary longitudinal stiffness degradation factor of 0·01, or 1% of the virgin stiffness. Thirdly, having a reduced matrix or fiber modulus, the next ply failure can be calculated. The failed ply will again be selectively degraded depending on whether or not the transverse strain is positive. If this second ply is the same ply that has failed by matrix degradation (from micro-cracking), fiber degradation will be the only path. Thus, this ply will have to be degraded twice. But if the first ply has failed as a result of fiber failure, it cannot fail again through matrix failure. This ply can only fail once. Finally, the process of progressive failures on a ply-by-ply basis will continue until the maximum load is reached, beyond which the load will reduce as additional plies fail. The ultimate load of the laminate is thus determined.

8. Comments on the baseline data provided Details of the lamina properties, lay-up configurations and loading of the laminates analysed are provided in Ref. 4. In the prediction of failure envelopes and stress/strain curves using our progressive failure model, our analysis is linear. For materials that exhibit non-linear stress/strain curves, we used the secant modulus as the linear modulus from the origin to failure. The failure strain is determined by the ratio of ultimate strength to the secant modulus. All other properties given by the editor of this exercise were used for the predicted results shown in all subsequent figures (Figs 6–19), with the following additional properties. • The degradation factors, however, were selected by us and are all listed in Table 3 for the transverse and shear moduli. • Additional properties are the fiber degradation and compressive strength exponent shown in Table 4. • The hygrothermal constants used for curing stresses and moisture absorption follow those recommended in Ref. 3, Table 4.4. These values differ from those provided by the

343 organizers of this study, but the effect on the resulting hygrothermal stresses is not significant.

9. Observations on predicted results Figures 6–8 show the failure envelopes of the UD laminae. The first and last failure are the same in the Figs 9–11 and Fig. 14 where more than one failure envelope has been plotted, the envelope that contains the origin will be a first-ply-failure (FPF) envelope within which no failure occurs. Envelopes beyond the FPF are the final failure or the last-ply-failure (LPF). Degradation occurs instantaneously when FPFis reached. If proportional loading continues to

Fig. 6. Biaxial failure stress envelope for 0° unidirectional lamina made of glass/epoxy composite under transverse and shear loading (y versus xy).

Fig. 7. Biaxial failure stress envelope for 0° unidirectional lamina made of T300/BSL914C composite under longitudinal and shear loading (x versus xy).

344

Fig. 8. Biaxial failure stress envelope for 0° unidirectional lamina made of glass/epoxy composite under longitudinal and transverse loading (y versus x).

increase beyond FPF, progressive ply failures occur until the LPF is reached. The envelopes plotted show only FPF and LPF and do not show the intermediate plies. An LPF envelope is not smooth because ply orientations vary depending on the combined stresses applied. The convexity of the envelopes is not violated if radial loading lines are drawn from the origins. At certain loading vectors, the predicted LPF jumps from one ply orientation to another. The jump gives the appearance of non-convexity.

Fig. 9. Biaxial failure stress envelope for (90°/ ± 30°/90°) laminate made of glass/epoxy composite under combined loading (y versus x).

345

Fig. 10. Biaxial failure stress envelope for (90°/ ± 30°/900) laminate made of glass/epoxy composite under combined loading (x versus xy).

Fig. 11. Biaxial failure stress envelope for ± 55° angle-ply laminate made of glass/epoxy composite under combined loading (y versus x).

346

Fig. 12. Stress/strain curves for ± 55° angle-ply laminate made of glass/epoxy composite under uniaxial tensile loading with y /x = 1/0.

Fig. 13. Stress/strain curves for ± 55° angle-ply laminate made of glass/epoxy composite under biaxial tensile loading with y /x = 2/1.

347

Fig. 14. Biaxial failure stress envelope for (0°/ ± 45°/90°) laminate made of AS4/3501-6 composite under combined loading (x versus y).

In Figs 15 and 16, where uniaxial and 2 : 1 biaxial stresses are imposed on [ /4] laminates, the successive failure plies are shown by ‘
’ on the stress strain curves. The specific ply that is associated with the ‘
’ is as follows: • For uniaxial tensile loading, shown in Fig. 15, the lowest ‘
’ was [90], followed by [ ± 45], and ultimately [0]. Each ply failed only once; i.e. each went from intact to degraded by micro-cracking. Thus when the ultimate stress was reached, all plies were saturated with micro-cracks. • For biaxial tensile loading, the lowest ‘
’ was [90], followed by [ ± 45], and then [0] microcracked, and ultimately [0] collapsed. This progression is different from the uniaxial loading by having the [0] ply fail twice. This failure scenario is explained in Fig. 5. In uniaxial loading, after all plies including [0] had micro-cracks, the degraded laminate could not carry any more load. The ultimate stress was thus reached. For biaxial loading, the [0] ply could continue to carry a load after microcracking. Thus fiber failure must set in before the ultimate stress is reached. In Fig. 13, where a [ ± 55] laminate is subjected to hydrostatic stress, the lower ‘
’ signified the FPF or formation of micro-cracks, and the higher ‘
’ the fiber failures at LPF.

348

Fig. 15. Stress/strain curves for (0°/ ± 45°/90°) laminate made of AS4/3501-6 composite under uniaxial tensile loading in y direction (y /x = 1/0).

Fig. 16. Stress/strain curves for (0°/ ± 45°/90°) laminate made of AS4/3501-6 composite under biaxial tensile loading with y /x = 2/1.

349 Thus each ply went through two failures, similar to the [0] ply of [ /4] subjected to 2 : 1 stress above. In Fig. 18, where the [ ± 45] lay-up is subjected to tension and compression, FPF coincides with LPF, implying that after micro-cracks set in the laminate could not carry any more load. The solid line represents strain in the y direction and the dashed line represents that in the x direction. The behavior is the same as imposing pure shear on [0/90], in which case there is no post-FPF load-carrying capability.

10. Concluding remarks The predictions of failure envelopes and stress/strain curves for various composite laminates are attached. The failures of laminates on a progressive basis can be found in Tsai’s work.3 Again we recognize that failure criteria are empirical. The progressive failure scenario described here can be applied to other failure criteria. Our favorite is the quadratic criterion for its accuracy, ease of use, flexibility, scalar representation, and so on. A criterion is only as good as the data available. More data for wide-ranging combinations of stresses seem to agree with the quadratic criterion than with any other criterion. It is also important to recognize that a failure criterion is often the only basis for interpolating and extrapolating strength under combined stresses where data are not available.

Fig. 17. Stress/strain curves for ± 45° angle-ply laminate made of glass/epoxy composite under biaxial tensile loading with y /x = 1/1.

350

Fig. 18. Stress/strain curves for ± 45° angle-ply laminate made of glass/epoxy composite under biaxial tensile loading with y /x = 1/–1.

Fig. 19. Stress/strain curves for (0°/90°) cross-ply laminate made of glass/epoxy composite under uniaxial tensile loading with y /x = 0/1.

351 Micro-cracking has been a mechanism of failure extensively studied in many investigations. This mechanism occurs under transverse tensile/longitudinal shear loading conditions. We use a selective degradation criterion to distinguish this from non-tensile conditions. Compression and shear failures are caused by crushing and buckling, and involve no micro-cracking. Then a ply is considered to have failed totally, as implemented by a fiber collapse. The FPF represents the ultimate load and is the one and only LPF. Our criterion is intended to describe intralaminar failures, in 1, 2 and 3 dimensions. It can be extended to interlaminar failures if normal and tangential strengths can be measured. From our experience with many existing composite materials, we conclude that progressive failure of a point-stress analysis is not as important as that of a structural analysis having nonhomogeneous stresses. A simple example would be the failure progression of a plate with an open hole subjected to a uniaxial tensile load. There will be a stress concentration, and ply failures will extend from the free edge of the hole into the interior of the laminate. Thus at each point within the laminate ply-by-ply failure may be modeled by the progressive model described in this paper. Then a ply failure can extend into neighboring points as the applied load increases. The process will continue until the applied load reaches a maximum, after which the load will drop as failure expands within plies and among points or elements. It will take a sophisticated finite-element analysis to track the progressive failures of a laminate under non-homogeneous stresses. But the scenario described here may provide a good starting point. We are not happy with the degree of empiricism that we have used in the progressive failure scenario. First of all, the failure process is extremely complicated. Our attempt is a much simplified approach. What we have not studied includes delamination and the limitations of the homogenization of micro-cracking. While new approaches will always be welcome, the best justification of failure criteria can be made from the standpoint of convenience and utility. The present exercise is an effective way to address this important subject. A follow-up to verify non-homogeneous stresses may be equally important. Test data on open-hole and plugged-hole tensile and compressive loading conditions are available and can serve as starting points for the next exercise.

Acknowledgement The work was partially supported by the National Science Foundation.

References 1 2 3 4

Hinton, M. J. and Soden, P. D., Predicting failure in composite laminates: the background to the exercise. Compos. Sci. Technol., 1998, 58(7), 1001. Tsai, S. W. and Wu, E. M., A general theory of strength for anisotropic materials. J. Compos. Mater., 1971, 5, 58–80. Tsai, S. W., Theory of Composites Design. Think Composites, Palo Alto, 1992. Soden, P. D., Hinton, M. J. and Kaddour, A. S., Lamina properties, lay-up configurations and loading conditions for a range of fibre-reinforced composite laminates. Compos. Sci. Technol., 1998, 58(7), 1011.

352 5

6

Perez, J. L. P. A., An integrated micro-macromechanics analysis of progressive failure in cross-ply composite laminates. Ph.D. thesis, Department of Aeronautics and Astronautics, Stanford University, June 1 992. Rosen, B. W., Mechanics of composite strengthening. In Fiber Composite Materials. American Society for Metals, Metals Park, Ohio, 1965, Ch. 3, pp. 37–75.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

353

CHAPTER 3.11

A strain-energy based failure criterion for non-linear analysis of composite laminates subjected to biaxial loading* William E. Wolfe and Tarunjit S. Butalia Department of Civil Engineering, The Ohio State University, Columbus, OH 43210, USA Abstract The non-linear responses of a lamina under uniaxial longitudinal tension and compression, transverse tension and compression and shear loadings are used to predict the stress/strain response, failure onset and progression in composite laminates subjected to in-plane biaxial loading. A piecewise cubic spline interpolation scheme has been used to represent the basic material properties along the lamina material axes. A laminated plate theory that includes an iterative incremental constitutive law to account for the non-linear behavior of the laminae is combined with a strain-energy based failure criterion for orthotropic materials. Unloading of the failed laminae and subsequent failure progression of the laminate are included in the model. Results for failure onset and progression are presented for a wide variety of unidirectional and symmetric laminates under biaxial loading. Keywords: failure criterion, non-linear analysis, composite laminates, biaxial loading

Notation [A] [C]k ˜ k [C] E11, E22 G12 i, j k Ki m1, m2, m6 Sij

Stiffness matrix of the laminate Stiffness matrix of the kth lamina relative to the lamina material axes Stiffness matrix of the kth lamina relative to the laminate axes Tangent moduli of elasticity Tangent shear modulus Indices which take values 1, 2 and 6 Identification for a lamina Material characterization factor Longitudinal, transverse and shear strain energy shape factors Components of compliance matrix

* This article represents the authors’ contribution to a worldwide exercise to confirm the state-of-the-art for predicting failure in composites, organised by Hinton and Soden.32

354 [S]k tk [T]k, [T]k i v12, v21 i

Compliance matrix of the kth lamina Thickness of the kth lamina Stress and strain transformation matrices Strain components in the lamina Tangent Poisson ratios Stress components in the lamina

1. Introduction Composite materials have been used in structures since the beginning of recorded civilization. Examples of mud and straw brick used by early Incan and Egyptian societies to the concrete and plywood used by modern societies are well known. The advent of resin matrix composite materials has led to the application of laminated composites in a wide variety of structures, particularly in the aerospace and automobile industries. This material is attractive to design engineers because of its high strength-to-weight ratio, superior corrosion resistance and the anisotropic material properties that can be tailored through variations in fiber orientations, lamina thicknesses, and stacking sequences. Prediction of biaxial failure strengths and modes for composite laminates has been investigated in the past with limited success. Comprehensive reviews of the technical literature for composite failure theories have been presented by many researchers.1–8 The majority of the failure theories for composite laminates, have been developed by extending failure models applicable to homogeneous isotropic materials so as to apply them to heterogeneous anisotropic materials. Examples of such failure theories include noninteractive theories like the maximum-stress9 and maximum-strain models, and interactive theories where quadratic,10–16 cubic,18 fourth19 and higher order polynomials are used. These failure theories, for the most part, are either stress or strain based. These theories would be quite reasonable for materials exhibiting linear stress/strain behavior. In general, they do not predict the failure modes. For materials exhibiting non-linear behavior, a failure theory that includes the stress as well as the strain state in the failure criterion would be a more favorable approach. The longitudinal, transverse and shear strain energies of an orthotropic material along its materials axes have been shown by Sandhu20 to be independent parameters. A failure criterion defined by the sum of suitable exponents of the longitudinal, transverse and shear strain energies, forms the basis of the failure theory proposed by Sandhu20,21 and is described in this work. The failure model presented in this paper is a restatement of the theoretical model presented by Sandhu.20,21 Since the development of Sandhu’s original model, the failure theory has been refined to include a variety of ply-unloading mechanisms. Additionally, the theory has been extended to applied stress or strain loading of composite laminates. Sandhu’s strain energy based failure theory, in conjunction with an incremental constitutive law, has shown good comparison with experimental investigations carried out on tubular specimens made of glass/ epoxy and graphite/epoxy, conducted by Sandhu at Air Force Flight Dynamics Laboratory, Dayton, Ohio.22,23 2. Laminate analysis A laminate is made up of unidirectional plies or laminae arranged in a particular stacking sequence chosen by the designer. The mechanical properties of a lamina under static loading

355 are obtained experimentally by conducting five tests (longitudinal tension and compression, transverse tension and compression and shear). In general, these tests reflect the non-linearity of lamina mechanical properties. The study described in this paper accommodates this nonlinearity of the laminae under longitudinal (tension and compression), transverse (tension and compression) and shear loadings. An incremental constitutive law has been chosen to account for the non-linearity of the mechanical properties of a lamina. Many of the non-linear theories presented earlier,24–27 do not predict accurate laminate response.20 In the study described in this paper, the non-linear properties are modeled using piecewise, cubic-spline interpolation functions,28–31 which have been shown by Sandhu20,21 to give good representations of stresses, strains and moduli over a broad range of the stresses and strains. It is assumed that: 1. The strain increment, di, depends upon the stress increment, dj, as well as the strain state, i, and 2. The strain increment, di, is proportional to the stress increment, dj. Expressing the incremental constitutive law in contracted notation gives: di = Sij(i) dj

(i, j = 1, 2, 3, . . . , 6)

(1)

For generalized plane stress, this reduces to di = Sij(i) dj

(i, j = 1, 2, 6)

(2)

where Sij is a function of the strain state. Assuming that the laminae remain orthotropic at all load levels, eqn (2) can also be written as: d = [S] d

(3)

where the vector of strain and stress increments, and the elastic compliance matrix are given by: dT = [d1 d2 d6]

(4)

dT = [d1 d2 d6]

(5)

S11 [S] = S21 S61

S12 S22 S62

S16 S26 S66

(6)

The components of the compliance matrix for an orthotropic material are given by: S11 = 1/E11S22 = 1/E22S66 = 1/G12 S12 = S21 = –v12 /E11 = –v21 /E22 S16 = S26 = S61 = S62 = 0

(7)

356 where E11 and E22 represent the tangent moduli of elasticity along the fiber and transverse to the fiber, respectively. G12 is the tangent shear modulus. v12 and v21 represent the tangent Poisson ratios for the orthotropic material. Simplifying using eqns (3)–(7) gives: d1 = d1(1  v12 B)/E11

(8)

d2 = d2(1  v21 /B)E22

(9)

d6 = d6 /G12

(10)

B = d2 /d1

(11)

where

It can be seen by referring to eqns (8) and (9) that the use of d1, or d2 representing the biaxial stress state (d1, d2), to determine E11 or E22 from the stress/strain curves, under simple loading conditions would be in error. Sandhu20 computed the equivalent strain increments as: d1 |eqv = d1 /(1  v12 B )

(12)

d2 |eqv = d2(1  v21 /B )

(13)

In order to write the equations for a multidirectional laminate, eqn (3) is written for each lamina as: dk = [S]k dk

(14)

where the subscript k refers to the kth lamina. Inverting eqn (14) gives: dk = [C]k dk

(15)

where [C]k = [S]–1 k is the stiffness matrix of the kth ply in the lamina material axes, {d}k and {d}k are the kth ply stress and strain increments with respect to the lamina material axes, respectively. The lamina material axes can be represented as 1 in the fiber and 2 in the transverse direction. Since the lamina material axes and the global laminate axes typically will not coincide, the stress and strain increments must be transformed from the lamina to the laminate (global) axes. The stress and strain increments in the lamina directions are related to the global axes by: dk = [T]k d¯ k

(16)

dk = [T]k d¯k

(17)

where [T]k and [T]k represent the stress and strain transformation matrix, respectively. Vectors {d} ¯ k and {d¯}k represent the laminate stress and strain increments. Substituting eqns (16) and (17) into eqn (15) yields: ¯ k d¯k d¯ k = [C]

(18)

357 ¯ k is the global stiffness matrix. In terms of [C]k, the global stiffness matrix is where the [C] given by ¯ k = [T]–1 [C] k [C]k [T]k

(19)

Further, assuming that the stress distribution through the thickness of each ply is uniform, the vector of in-plane stress resultant increments {dN } can be expressed as:

冘 n

dN =

d¯ ktk

(20)

k=1

where n is the total number of plies in the laminate and tk is the thickness of the kth lamina. Substituting eqn (18) into eqn (20) gives:

冘 n

dN =

¯ k d¯k tk [C]

(21)

k=1

Assuming the strain increments to be the same for all the plies, i.e. {d¯}k = {do}, eqn (21) can be written as: dN = [A]do

(22)

冘

(23)

where n

[A] =

¯ k tk [C]

k=1

For the (n + 1)th load increment, inverting eqn (22) yields: don + 1 = [A]–1 n + 1 dNn + 1

(24)

Here [A]–1 n + 1 represents the average compliance properties of the laminate for the (n + 1)th load increment. But [A]–1 n + 1 is not known when (n + 1)th load increment is applied. Hence the model employs an iterative procedure (see Fig. 1) which starts by approximating don + 1 = [A]–1 n dNn + 1

(25)

The strain increments {do}n + 1, evaluated from eqn (25) are substituted into eqn (18) to obtain the stress increments {d} ¯ k in the global laminate axes. Then {d}k, {d}k, d2 |eqv and d2 |eqv are calculated using eqns (16), (17), (12) and (13), respectively. These incremental stresses and strains are added to the stresses and strains at the nth load increment to obtain updated lamina stresses and strains. These new estimates of stress and strain are then used to determine the average elastic properties of the plies. A new [A]–1 is then calculated. This procedure is repeated until the difference in values of {do}n + 1 between two consecutive cycles is small

358

Fig. 1. Schematic representation of iterative procedure to evaluate {do}n + 1.

enough to achieve the order of approximation desired. Hence, starting with eqn (25) and employing an iterative procedure, the sequence of approximations converges to {do}n + 1. 3. Strain energy based failure criterion Non-linear response of the laminate is evaluated using the incremental loading procedure outlined earlier. As the loading is increased, some or all of the laminae reach a critical state at which they begin to fail. Cumulative failure of laminae leads to the eventual failure of the laminate. Final failure of the laminate occurs when the laminate is unable to carry the imposed loads. The failure criterion for a lamina exhibiting non-linear stress/ strain behavior, may be expressed as: f(, , ) = 1 where f is a scalar function, and  is factor which accounts for material characteristics.

(26)

359 The combined stress and strain state failure criterion based on strain energies of a nonlinear orthotropic material proposed by Sandhu,20,21 is used in this study. For an orthotropic material, the strain energies under uniaxial tension (or compression), transverse tension (or compression) and shear are all independent parameters. A general form of the strain energy based failure criterion for non-linear orthotropic materials may be expressed as:

冘 冋冕 册

1 Ki i = 1, 2, 6

i di

mi

(27)

=1

i

where i is the current strain state, mi define the shape of the failure surface in strain energy space and Ki may be defined for the uniaxial tension (or compression), transverse tension (or compression) and shear as: Ki =

冋冕 册 i di

mi

(28)

(1 = 1, 2, 6)

ui

where ui is the ultimate failure strain. Combining eqns (27) and (28) gives: 兰 1 d1

1

兰 1 d1

u1

m1

兰 2 d2

+

2

兰 2 d2

u2

m2

兰 6 d6

+

6

兰 6 d6

m6

=1

(29)

u6

The factors m1, m2 and m6 determine the shape of the failure surface in strain energy space. Theoretically, m1 is different in tension and compression and the same holds true for m2. For m1 = m2 = m6 = 1, the failure surface in the non-dimensionalized strain energy space is pyramidal, while for m1 = m2 = m6 = 2, it is spherical. For each material system, the experimental values of the non-dimensionalized strain energies at failure, will be compared with the numerical predictions, under biaxial loading of unidirectional laminae or uniaxial/ biaxial loading of multidirectional laminates. This technique will define the numerical values of the shape factors to be used for that material system. Once the values for the shape factors have been evaluated, they remain the same for all laminate configurations. Theoretically, the values of mi can take on any real value greater than zero. However, the approximate upper bound value for mi of 2 when m1 = m2 = m6, corresponds closely to the maximum stress criterion. Due to the unavailability of the experimental data for the laminates analyzed in this study, we have chosen m1 = m2 = m6 = 1 as suggested by Sandhu.20 More appropriate shape factors for the four composite materials systems listed in Table 1 will be determined when the experimental biaxial loading data become available.

4. Failure prediction and progression At the end of each load increment, the stress/strain state of each lamina is checked against the strain energy failure criterion. The percentage contribution of the longitudinal, transverse and shear strain energy terms to the failure of each ply is evaluated. When the sum of the three percentages equals 10°, the lamina fails. Two types of failure modes are considered, i.e. fiber

360 Table 1 Properties of unidirectional composite material systems33.

AS4/3501-6

T300/BSL914C

E-glass/ LY556

E-glass/ MY750

1950a 1480 48 200a 79a 1·38a 1·175 0·436 2·0a 2·0a

1500 900 27 200 80a 1·087 0·652 0·245 1·818 4·0a

1140 570 35 114 72a 2·132 1·065 0·197 0·644 3·7a

1280 800 40 145a 73a 2·807 1·754 0·246 1·2a 4·0a

Longitudinal tensile strength (MPa) Longitudinal compressive strength (MPa) Transverse tensile strength (MPa) Transverse compressive strength (MPa) In-plane shear strength (MPa) Longitudinal tensile failure strain (%) Longitudinal compressive failure strain (%) Transverse tensile failure strain (%) Transverse compressive failure strain (%) In-plane shear failure strain (%) a

Refers to non-linear stress/strain behavior of the lamina.

failure and matrix failure. Having established that the failure of a particular lamina has occurred, it is assumed to be fiber failure if 兰 1 d1

1

兰 1 d1

u1

m1


100

冒冘

i = 1, 2, 6

兰 i di

i

兰 i di

m1

≥ 10

(30)

ui

Having established the failure of a ply and the need to unload that ply, it is then necessary to look into the mechanisms by which loads carried by that ply could be re-distributed among the remaining plies of the laminate. Three types of unloading techniques have been incorporated in the model, i.e. sudden unloading, gradual unloading, and perfectly plastic behavior.31 In the sudden unloading pattern, the lamina loads are immediately redistributed to the undamaged plies. For fiber failure, the lamina unloads completely while the laminate loads are maintained. For matrix failure, the transverse and shear loads of the affected lamina are set to zero and its properties are modified to reflect zero transverse and shear moduli as well as Poisson’s ratio, while the laminate loads are maintained. The gradual unloading pattern allows for unloading of the affected ply in small steps as the laminate loads increase beyond first ply failure. In the third approach, the lamina continues to carry the failure loads but does not take on additional loads. Choice of the unloading mechanism depends on the particular material system being used. The sudden unloading method has been used for the numerical results presented in this paper. A comparison of the various unloading patterns for different laminate material systems will be presented in the next phase of the failure exercise. First ply failure could lead to subsequent failures depending on the material type and stacking sequence as well as thickness of the individual laminae. Subsequent failure is defined in the same way as the first ply failure and the unloading process of the affected plies remains the same. Laminate failure is said to occur when all the laminae undergo complete failure or when the matrix [A] in eqn (23) becomes singular.20

361 It should be observed that, the strain components for the biaxial stress states are used to satisfy eqns (29) and (30), and the equivalent strain components given by eqns (12) and (13) are used to evaluate the tangent moduli. Further we assume that, without experimental evidence to the contrary, no strain component can be greater than the ultimate strain measured during simple load conditions. Thus, at the end of each load increment, in addition to the strain energy based failure criterion, the calculated strains are checked against the ultimate strains. The ply is assumed to have failed if the calculated strains exceed the ultimate strain in the longitudinal direction (fiber failure) or in the transverse or shear direction (matrix failure).

5. Laminate failure predictions The failure model described earlier has been used to investigate the failure of composite laminates made up of four different material systems, namely, AS4/3501-6 Epoxy, T300/ BSL914C Epoxy, E-Glass/LY556/HT907/DY063 Epoxy and E-Glass/MY750/HY917/ DY063 Epoxy. (Details of the lamina properties and lay-up configurations and loading of the laminates analyzed are provided in Ref. 33.) The properties of these composite material systems are summarized in Table 1. The effect of thermal residual stresses has not been included in the numerical failure predictions.

5.1. AS4/3501-6 laminates A 0° unidirectional lamina of AS4/3501-6 exhibits linear stress/strain behavior up to failure under longitudinal compressive and transverse tensile loading. A small amount of nonlinearity is observed under longitudinal tension and transverse compression. The shear stress/strain behavior is highly non-linear. It can be seen from Table 1 that the AS4/3501-6 material has the highest longitudinal and transverse strengths as compared to the other material systems included in this study. Using the properties given in Table 1, the behavior of a [90°/ + 45°/– 45°/0°]s laminate was studied under x– y biaxial loading. The laminate had a total thickness of 1·1 mm, with all layers having equal thickness. The computed initial and final failure envelopes are shown in Fig. 2. It is observed that the initial and final failure envelopes are symmetric about the x– y line. Also indicated in Fig. 2 are the intermediate (or subsequent) failure conditions. Under longitudinal (x) tensile and transverse (y) tensile loading, we can distinguish separate initial, intermediate and final failure of the plies. Initial failures also occurred at loads less than those resulting in final failure for that portion of the second quadrant where x > |– y | and in the fourth quadrant when y > | – x |. For all other loading patterns, the laminate fails at first ply failure. Maximum strength of the laminate is obtained when x and y are equal as indicated by points B and F in Fig. 2. Minimum strength of the laminate is observed to occur when x and y are equal in magnitude but are of opposite signs as indicated by points D and H. For tensile (longitudinal and transverse) loading, initial failure occurs due to transverse tensile failure in the 90° and 0° plies, for | y /x | < 1 and | y /x | > 1, respectively. Final failure is predominantly by longitudinal tensile failure of the 90° and 0° plies for | y /x | > 1 and | y / x | < 1, respectively. For the x = y loading path, the first ply failure by transverse tension

362 occurs simultaneously in the 90°, + 45° and 0° plies. This is followed by the longitudinal tensile failure of these plies resulting in laminate failure (point B). For y = – y loading paths, the least strength of the laminate is observed. This is due to the fact that the laminate under consideration is quasi-isotropic and under x /y = –1 loading, it experiences pure shear. For longitudinal tensile and transverse compressive loading, the final failure generally occurs in the 90° plies by a combination of longitudinal compression and transverse tension failure. For these loading paths, as the ratio | x /y | increases, the contribution to failure due to longitudinal compression increases while the transverse tensile contribution to failure reduces. Similar observations can be made for longitudinal compressive and transverse tensile loading of the laminate. The initial and final failure envelopes coincide for compressive longitudinal and transverse loading of the laminate. For equal longitudinal and transverse compressive loading, the 90°, ± 45°, and 0° plies fail (point F) in longitudinal and transverse compression. Further, in the third quadrant, longitudinal compressive failure occurs in the 90° plies for | y /x | > 1 and in the 0° plies for | y /x | < 1. For the longitudinal and transverse compressive loading combinations, micro-buckling of the fibers could result in a significant reduction in laminate strength. However, in the current study this micro-mechanical effect has been neglected. The stress/strain curve for transverse loading (y /x = 1/0) of the [90°/ + 45°/– 45°/0°]s laminate is shown in Fig. 3. The initial transverse tensile failure occurs in the 0° plies at

Fig. 2. Biaxial failure stresses for [90°/ + 45°/– 45°/0°]s laminate.

363

Fig. 3. Stress/strain curves for [90°/ + 45°/– 45°/0°]s laminate (y /x = 1/0).

225 MPa (point C1). This results in a slight increase of laminate strains due to unloading of the 0° plies in the transverse direction. Subsequent failure (point Cs) of the laminate at y = 454 MPa is predicted in the 45° plies, where the contributions to failure are in longitudinal tension (4·8%), transverse tension (53·0%) and shear (42·2%). Unloading of the + 45° plies leads to a large increase in laminate strains. The final failure of the laminate occurs by the failing of the 0° plies in longitudinal compression at a transverse tensile laminate stress of 513 MPa (point C). The stress/strain curves are piece-wise linear, with sudden jumps in laminate strains at lamina failure locations (points C1 and Cs). Figure 4 shows the stress/strain curve for the [90°/ + 45/– 45°/0°]s laminate, subjected to a loading path of y = 2x. Similar to Fig. 3, the stress/strain behavior is piece-wise linear. Initial transverse tensile failure of the 0° plies is predicted at y = 246 MPa (point KI). Unloading of this ply in the transverse direction leads to a slight reduction in laminate longitudinal strain (x) and a small increase in laminate transverse strain (y) which would be expected. As the laminate loading is further increased, the ± 45° plies fail at y = 321 MPa, due to transverse tensile failure of the ply (point KS1). This leads to unloading of the ply with a similar unloading pattern to that of the 0° plies. Continued loading of the laminate leads to transverse tensile failure of 90° plies at y = 506 MPa (point KS2). At this load level, the 90° plies unload resulting in an increase of strain x while strain y decreases slightly. Final failure of the laminate at 702 MPa is caused by the longitudinal tensile failure of the 90° plies.

364

Fig. 4. Stress/strain curves for [90°/ + 45°/– 45°/0°]s laminate (y /x = 2/1).

5.2. T300/BSL914C laminates A T300/BSL914C unidirectional lamina exhibits linear stress/strain behavior for longitudinal (tension and compression) and transverse (tension and compression) loading. The shear stress/ strain curve is observed to be highly non-linear. It can be seen from Table 1 that the transverse tensile strength of this material system is very low (27 MPa). A 0° unidirectional T300/BSL914C laminate under longitudinal (x) and shear (xy) loading was studied. Initial ply failure corresponds to the final laminate failure. The failure envelope for the laminate is presented in Fig. 5. The three bracketed terms at each load combination refer to the percentage contribution of longitudinal, transverse and shear strain energies to failure. Failure under longitudinal tension, longitudinal compression and shear are predicted at the strength values obtained experimentally. The behavior of the laminate for longitudinalshear loading is uncoupled except between x /xy ranging from 6·25 to 25 and – 6·25 to –12·5. At x /xy = –9, the contributions to failure from longitudinal compression and shear are almost equal, while the longitudinal tensile and shear contributions to failure are equal at approximately x /xy = 15. For tensile (or compressive) longitudinal and shear loading of the laminate, as the ratio | x /xy | is increased, the percentage longitudinal tensile (or compressive) contribution increases while the shear contribution decreases.

365

Fig. 5. Biaxial failure stresses for 0° laminate.

5.3. E-Glass/LY556 laminates The mechanical properties used to predict the behavior of this material system are also shown in Table 1. It is noted that the shear stress/strain curve is highly non-linear while longitudinal (tensile and compressive) and transverse (tensile and compressive) stress/strain responses are linear. Further, the longitudinal (tensile and compressive), transverse compressive and shear strengths for E-Glass/LY556 are the least of all the material systems listed in Table 1. Consider a 0° unidirectional E-Glass/LY556 laminate subjected to a combination of transverse and shear loading. The initial and final failure envelopes are identical as shown in Fig. 6. For all loading paths, the percentage contribution to failure from longitudinal failure of the ply is zero, as would be expected. As the ratio | y /xy | increases, the shear contribution to failure reduces while the transverse failure contribution increases. For transverse tension and shear, the failure is dominated by shear for | y /xy | < 0·2, and transverse tension for | y /xy | > 1·5. When the transverse loading is compressive, shear response dominates for | y / xy | < 0·5, while the compressive transverse failure dominates for | y /xy | > 3. The transverse and shear contributions to failure, are equal at ratios of y to xy of approximately 1·25 and –2·75. The response of a [90°/ + 30°/–30°]s laminate with total thickness of 1·992 mm, under two types of biaxial loading, i.e. x– y and x– xy, was investigated. The thicknesses of the 90° and ± 30° plies are 0·172 mm and 0·412 mm, respectively. In general, for both these biaxial loading combinations, the final failure of the laminate occurred at a much higher stress than that of first ply failure. Intermediate ply failures were also observed. Figure 7 (x– y) and Fig. 9 (x– xy) show the final failure envelopes drawn according to the scale provided for the failure exercise. Figure 8 (x– y) and Fig. 10 (x– xy) represent the initial, subsequent and final failures of the laminate, drawn for clarity with an enlarged scale. For longitudinal and

366

Fig. 6. Biaxial failure stresses for 0° laminate.

transverse loading of the laminate, the failure envelopes are shown in Fig. 7 and 8. For a tensile longitudinal loading, the initial failure is due to the transverse tensile failure of the 90° plies at a low stress level of x = 59 MPa. This is followed by the longitudinal compressive failure of the 90° plies at x = 207 MPa. The final failure occurs due to longitudinal tensile (10·7%), transverse compressive (49·4%) and shear (39·9%) failure of the ± 30° plies (point J). For a compressive longitudinal loading, the laminate fails at an initial failure stress of x = –186 MPa, due to the transverse compressive failure of the 90° plies (point D). A transverse tensile loading, causes first ply failure at y = 64 MPa due to the transverse tensile failure of the ± 30° plies, while the final failure is caused by the 90° plies failing in longitudinal tension (44·6%) and transverse compression (55·4%) at y = 132 MPa (point A). However, for a compressive transverse loading, the laminate fails at the initial ply failure (y = –120 MPa, point A) in the 90° plies, caused by longitudinal compressive (74·2%) and transverse tensile (25·8%) failure. For a longitudinal compressive and transverse tensile loading of the laminate, the curve ABCD is obtained. The segment AB represents initial transverse tensile failure of the ± 30° plies and final longitudinal and transverse failure of the 90° plies. For segment BC, the ± 30° plies fail due to longitudinal compressive, transverse tensile and shear failure. The curve CD corresponds to transverse tensile failure of 90° plies. For compressive longitudinal and transverse loading, the curve DEFGHI is obtained. Curve DE is an extension of curve CD at point D and hence represents transverse tensile failure of the 90° plies. The steep curve EF represents an initial transverse tensile failure of 90° plies followed by final failure due to compressive transverse and longitudinal failure of the ± 30° plies. As one proceeds from point F to point G, the final failure contribution due to longitudinal and shear failure modes decreases while the transverse failure contribution increases. The steep curve GH represents the initial (and final) failure of the ± 30° plies due to transverse compressive failure. The segment HI represents the failure of the 90° plies in transverse tension and longitudinal compression. For longitudinal tensile and transverse compressive loading, the final failure envelope is illustrated by curved segment IJ. In this segment, the failure is always initiated in the 90° plies by predominantly transverse tensile

367 failure, subsequent failure occurs due to the longitudinal failure of the 90° plies and the final failure is predicted by longitudinal tensile, transverse compressive and shear failure of the ± 30° plies. The laminate response to tensile longitudinal and transverse loads was carefully studied. The initial failure is due to the transverse tensile failure of the 90° plies for x /y > 1·26 and one observes intermediate as well as final lamina failures (segment JKLMNO). For x /y < 1·26, the initial failure is caused by the transverse tensile failure of the ± 30° plies and there are no intermediate failures but only a final failure (segment OPQRSA) of the laminate beyond the initial failure state. Some important observations made for this loading case are:

Fig. 7. Biaxial final failure stresses for [90°/ + 30°/–30°]s laminate.

368 (a) Segment JK represents initial transverse tensile failure of 90° plies, followed by the longitudinal compressive failure of the 90° plies, while the final failure is predicted due to transverse compressive and longitudinal tensile failure of ± 30° plies. (b) The sudden drop in segment KL denotes the initial transverse tensile failure of 90° plies, followed by the longitudinal compressive failure of the 90° plies as well as the transverse tensile failure of the ± 30° plies. (c) The linear segment LM represents the initial transverse tensile failure of 90° plies, subsequent transverse tensile failure of ± 30° plies, and final failure by longitudinal compressive failure of 90° plies. At point M, the initial failure occurs due to the transverse tensile failure of the 90° plies as well as the transverse tensile failure of ± 30° plies, and the final failure is due to the longitudinal tensile failure of ± 30° plies. (d) Segment MN denotes initial transverse tensile failure of 90° plies, subsequent transverse tensile failure of ± 30° plies, and final failure by longitudinal tensile failure of ± 30° plies. (e) Segment NO represents initial transverse tensile failure of 90° plies, subsequent transverse tensile failure of ± 30° plies, and final failure by longitudinal tensile failure of 90° plies. (f) Point O corresponds to x /y = 1·26. For segment OP, we observe a sudden reduction in laminate strength. The segment OP represents transverse tensile failure of the ± 30° plies.

Fig. 8. Biaxial failure stresses for [90°/ + 30°/–30°]s laminate.

369

Fig. 9. Biaxial final failure stresses for [90°/ + 30°/–30°]s laminate.

(g) The line segment PQ indicates initial transverse tensile failure of the ± 30° plies and final transverse tensile failure of the 90° plies. (h) Segment QR represents initial transverse tensile failure of ± 30° plies and final transverse tensile as well as longitudinal tensile failure of the 90° plies. (i) In segment RS, the same kind of failure pattern as segment QR is observed but the contribution of longitudinal tension to final failure of the 90° plies increases as we move from point R to point S. (j) The segment SA indicates the same failure pattern as segment QR but in this segment the longitudinal tensile contribution to final failure of the 90° plies decreases slightly as we move from point S to point A.

Fig. 10. Biaxial failure stresses for [90°/ + 30°/–30°]s laminate.

370 Figures 9 and 10 show the failure envelopes for the [90°/ + 30°/–30°]s laminate under x– xy biaxial loading. Applying only shear loading, the failure is initiated at xy = 54 MPa due to transverse tensile failure of the ± 30° plies. This is followed by the longitudinal compressive failure of the ± 30° plies at xy = 94 MPa (point D). The failure under longitudinal tensile (x /xy = 1/0) and compressive (x /xy = –1/0) loading paths has been discussed earlier. A compressive longitudinal and shear loading results in the final failure envelope identified in Fig. 9 and Fig. 10 as the curved segment ABCD. Initial and final failure stresses are both depicted by the segment ABC. For the segment CD, the first failure occurs predominantly as transverse tensile failure of the + 30° plies while the final failure is due to the longitudinal compressive failure of the + 30° plies. The segment AB is that portion of the failure envelope for which the 90° plies fail in transverse compression while the segment BC represents the failure envelope for 30° plies failing in transverse tension. In segment DL, initial transverse tensile failure of the + 30° plies is followed by final longitudinal compressive failure of the + 30° plies. In segment LE, the initial failure is due to transverse tensile failure of the + 30° plies, subsequent failure is due to combined transverse tensile and shear failure of the 90° plies, and the final failure occurs due to longitudinal compressive failure of the + 30° plies. For x = 0·5 xy (point E), initial transverse tensile failure occurs in the + 30° plies, subsequent failure in the 90° plies results from transverse tensile failure and final failure of the laminate occurs due to transverse compressive failure of the –30° plies. At point F (x = xy), a much lower strength of the laminate is calculated. At this point, the initial failure is due to the transverse tensile failure of the + 30° plies, leading to immediate failure of the laminate without any subsequent ply failures. The segment GH represents initial transverse tensile failure of 90° plies, subsequent transverse tensile failure of + 30° plies and the final transverse and longitudinal failure in the –30° plies. Line segment HI represents the same kind of failure pattern as segment GH but intermediate failure of the + 30° plies is caused by the maximum transverse tensile strain being exceeded in these plies. The segment IJ is the result of an initial transverse tensile failure in the 90° plies, followed by longitudinal compressive failure of the same plies, and a final failure due to longitudinal, transverse and shear failure of the –30° plies.

5.4. E-Glass/MY750 laminates The E-Glass/MY750 material system exhibits a slight non-linearity for transverse compressive stress/strain behavior while the shear stress/strain behavior is highly non-linear. Four types of laminates made of E-Glass/MY750 were studied, i.e. 0° unidirectional [ + 55°/–55°]s’ [ + 45°/– 45°]s and [0°/90°]s laminates. Figure 11 shows the failure envelope for a 0° unidirectional laminate subjected to longitudinal and transverse loading. The initial and final failure envelopes are identical. The longitudinal tensile, transverse tensile and transverse compressive strength values for EGlass/ MY750 noted in Table 1 were obtained from simple longitudinal tensile, transverse tensile and transverse compressive loading tests on the unidirectional laminate. However, on loading the 0° unidirectional laminate under longitudinal compression (x /y = –1/0), it is observed that the laminate fails at x = – 403 MPa (point B) which is only about half the longitudinal compressive strength of the material. This is due to the transverse tensile strain, which is caused by Poisson’s effect, reaching its maximum allowable value of 0·246%. Since, in the

371

Fig. 11. Biaxial failure stresses for 0° laminate.

experimental data provided, the Poisson’s ratio, v12, was not given as a function of the strain, a constant Poisson’s ratio was assumed in the analysis. However, the theoretical model developed here can account for a variable Poisson’s ratio under tensile and compressive loadings. Incorporation of the experimental strain versus Poisson’s ratio (v12) data in the analysis and/or the relaxation of the condition that no strains can be greater than the ultimate measured strains, would likely to lead to laminate failure near 800 MPa, the longitudinal compressive strength of E-Glass/MY750. For tensile longitudinal and transverse loading, the laminate fails due to transverse tensile failure of the plies for the segment CD. For the segment DE, the failure occurs due to a dominant longitudinal tensile failure and a small amount of transverse compressive failure of the plies. If only a longitudinal tensile loading is applied, the tensile failure strength of 1280 MPa is obtained as the failure stress. However, if in addition to the longitudinal loading, even a small amount of transverse compressive stress is applied, the failure occurs at a much lower stress due to longitudinal tensile as well as transverse compressive failure of the plies (e.g. for x = – 0·01y , a failure stress of 1020 MPa is predicted (point F) due to longitudinal tensile (61·5%) and transverse compressive (38·5%) failure of the 0° unidirectional laminate). In general, for longitudinal tensile and transverse compressive loading, a failure envelope represented by segment FG is obtained. In this region, as the ratio x /y decreases, the longitudinal tensile contribution reduces while transverse compressive contribution increases. At approximately x = –20y, the contributions from

372

Fig. 12. Biaxial failure stresses for [ + 55°/–55°]s laminate.

longitudinal and transverse failure become equal. When both the longitudinal and transverse loadings are compressive, the failure is dominated by transverse compression (segment GH). For the segment HA, failure is dominated by longitudinal compression. For compressive longitudinal and tensile transverse loading paths, the failure of the 0° unidirectional laminate is predominantly due to a large contribution of transverse tensile failure and a small amount of longitudinal compressive failure (segment BC). Let us now look at a [ + 55°/–55°]s angle ply laminate subjected to longitudinal and transverse loading. The total thickness of the laminate is 1 mm with all the plies having the same thickness. The initial and final failure envelopes coincide as shown in Fig. 12. The laminate is observed to exhibit the highest strength in the vicinity of point I, which corresponds to the stress path x /y = –1/–2. For a longitudinal tensile loading, a low failure stress of 65 MPa (point B) is predicted due to transverse tensile failure in the laminae. If the loading is longitudinally compressive, a much higher ultimate stress of 156 MPa (point G) is predicted. Failure is the result of transverse compressive (54%) and shear (46%) failure of the ± 55° plies. A high laminate strength of y = 253 MPa (point F) is observed for transverse tensile loading of the laminate. This failure is primarily by shear (74·8%) failure of ± 55° plies. Longitudinal tensile (6·4%) and transverse compressive (18·8%) stresses also

373 contributed to failure. A compressive transverse loading results in a much lower laminate strength of y = –188 MPa (point A) due to the longitudinal tensile (6·4%), transverse compressive (62·5%) and shear (31·1%) failure of the ± 55° plies. For longitudinal compressive and transverse tensile loading of the laminate, it is observed that as the ratio | x /y | increases, the longitudinal tensile and shear contributions reduce, while transverse compressive failure contribution increases. This is illustrated by curve FG. As one moves from point G to point H, the contributions of longitudinal and transverse compression increase while the shear failure contribution reduces rapidly. For the curve HI, the ply transverse compressive failure strain is exceeded, causing failure of the laminate. As one moves from point I to point A, the transverse compressive and shear contributions increase while the longitudinal compressive contribution reduces, becomes zero and then becomes longitudinal tensile failure (point A). The segment AB represents the failure envelope for tensile longitudinal and compressive transverse loading of the laminate. Along this segment as the ratio x /y increases, the shear failure contributions decrease while the transverse failure contribution increases. For tensile longitudinal and transverse loading, the failure in segment BJ is governed by transverse tensile failure. From point J to point C, an increase in the shear failure contribution is observed. From point C to point D, the shear failure contribution reduces while transverse failure contribution increases. Moving from point D to point E, the shear failure contribution increases rapidly while the transverse failure contribution reduces rapidly. In the segment EF, as the ratio x /y decreases, the shear failure contribution reduces slightly while the transverse failure contribution increases slightly. The stress/strain response for the [ + 55°/–55°]s laminate subjected to transverse tensile loading is shown in Fig. 13. As explained earlier, the laminate failure occurs at y = 253 MPa due to longitudinal tensile (6·4%), transverse compressive (18·8%) and shear (74·8%) failure. The stress/strain laminate response is non-linear and the first ply failure leads to failure of the

Fig. 13. Stress/strain curves for [ + 55°/–55°]s laminate (y /x = 1/0).

374

Fig. 14. Stress/strain curves for [ + 55°/–55°]s laminate (y /x = 2/1).

laminate. Figure 14 shows the laminate stress/strain response for y /x = 2/1. The stress/strain curve is linear up to failure but a much lower failure stress of y = 111 MPa is predicted. This occurs due to the transverse tensile failure of the ± 55° plies. Consider a 1 mm thick [ + 45°/–45°]s E-Glass/MY750 laminate, with all plies having the same thickness. For equal longitudinal and transverse tensile loading, a laminate strength of y = x = 68·2 MPa is predicted. The stress/strain response is shown in Fig. 15. It is observed that the stress/strain behavior is linear up to failure, and the initial ply failure which also corresponds to the ultimate laminate failure is due to the transverse tensile failure of the plies. If the loading path is chosen such that we apply an equal magnitude longitudinal tensile and transverse compressive biaxial loading (y /x = 1/–1), then the non-linear stress/strain response shown in Fig. 16 is predicted. Since the [ + 45°/– 45°]s laminate is subjected to pure shear loading, it would be expected that the failure would occur at the shear strength of the material (73 MPa). This is observed in Fig. 16. The plies fail due to shear failure at 72·9 MPa and the calculated laminate failure strains are half that of the ply shear failure strains. A [0°/90°]s laminate with total thickness of 1·04 mm (all plies of same thickness), was studied under longitudinal tensile loading. The stress/strain response predicted is shown in Fig. 17. The stress/strain behavior is essentially linear except at ply failure locations where a sudden increase in laminate strains is observed due to the unloading of the failed plies. Initial

375

Fig. 15. Stress/strain curves for [ + 45°/– 45°]s laminate (y /x = 1/1).

Fig. 16. Stress/strain curves for [ + 45°/– 45°]s laminate (y /x = 1/–1).

376 ply failure at 76·5 MPa is caused by the transverse tensile strain in the 90° plies exceeding the maximum allowable transverse tensile failure strain of 0·246%. The 0° plies are under transverse compressive strain but the non-slip condition at the 0·90° ply interface causes a net transverse tensile stress to occur in the 0° plies. The final failure of the laminate occurs at 293 MPa due to the transverse failure of the 0° plies. 6. Conclusions In the exercise described in the present study, the stress/strain responses and failure stresses of eight composite specimens made from four different materials were predicted. Response to the loads imposed was seen to be a function of the composite material, ply configuration and stress path. In each case studied, the relative contribution of the longitudinal, transverse and shear strain energies to the overall strain energy was determined incrementally for the applied load. Using the incremental approach proposed by Sandhu allowed for the updating of material properties as necessary to achieve convergence of ply stress and strain. In some cases assumptions were made that might not be necessary when a more complete experimental data set becomes available. One of the pieces of information that should prove to be of particular use is the Poisson’s effect as a function of strain. The model incorporates strain dependent relationships between longitudinal and transverse strains which, depending on the predicted failure mechanism, may measurably affect the computed ultimate strengths. The relative contributions of the longitudinal, transverse and shear strain energy to the overall value are determined by three shape factors (m1, m2 and m6) which are material properties. In

Fig. 17. Stress/strain curves for [0°/90°]s laminate (y /x = 0/1).

377 the analysis presented, the shape factors were all taken to be equal to one. Experimental data that will allow us to improve on this simplifying assumption will be available in the next phase of the study. We also assumed that, upon failure of an individual lamina, all load being carried by that lamina would be immediately redistributed to the remaining unfailed laminae. The failure model allows for two additional ply unloading mechanisms. Our studies have shown that these other assumptions of load transfer lead to different ultimate failure load predictions. Additional work needs to be done to identify the most appropriate load transfer mechanisms. Further, it has been assumed that no strain components can exceed the ultimate strain components measured during simple loading conditions. This assumption needs to be investigated. An energy-based failure criterion suitable for describing the behavior of laminated composites has been described. This method evaluates the non-linear response to the applied load on each ply in the subject laminate. By combining the computed longitudinal, transverse and shear strain energies and then comparing the calculated values with those obtained from measurements made in laboratory tests on samples of simple configuration, the analyst is able to predict the onset and progression of failure in laminates of arbitrary thickness and ply configuration. Acknowledgements The assistance provided by Drs Raghbir S. Sandhu and Ranbir S. Sandhu is gratefully acknowledged. Assistance received from Aerospace Structures Information and Analysis Center, Wright-Patterson Air Force Base, Dayton, Ohio is appreciated. Dr Raghbir S. Sandhu made available upgraded versions of his programs and provided the authors with advice and guidance at numerous points in the implementation of the model and the preparation of this paper. Financial support from The Ohio State University for T. S. Butalia in the form of a Presidential Fellowship is greatly appreciated. The authors also express their gratitude to Ms Sharon Morland of The Ohio State University for her help in preparation of the manuscript. References 1 Franklin, H. G., Classical theories of failure of anisotropic materials. Fiber Sci. Technol., 1968, 1, 137–150. 2 Sandhu, R. S., A survey of failure theories of isotropic and anisotropic materials, AFFDL-TR72–71, AD 756889, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, OH, 1972. 3 Sendeckyj, G. P., A brief survey of empirical multiaxial strength criteria for composites. In ASTM STP 497, 1972, pp. 41–51. 4 Wu, E. M., Phenomenological anisotropic failure criteria. In Treatise on Composite Materials, ed. L. J. Broutman, R. H. Kroch and G. P. Sendeckyj. Academic Press, 1973, pp. 353–432. 5 Wu, E. M., Strength and fracture of composites. In Composite Materials, Vol. 5: Fatigue and Fracture, ed. L. J. Broutman. Academic Press, 1974, pp. 191–248. 6 Vicario, A. A. and Toland, R. H., Failure criteria and failure analysis of composite structural components. In Composite Materials, Vol. 7: Structural Design and Analysis, Part I, ed. C. C. Chamis. Academic Press, 1975, pp. 52–97. 7 Rowlands, R. E., Strength (failure) theories and their experimental correlation. In Handbook of Composites, Vol. 3: Failure Mechanics of Composites, ed. G. C. Sih and A. M. Skudra. NorthHolland, 1984, pp. 71–125.

378 8 Hart-Smith, L. J., The role of biaxial stresses in discriminating between meaningful and illusory composite failure theories. Compos. Struct., 1993, 25, 3–20. 9 Jenkins, C. F., Materials of construction used in aircraft and aircraft engines. Report to Great Britain Aeronautical Research Committee, 1920. 10 Azzi, V. D. and Tsai, S. W., Anisotropic strength of composites. Exp. Mech., 1965, 5, 283–288. 11 Fisher, L., Optimization of orthotropic laminates. Engineering for Industry, 1967, 89, 399–402. 12 Chamis, C. C., Failure criteria for filamentary composites. In ASTM STP 460, 1969, pp. 336–351. 13 Marin, J., Theories of strength for combined stress on non-isotropic materials. J. Aero. Soc., 1957, 24, 265–268. 14 Malmeister, A. K., Geometry of theories of strength. Polym. Mech., 1966, 2, 324–331. 15 Hoffman, O., The brittle strength of orthotropic materials. J. Compos. Mater., 1967, 1, 200–206. 16 Tsai, S. W. and Wu, E. M., A general theory of strength for anisotropic materials. J. Compos. Mater., 1971, 24, 58–80. 17 Cowin, S. C., On the strength anisotropy of bone and wood. Trans. ASME J. Appl. Mech., 1979, 46, 832–838. 18 Huang, C. L. D., Strength coefficients of the cubic criterion for graphite composites. J. Engng Mater. Technol., 1990, 112, 227–230. 19 Golfman, Y., Strength criteria for anisotropic materials. J. Reinf. Plast. Comp., 1991, 10, 542–556. 20 Sandhu, R. S., Ultimate strength analysis of symmetric laminates, AFFDL-TR-73-137, AD 779927, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, OH, 1974. 21 Sandhu, R. S., Nonlinear behavior of unidirectional and angle ply laminates. J. Aircraft, 1976, 13, 104–111. 22 Sandhu, R. S., Monfort, J. B., Hussong, F. E. et al. Laminate tubular specimens subjected to biaxial stress states (glass/epoxy), AFFDL-TR-73-7, Vol. I, Air Force Flight Dynamics Laboratory, WrightPatterson Air Force Base, OH, 1973. 23 Sandhu, R. S., Monfort, J. B., Hussong, F. E. et al. Laminate tubular specimens subjected to biaxial stress states (graphite/epoxy), AFFDL-TR-73-7, Vol. II, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, OH, 1975. 24 Petit, P. H. and Waddoups, M. E., A method of predicting the nonlinear behavior of laminates composites. J. Compos. Mater., 1969, 3, 2–19. 25 Hahn, H. T. and Tsai, S. W., Nonlinear elastic behavior of unidirectional composite laminae. J. Compos. Mater., 1973, 7, 102–108. 26 Hahn, H. T., Nonlinear behavior of laminates composites. J. Compos. Mater., 1973, 7, 257–271. 27 Hashin, Z., Bagchi, D. and Rosen, B. W., Nonlinear behavior of fiber composite laminates, NASA CR-2313, 1973. 28 Walsh, J. L., Ahlberg, J. H. and Nilson, E. N., Best approximation properties of the spline fit. J. Math. Mech., 1962, 2, 225–233. 29 Ahlberg, J. H., Nilson, E. N. and Walsh, J. L. Theory of Splines and Their Applications. Academic Press, New York, 1967. 30 Sandhu, R. S., Computer program NOLAST for nonlinear analysis of composite laminates, AFFDL-TR-76-1, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, OH, 1976. 31 Sandhu, R. S., Analytical-experimental correlation of the behavior of 0°, ± 45°, 90° family of AS/ 3501–5 graphite-epoxy composite laminates under uniaxial tensile loading, AFFDL-TR-79-3064, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, OH, 1979. 32 Hinton, M. J. and Soden, P. D., Predicting failure in composite laminates: the background to the exercise. Compos. Sci. Technol., 1998, 58(7), 1001. 33 Soden, P. D., Hinton, M. J. and Kaddour, A. S., Lamina properties, lay-up configurations and loading conditions for a range of fibre reinforced composite laminates. Compos. Sci. Technol., 1998, 58(7), 1011.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

379

CHAPTER 3.12

The strength of multilayered composites under a plane-stress state Peter A. Zinoviev,a* Sergei V. Grigoriev,a Olga V. Lebedevab and Ludmilla P. Tairovaa a

Bauman’s Moscow State Technical University, 2nd Baumanskaya Street 5, Moscow, 107005 Russia lnstitute of Composite Technologies, Orevo, Dmitrovskii raion, Moskovskaya obl., 141814 Russia

b

Abstract The paper presents a model that describes the deformation and failure processes of multilayered hybrid composites in a state of plane stress. The model, which can be considered as a structural– phenomenological one, is a coupled deformation/failure model (DFM). According to the model, except for two common states of the composite laminate – the initial state (monolithic material) and the final state (completely broken material) – there exists a group of intermediate states, namely the material with cracks. Within the group, we can distinguish two further sub-groups of material states: (1) with open cracks and (2) with closed cracks. The total number of states equals eight (four in the ‘open-crack’ group). The algorithm of successive stress loading is a natural one for the model. The low shear modulus of modern polymeric composites often results in significant changes in the initial reinforcing angles of the plies as a consequence of shear strains, which is why the model considers the effects of the ‘structural non-linearity’ of the laminate. The algorithm is capable of predicting both stress/strain curves and failure envelopes for multilayered composites under a variety of loading conditions. Theoretical predictions are presented for a number of cases provided by the organisers of the failure exercise Keywords: composite laminates, failure, theoretical model, stress/strain curves, failure envelopes, predicted strength

Nomenclature Ex, Ey E1, E2, G12 ˜ 12 E˜ 2, G F + 1, F–1 F + 2, F–2

effective moduli of elasticity (Young’s moduli) of the laminate in the x and y axes longitudinal, transverse and shear moduli of the ply unloading transverse and shear moduli of the ply ultimate longitudinal tensile and compressive stresses in the ply ultimate transverse tensile and compressive stresses in the ply

* To whom correspondence should be addressed.

380 F12 [G] [G 0] H h¯ [T1] v12 x, y 1, 2

ultimate shear stress in the ply stiffness matrix of the laminate stiffness matrix of the ply thickness of the laminate relative thickness of the ply transformation matrix major Poisson’s ratio of the ply coordinate axes of a composite laminate coordinate axes of a unidirectional ply

Greek symbols x, y, xy average strains in the laminate along the x, y axes and under shear 1, 2, 12 longitudinal, transverse and shear strains in the unidirectional ply ¯ 2 modified transverse strain in the ply x, y, xy average stresses in the laminate along the x, y axes and under shear 1, 2, 12 longitudinal, transverse and shear stresses in the unidirectional ply *2, *12 largest algebraic values of transverse and shear stresses in the ply during its deformation

1. Introduction The strength of laminated composites has been the object of much attention of scientists for a considerable time. A part of the literature familiar to the authors, which is directly related to the subject of the present paper, is given in the References. This is a central problem, interesting to both academic scientists and practical engineers. Scientists felt, time and again, that the problem had been completely solved. We recall, for example, a tensor-polynomial criterion proposed by Goldenblat and Kopnov,1 which has received wide recognition, followed by the work by Tsai and Wu.2 There is no need now to analyze the flaws in this very significant step in describing the strength of anisotropic solids; however, satisfaction with efforts to describe strength phenomenologically with tensor polynomials of high order has long evaporated, and the problem remains unsolved. However, enthusiasts still hope to find a universal approach. Failure processes in real composite are of a great variety, which is why papers describing the strengths of particular types of composite are of special interest. The present paper deals with a model for describing the deformation and failure processes of multilayered hybrid composites under a plane stress state. The model was first proposed by Zinoviev and Tarakanov:3 it was then carefully checked experimentally,4 and has subsequently been further improved by the authors.5–7

2. What should a model describe? The behaviour of modern composites under a variety of loading conditions has been much investigated. Some of work is listed below.8–14 In our opinion, the available results (i.e.

381 primarily experimental results) can be considered as being neither exhaustive nor sufficient for an adequate theoretical description. Nevertheless, the main features of composites under load are well determined. While not claiming exhaustiveness, we list some of these features. Beginning from a very low load level, irreversible changes arise and grow in modern composites: cracks appear in the matrix, imperfections in the fibre/matrix interface grow, individual fibres break, etc. These processes significantly govern not only the nature of the deformation process in the composite, but they are also responsible for the process commonly termed final failure. It is evident that conventional theories which consider two completely independent successive processes, deformation and failure, (e.g. the linear theory of the elastic behaviour of composites plus a tensor-polynomial criterion of initial/final failure) do not have serious hope of success. Coupled deformation/failure models (DFMs) seem to offer better proposals. Figure 1 schematically shows a typical stress/strain curve for an angle-ply composite under loading, unloading and repeated loading. The figures on the curve correspond to the following stages: 1. 2. 3. 4. 5. 6. 7. 8.

stiffness properties at the first ‘linear’ part of the curve conditions for appearance of ‘the first-ply failure’ – the ‘knee’ of the curve stiffness characteristics of the material following the first stage behaviour under unloading residual strains behaviour of the composite under repeated loading behaviour of the material following the point in excess of the beginning of unloading conditions of final failure.

In the general case, a theoretical model must describe the stages mentioned under a complex state of stress, bending, complex loading, combination of force and thermal factors. Unfortunately, this is not a complete list of factors, which makes the problem more difficult.

Fig. 1. Stress/strain curve of an angle-ply composite.

382

Fig. 2. Stress/strain curve of ± 60° angle-ply glass fiber reinforced plastic under biaxial tension, ‘unstable deformation’: experimentai (solid line) and predicted (broken line) data.

We note several more features. Many modern composites consist of rigid, high-strength fibres bonded with a ductile polymeric matrix. When loading the composite in directions other than the fibre direction, significant changes may arise in the fibre orientation. This is a manifestation of the effects of geometrical non-linearity at the level of the composite internal structure which we term ‘structural non-linearity’. Increase of the composite tangent stiffness in the final stages of deformation (Fig. 1, segment 7) is an example of such structural nonlinearity, which is not typical for common materials. Another effect related to the above is the unstable deformation of composites. This is an uncommon mode of composite failure, when strains increase without increasing loads. Figure 2 illustrates experimental stress/strain curves for an angle-ply glass-fibre-reinforced plastic ( ± 60°) under uniform biaxial loading x /y = 1/1.5 The term ‘failure’, as such, should be expanded as applied to these materials. A model described below is the simplest model which gives a means of describing the phenomena mentioned above. The model can be considered as a structural–phenomenological one. The base element of the model is an individual unidirectional ply.

3. The behaviour of an isolated unidirectional ply: The strength criterion The behaviour of the unidirectional lamina is considered to be linearly elastic up to failure. The strains 1, 2, 12 can be determined if one knows the stresses 1, 2, 12 applied to the ply and the stiffness characteristics of the ply, E1, E2, G12, v12. Directions 1 and 2 are for the axes of the Cartesian co-ordinates in the ply plane, axis 1 being directed along the fibres (Fig. Al in Appendix). It has been noted in some papers that stress/strain curves of the unidirectional material (particularly under shear) could not be completely linear. The real stress/strain curves can be taken into account in computation algorithms, but, as correctly pointed out by the authors of Ref. 15, in many cases the results of considering the non-linear effects do not justify the effort expended. Within the model, the ply remains elastic when the following conditions are fulfilled F–1 ≤ 1 ≤ F + 1, F–2 ≤ 2 ≤ F + 2, | 12 | ≤ F12

(1)

383 Here F + 1, F + 2 are ultimate tensile stresses along and transverse to the fibres, respectively: F–1, F–2 are the equivalent characteristics in compression; and F12 is the ultimate in-plane shear stress. The conditions in eqn (1) determine a failure surface in the shape of a rectangular parallelepiped in the coordinates 1, 2, 12. When the stress in an isolated ply reaches any of the mentioned ultimate values, the ply fails. Conditions (1), in addition to their obvious simplicity, have a number of appreciable advantages in comparison with other strength criteria, namely: • the accuracy of approximation of known experimental data (even in the worst case) is not inferior to that obtained with other criteria • criterion (1) clearly shows the causes of ply failure. This is convenient for developing the models of ply behaviour after the first failure • many studies have shown that a variety of essentially different failure mechanisms can take place within the ply, e.g. matrix cracking, break-up of the fibre/matrix interface, rupture of the fibres, buckling of the fibres, etc. During the loading process, abrupt changes in the modes of failure always take place. This is why a smooth strength surface seems to be less physically convincing than the simple rectangular parallelepiped each side of which corresponds to a single failure mechanism. It is well known that modern unidirectional composites are highly anisotropic materials, e.g. the longitudinal tensile strength of unidirectional composites exceeds the transverse tensile strength of the material by approximately two orders of magnitude. The experimental failure surface of such material is a highly elongated surface of a ‘thin pencil’ type, and it is probably not very constructive to debate the form of the edges of this long, thin pencil. In our opinion, it is of prime importance that the strength criteria can adequately describe different modes of failure in composites rather than refine the form of the elongated failure surface. The ‘simple’ criterion of maximum stresses copes with the task.

4. The behaviour of individual unidirectional ply within the laminated composite We assume that the unidirectional ply within the composite laminate deforms as shown in Fig. 3. Deformation of the ply (both under tension and compression) along the fibre direction is completely elastic (Fig. 3(a)). When longitudinal stresses reach their ultimate values, F + 1 or F–1, the ply is assumed to be broken. The behaviour of the ply in the transverse direction is much more complicated (Fig. 3(b)). The ply is monolithic and elastic under tension in the transverse direction within the first stage (the 0–1 segment). The process of cracking the matrix begins at point 1 and progresses within the 1–2 segment of the diagram. The isolated unidirectional ply fails at point 1. Unloading from any point of the segment 1–2 occurs with the unloading modulus E˜ 2 which equals the secant modulus of the diagram. Residual strains are zero, suggesting that the cracks are completely closed. This is why on further compression of the ply (segment 3–4 of the diagram), the modulus of the ply completely regains its initial value. Repeated deformation of the unidirectional ply under transverse tension (positive 2 values) follows along the 3–2 segment and further along the 2–2 segment of the diagram. The deformation diagram in

384

Fig. 3. Behaviour of a unidirectional ply within a multilayered composite (a model).

Fig. 3(b) is plotted as a function of the modified strain ¯ 2 = 2 + v121 (Appendix Al). The second equation from eqn (A1) with consideration of eqn (A2) can be written as

2 =

E1 1  v12v21

冉

v211 +

冊

v12 E2 2 = ¯ 2 v21 1  v12v21

The last equation is Hooke’s law written in complex or reduced form. Suppose that 2 = 0 in the last equation. In this case it is always possible to increase 2 at the expense of increasing 1 up to any 2 value including F + 2 when the cracks appear. Thus, the modified strain, ¯ 2, is the special strain which enables us to consider the effect of deformation in the fibre direction on the process of matrix cracking. It follows from Fig. 3(b) that the unloading modulus, E˜ 2, is expressed as follows:

冋 册

¯ *2 v212 E˜ 2 = + *2 E1

–1

=

*2 ¯ *2 + v212*2 /E1

(2)

where the starred values are the largest algebraic values during the history of deformation. If the stress 2 reaches its ultimate value F–2 (which is independent on the history of previous loading), the ply is then considered to be broken. The ply behaviour under shear is, in many ways, similar to that under deformation in the direction transverse to the fibre direction (Fig. 3(c)). The stress/strain curve of the ply is linear elastic within the 0–1 segment. Segment 1–2 corresponds to the stage of matrix cracking. ˜ 12 = 12 / *12. Unloading process (segment 2–3) takes place with the unloading shear modulus G The process of shear deformation does not depend on the sign of the stress 12, which is why ˜ 12. ply deformation within the 3–4 segment also occurs with the unloading modulus G

385

Fig. 4. Stresses vs average strain, x, in ± 42° angle-ply glassfiber-reinforced plastic (x) and constitutive unidirectional plies (1, 2 and 12).

Repeated deformation of the ply under positive 12 values follows along the 4–3–2 segment and further along the 2–2 segment where the process of matrix cracking resumes. It is worth mentioning that the conditions of ply unloading in one component of stresses need not be related to unloading of the whole laminate. The change in the ratio between ply stiffnesses (along the fibres, transverse to the fibres, and in the ply plane) may result in unloading in individual stress components even if the mean stresses on the laminate increase. Figure 4 illustrates the plots of the changes in ply stresses 1, 2, 12 under uniaxial tension of an angle-ply glass-fibre-reinforced plastic (GRP) with  = ± 42°. The plots were predicted with the theoretical model described below and the following initial data: E1 = 46·5 GPa, E2 = 7 GPa, G12 = 7 GPa, v12 = 0·25, F + 1 = 1600 MPa, F–1 = 500 MPa, F + 2 = 40 MPa, F–2 = 200 MPa, F12 = 60 MPa. Here, at x 0·625%, shear stresses reach their ultimate value, F12, and cracks appear in the matrix. This is followed by unloading of the plies in the transverse direction (¯2 < 0). So, the models of composites for describing the processes of deformation under increasing loads should take into consideration the possibilities of ply unloading in one or more components of the stress state. The processes of matrix cracking under shear and transverse tension are interrelated. Assume that the cracks, whatever the cause of their appearance, simultaneously affect ply behaviour under both shear and transverse tension. The stresses 2 and *12 correspond to the beginning of matrix cracking in the ply. In the process, two variants are possible: | *12 | = F12, *2 ≤ F + 2

(3)

or *2 = F + 2, | *12 | ≤ F12 Thus, according to the ply model, except for two common ply states – the initial state (monolithic material) and the final one (completely broken material) – there exists a group of intermediate ply states: the material with cracks. Within that group, we can distinguish

386 between two further sub-groups of material state: (1) with open cracks and (2) with closed cracks. The sign of 2 governs the sub-groups: if 2 is positive, the cracks are open, and if 2 is negative (compression), the cracks are closed. Within the ‘open cracks’ sub-group, the model distinguishes four ply states differing in the values of strains ¯ 2 and | 12 | as well as the signs of their increments ¯2 and  | 12 |. Take the diagram in Fig. 3(b); here segment 1–2–2 corresponds to active (or ‘pioneer’) deformation followed by crack formation (in the case, ¯ 2 < ¯ *2 and ¯2 > 0). In contrast, ply deformation within segment 2–3 (unloading and repeated loading), ¯ 2 < ¯ *2, is not followed by new irreversible changes in the ply. Analogous segments can easily be seen in the shear diagram (Fig. 3(c)). The total number of possible deformation stages of the ply under shear and transverse tension equals four in the open-crack sub-group. If the ply is under transverse compression (2 < 0) and the cracks are closed, then, depending on the stage of shear deformation, the model distinguishes two more ply states. The first corresponds to active crack formation under shear ( | 12 | = | *12 |,  | 12 | > 0), and the second to deformation within the ‘developed’ strain range ( | 12 | < | *12 | ). The column matrix of the parameters of the effective ply stiffness {1, 2, 12} is defined as follows: 1 =

E1 , E 01

2 =

E2 , E 02

12 =

G12 G 012

(4)

Here E 01, E 02, G 012 are initial, and E1, E2, G12 are current values of tangent moduli. Depending on the sign of the stress, 2, the values of the strains, ¯ 2 and 12, and the sign of their increments, the column matrix for the cracked ply has one out of six possible values given in Table 1. At the beginning of deformation (the ply is monolithic) the column matrix is {1, 2, 12} = {1, 1, 1}, while at the stage of the completely broken ply (the conditions

Table 1 Parameters of the effective stiffness of the cracked ply. Parameters of the strain state State of the ply

Open cracks 2 > 0

In the transverse direction

Under shear

¯ 2 < ¯ *2

| 12 | < | *12 |

¯ 2 /¯*2 ¯2 > 0 ¯ 2 < ¯ *2

Closed cracks 2 < 0

˜ 2 < 0

1

2

12

1

E˜ 2 /E 02

˜ 12 /G 012 G

1

0

˜ 12 /G 012 G

1

0

0

1

E˜ 2 /E 02

0

1

1

0

1

1

˜ 12 /G 012 G

| 12 | = | *12 |  | 12 | 0} | 12 | < | *12 |

387 1 = F + 1, or 1 = F–1, or 2 = F–2 are fulfilled) all its components are zero: {1, 2, 3} = {0, 0, 0}. Thus the total number of ply states equals eight. In the calculations, it is convenient to take non-zero but close values instead of zero values for the coefficients 2 and 12 in Table 1 (for example, 10–3). It is assumed that the Poisson ratio, v12, remains constant during deformation and that the following relationship is true for tangent moduli of elasticity: E1v21 = E2v12. In addition, by assuming that the 1 parameter equals unity for all ply states up to complete failure, one has {12} = [G 0]{12}

(5a)

or 1 1 2 = 1  v012v021 12

E 011 v021E 012 0

v012E 022 E 022 0

0 0 (1  v012v021)G 01212

1 2  12

(5b)

The components of the stiffness matrix in eqn (5b) depend upon the current stress/strain state of the ply and the history of its deformation (Table 1). Note that the structure interpretation of ply states given above (e.g. the ideas of open and closed cracks) is very schematic and simplified. One need not resort to it but consider the approach described as a phenomenological model of deformation processes in unidirectional plies within the multilayered composite material.

5. Calculation algorithm The model described in Section 3 was applied for the analysis of deformation and failure processes in multilayered composites including several plies of different orientation. Stress/ strain relationships for the unidirectional ply and the laminated composite are well-known (e.g. Ref. 16) and given in the Appendix. Assume that the plies are ideally bonded at all stages of the deformation process, i.e. strains of the plies in the global co-ordinate system (co-ordinate system of the laminate) are the same and equal to mean strains of the laminate (Fig. A1 in Appendix). The algorithm of successive stress loading is the natural one for the model. Suppose that at the nth step of loading mean stresses in the laminate increase by the step {xy}n = {x, y, xy}. We define the corresponding increments of mean strains in the laminate: {xy}n = [G ]–1 n  1{xy}n

(6)

where the stiffness matrix of the laminate [G ] was calculated at the previous step from eqn (A11) in Appendix. Following the nth loading step, complete mean strains in the laminate are: {xy}n = {xy}n  1 + {xy}n. Determine corresponding ply strains in the co-ordinate system related to (i)T (i) the ply (1(i), 2(i)). For the ith ply one obtains: {12}(i) n = [T1]n  1{xy}n and {12}n = (i) (i) {12}n  {12}n  1. Here the transformation matrix [T1](i) n  1 was found at the previous step from eqn (A6). Calculate the increments of stresses and full stresses in the plies: 0 (i) (i) {12}(i) n = [G ]n  1{12}n ,

(i) {12}(i) = {12}(i) n  1 + {12}n

388 Now one knows all parameters of stress/strain states of the plies, the parameters are necessary to define the matrix of the effective stiffness for each ply according to the model (Table 1): (i) (i) (i) {}(i) n = f ({12}n , {12}n , {12}n ). The parameters of the effective stiffness of the plies are used to form stiffness matrices [G 0](i) n according to eqn (5b). With stiffness matrices of the plies [G 0](i) n we can calculate refined values of stress 0 (i) (i) increments in the plies (these have superscript i): {12}(i) nI = [G ]n {12}n and full stresses in the plies: (i) (i) {12}(i) nI = {12}n  1 + {12}nI .

The low shear modulus of modern polymeric composites often results in significant changes in the initial reinforcing angles of the plies as a consequence of shear strains. As mentioned above, this type of non-linearity related to the changes in geometrical parameters of the laminate is termed a structural non-linearity. The structural non-linearity can be taken into (i) (i) account by correcting the angles of the plies: (i) n = n  1   12n /2. (i) New values of the reinforcing angles, n , are used in calculation of the transformation (i) matrix [T1](i) n  1 (eqn (A6)), and matrix [T1]n  1, in its turn, is used in calculation of the stiffness matrix of the laminate, [G ]n, for the nth step of loading (eqns (A1.1) and (A7)):

冘 M

[G ]n =

0 (i) (i)T ¯ (i) [T1](i) n [G ]n [T1]n h

i=1

We must now check whether the matrix [G ]n, is positively determined. To do this, we need only be sure that g11 ≥ 0, g22 > 0, g66 > 0. We then move to the next step of loading. Loading (stress) steps continue to increase until the laminate is broken. The multilayered composite is considered to be broken if only for a single ply the following condition is fulfilled: (i) (i) 1 = F + 1,

or

(i) (i) 1 = F –1,

or

(i) (i) 2 = F –2

Experiments5 show that this suggestion is well justified at least for composite laminates including a moderate number of the plies with different reinforcing angles (up to 4–5 plies). In the analysis, it is important that some constraints are imposed regarding the laminate deformability. The laminated material is believed to be broken if: • linear strains in a single ply, 1 and 2, exceed their ultimate values • linear strains in the laminate, x and y , exceed their ultimate values • shear strains in any ply, 12, or the laminate, xy , exceed their ultimate values. Ultimate strain values are selected on the basis of experimental studies. Ultimate strains determined from the tests in uniaxial tension of ± 45° laminates (this is the most ‘yielding’ structure) can serve as initial approximation for x and y. Modern composite laminates of ± 45° structure usually show ultimate strains under uniaxial tension around 10–15%. These values can be used as initial estimates for x and y . Note that one should well understand final objectives of the theoretical prediction and have some experience in predicting the behaviour of composite laminates. For example, a designer of a composite construction restricts its strains with assigned values of x , y and xy . These values will then be taken as the ultimate ones, and theoretical calculations will determine the

389

Fig. 5. Ultimate axial stresses, x, vs reinforcing angle, °, for angle-ply glass-fiber-reinforced plastics under biaxial tension. Predicted data are shown by solid lines (final failure) and broken line (initial failure); experimental data belong to K. Kawata (䊏)11 and M. Uemura (䊊).12

conditions (the state of stresses) when the constraints mentioned are fulfilled. In case one solves the problem of the detailed non-linear analysis of the laminate behaviour, the variation in the values of ultimate strains enables one to realise the most successive investigation. The algorithm described above can be used for predicting both stress/strain curves and failure envelopes for multilayered composites under a variety of loading conditions. Figures 5 and 6 give the examples of predicted results obtained with the model described.

Fig. 6. Failure envelopes of quasi-isotropic glass fiber reinforced plastic ( ± 30°/90°).16 Experimental data are shown by the circles (average values), predicted data are shown by solid lines (final failure of the laminate) and broken lines (appearance of the cracks in the plies); the causes of initials and final failure are given.

390 The model can be easily expanded to accommodate both mechanical and thermal loads.17 Thermal loads may arise, for example, on cooling the laminates from the curing temperature to the operating temperature, or from a standard temperature to cryogenic temperatures. Initial stresses arising on cooling can be taken into account with the help of the present model.18 A computer program was developed on the basis of the model described above. This is the STRAN software.19 STRAN enables one to perform the detailed analysis of the deformation and failure processes of composite laminates. The program can predict and draw stress/strain curves for the laminate under uni-axial and bi-axial loading as well as the failure envelopes. In the process, STRAN indicates the causes of material cracking and final failure. One may restrict his analysis to the calculation of effective characteristics of the laminate (moduli of elasticity, shear modulus, ultimate stresses). STRAN uses both stress and strain loading of the laminates. 6. Theoretical results The theory described above was applied to predict the behaviour of the laminates supplied by the organisers of the ‘Failure Prediction of Composite Laminates’ exercise.20,21 In the analysis, the lamina properties of the four materials (T300/BSL914C, AS4/3501-6, E-glass/LY556 epoxy and E-glass/MY750 epoxy) were taken from the data provided. The non-linear stress/ strain curves of unidirectional laminae provided by the organisers were not used. Four elastic constants (E1, E2, G12, and v12) and five strength values (F + 1 = XT , F–1 = XC , F + 2 = YT , F–2 = YC , and F12 = S12) were used as initial data for calculations. Figures 7–9 give the failure stress envelopes predicted for unidirectional laminae. Figure 7 shows the biaxial failure stress envelope for the E-glass/LY556 unidirectional lamina

Fig. 7. Biaxial failure stress envelope under transverse and shear loadmg (y  xy) for the E-glass/ LY556/HT907/DY063 unidirectional lamina.

391

Fig. 8. Biaxial failure stress envelope under longitudinal and shear loading (x  xy) for the T300/BSL 914C unidirectional lamina.

subjected to combined transverse and shear loading (y  xy). Three modes of failure are predicted in the lamina depending upon the state of stresses. They are transverse tension, inplane shear and transverse compression. Figure 8 illustrates the failure stress envelope for the T300/BSL 914C lamina under combined longitudinal and shear loading (x  xy). Again, three modes of failure are distinctly shown: longitudinal tension, longitudinal compression and in-plane shear. The failure envelope for the E-glass/MY750 lamina under direct biaxial tension (y ~ x) is given in Fig. 9. Here four modes of failure are predicted, namely longitudinal tension, longitudinal compression, transverse tension and transverse compression. All three failure

Fig. 9. Biaxial failure stress envelope under longitudinal and transverse tension (y  x ) for the E-glass/MY750/HY907/DY063 lamina.

392 envelopes for the unidirectional laminae are rectangular. The rectangular form of the failure surfaces is governed by the strength criterion used in the model described above, i.e. the criterion of maximum stresses (eqn (1)). Figures 10 and 11 show the failure envelopes of the 90°/ ± 30°/90° E-glass/LY556 epoxy laminate under combined direct stresses (y ~ x) and combined direct and shear stresses (x ~ xy), respectively. In the calculations, the laminate was loaded in 80 trajectories (paths) y /x = const and x /xy = const. Both initial and final failure envelopes are shown. Different modes of failure are distinctly seen in the figures. Under biaxial direct loading of the laminate (y ~ x) in the 1st, 2nd and 4th quadrants (Fig. 10), one observes initial failure due to transverse tension (F + 2) and in-plane shear (F12) followed by final failure due to longitudinal tension (F + 1), longitudinal compression (F–1) and transverse compression (F–2). For example, in case of loading the laminate along the trajectory y /x = 2/1·5, the laminate first cracks due to transverse tension in ± 30° plies, then the same cracks appear in 90° plies, and at last final laminate failure is caused by longitudinal tension of 90° plies. When loading along the trajectory y /x = –1/1·5, first cracks appear in 90° plies due to transverse tension, then the cracks appear in ± 30° plies due to shear, and final failure occurs due to longitudinal compression of 90° plies. As is seen,

Fig. 10. Biaxial failure stress envelope under longitudinal and transverse tension (y  x) for the E-glass/LY556/HT907/DY063 (90°/ ± 30°/90°) laminate. Ply orientation: 90°/–30°/ + 30°/ + 30°/ –30°/90°; ply thickness, h: h90 = 0·172 mm, h + 30 = 0·412 mm, h–30 = 0·412 mm; total laminate thickness H = 2 mm.

393

Fig. 11. Biaxial failure stress envelope under longitudinal tension and shear (x  xy) for the E-glass/ LY556/HT907/DY063 (90°/ ± 30°/90°) laminate. For ply orientation and ply thickness see Fig. 10.

corner points both on the initial and final failure envelopes correspond to the changes in the modes of failure. Analogous stages can be observed for the laminate under combined longitudinal and shear loading (Fig. 11). Take trajectory x /xy = –1/1. First cracks appear in ± 30° plies as a result of transverse tension and cracks successively appear in 90° and + 30° plies as a result of shear; final failure occurs by longitudinal compression of –30° plies. Trajectory by trajectory, one is able to examine initial and final failure stages of the laminate along with the stresses responsible for specific modes of failure. Figures 12–14 are for the 0°/ ± 45°/90° AS4/3501-6 quasi-isotropic laminate. Figure 12 illustrates the predicted failure envelope for the laminate under biaxial loading. In most of the compression–compression quadrant, the initial and final failure are identical whereas in the other three quadrants, failure took place in a number of stages. The modes of failure are marked in Fig. 12. Take the first quadrant, for example, y /x = 2/5; prior to final failure, one can observe intensive cracking of the laminate due to successive transverse tension in 0°, + 45° and 90° plies; final failure is caused by longitudinal tension of 90° plies. For the trajectory y /x = –2/5, the laminate cracks first by transverse tension in the 0° plies, then by in-plane shear in the ± 45° plies, and finally failure takes place as a result of longitudinal tension in the 90° plies. Figures 13 and 14 show predicted stress/strain curves under uniaxial tension (y /x = 1/0) and biaxial (y /x = 2/1) loading for the 0°/ ± 45°/90° AS4/3501-6 quasi-isotropic laminate. The curve in Fig. 13 demonstrates that initial deformation of the laminate is linearly elastic, until, at y = 241 MPa, the transverse tensile stresses in the 0° plies reach their ultimate values (F + 2). The first cracks appear in the laminate, and its effective Young’s modulus, Ey decreases slightly. Further laminate cracking occurs at y = 466 MPa when cracks appear in the ± 45° plies as a consequence of in-plane shear (12 = F12). The effective Young’s modulus of the laminate decreases once more, and final failure takes place at y = 728 MPa because of longitudinal tension in the 90° plies (1 = F + 1).

394

Fig. 12. Biaxial failure stress envelope under longitudinal and transverse tension (y  x) for the AS4/ 3501-6 (0°/ ± 45°/90°) laminate.

Fig. 13. Stress/strain curves under uniaxial tensile loading in y direction (y /x = 1/0) for the AS4/3501-6 (0°/ ± 45°/90°) laminate.

Fig. 14. Stress/strain curves for y /x = 2/1 for the AS4/3501-6 (0°/ ± 45°/90°) laminate.

395

Fig. 15. Biaxial failure stress envelope (y ~ x) for the E-glass/MY750/HY917/DY063 angle-ply ( ± 55°) laminate.

Fig. 16. Stress strain curves under uniaxial tensile loading in y direction (y /x = 1/0) for the E-glass/ MY750/HY917/DY063 angle-ply ( ± 55°) laminate.

Fig. 17. Stress/strain curves for y /x = 2/1 for the E-glass/MY750/HY917/DY063 angle-ply ( ± 55°) laminate.

396

Fig. 18. Stress strain curves under uniaxial tensile loading in x direction (y /x = 0/1) for the E-glass/ MY750/HY917/DY063 cross-ply (0°/90°) laminate. Ply orientation: 0°/90°/0°; ply thickness: h0 = 0·26 mm, h90 = 0·52 mm, total laminate thickness H = 1·04 mm.

Fig. 19. Stress/strain curves for y /x = 1/1 for the E-glass/MY750/HY917/DY063 angle-ply ( ± 45°) laminate.

397

Fig. 20. Stress/strain curves for y /x = 1/–1 for the E-glass/MY750/HY917/DY063 angle-ply ( ± 45°) laminate.

Initial deformation of the laminate under biaxial loading (y /x = 2/1; Fig. 14) is also linearly elastic. Successive laminate cracking in 0°, ± 45° and 90° plies (2 = F + 2) results in a slight decrease in the slope of the curve, and final laminate failure takes place at y = 825 MPa when longitudinal tensile stresses in the 90° plies reach their ultimate values (1 = F + 1). Figures 15–17 are for the ± 55° E-glass/MY750-epoxy laminate. Figure 15 shows the failure envelope for the laminate subjected to a variety of biaxial direct stresses (y ~ x). As before, one is able to identify the stages of initial and final failures and corresponding stresses in the plies. The modes of failure are marked on the envelope. The zone above FMA line is a zone of unstable deformation (not shown here). Therefore, the predicted envelope is not closed. Figure 16 represents the stress strain curve for the ± 55° angle-ply laminate under uniaxial tensile loading in the y direction (y /x = 1/0). The predicted curve has the well-marked knee after first-ply failure, i.e. the appearance of the cracks in the plies as a result of shear stresses (12 = F12). Following first-ply failure, the effective modulus of elasticity, Ey , decrease by approximately 1·5 times compared to its initial value. Final failure of the laminate takes place at y = 383 MPa and is governed by transverse compression in ± 55° plies. Figure 17 shows the stress/strain curves of the ± 55° angle-ply laminate for y /x = 2/1. The curve has a single knee (y = 112 MPa) caused by transverse tension of the plies. Final failure occurs at y = 880 MPa when the plies fail in longitudinal tension (1 = F + 1). The behaviour of 0°/90° cross ply laminate made of E-glass/MY750 epoxy composite under uniaxial tensile loading (y /x = 0/1) is predicted in Fig. 18. The theoretical stress/strain curve shows that the laminate fails in three stages. First fail the 90° plies (y = 77·8 MPa)

398 resulting in slight decrease of the effective modulus Ex. This is followed by failure of the 0° plies (x = 315 MPa) through transverse tension (2 = F + 2), and total failure of the laminate takes place at y = 660 MPa when 0° plies fail in longitudinal tension (1 = F + 1). Finally, stress/strain curves for the ± 45° angle ply E-glass/MY750 epoxy laminate under biaxial loading are given in Figs 19 and 20. Stress/strain curves for biaxial tension (y /x = 1/1) are shown in Fig. 19. The first-ply failure at x = 68·4 MPa caused by transverse tension (2 = F + 2) results in a decrease in the slope of the curve, and total failure of the laminate takes place as a consequence of longitudinal tension of the cracked plies (1 = F + 1). Figure 20 illustrates stress/strain curves for the case y /x = 1/–1. As is seen, the curves are linear up to x = –73·0 MPa when first-ply failure occurs through in-plane shear. Total failure of the laminate takes place at x = –90·1 MPa because of transverse compression (2 = F2). Figures 7–20 clearly demonstrate that the developed theoretical model predicts initial linear parts of the stress strain curves, the first-ply failure (appearance of the ‘knees’ on the curves), reduction of laminate stiffness characteristics after material cracking (stages 1–3 in Fig. 1, Section 2). With the model one is able to examine coupled deformation and failure processes in composite laminates and determine the stresses responsible for specific modes of failure. References 1 Goldenblat, I. I. and Kopnov, V. A., Anisotropy of Structural Materials. Mashinostroenie, Moscow, 1968 (in Russian). 2 Wu, E. M., In Composite Materials, Vol. 2, Mechanics of Composite Materials, ed. G. P. Sendecky. Academic Press, New York and London, 1974, pp. 353–431. 3 Zinoviev, P. A., Tarakanov, A. I., On the non-linear deformation of laminated composite materials, Primenenie Plastmass v Mashinostroenii. Trudy MVTU, 1978, N16, pp. 72–80 (in Russian). 4 Zinoviev, P. A., Tarakanov, A. I. and Fomin, B. Ya., Deformation and Failure Processes in Composites under Biaxial Tension, Primenenie Plastmass v Mashinostroenii. Trudy MVTU, 1982, Nl9, pp. 33–58 (in Russian). 5 Alfutov, N. A., and Zinov’ev, P. A., Deformation and failure of fibrous composites with brittle polymeric matrix under plane stress. In Mechanics of Composites, ed I. F. Obraztsov and V. V. Vasiliev. MIR Publishers, Moscow, 1982, pp. 166–185. 6 Zinoviev, P. A. and Lebedeva, O. V., Deformation and failure processes in laminated composites under plane stress state. In Proceedings of the Vl National Congress on Theoretical and Applied Mechanics. Varna, Sofia, Bulgaria, 1989, pp. 11–14 (in Russian). 7 Zinoviev, P. A., Lebedeva, O. V. and Sokolova, T. V., Modelling failure processes in glass fiber reinforced plastics under plane stress state. In Proceedings of the V National Conference on Mechanics and Technology of Composite Materials. Varna, Sofia, Bulgaria, 1988, pp. 119–123 (in Russian). 8 Soden, P. D., Leadbetter, D., Griggs, P. R. and Eckold, G. C., The strength of a filament wound composite under biaxial loading. Composites, 1978, 9, 247–250. 9 Eckold, G. C., Leadbetter, D., Soden, P. D. and Griggs, R. P., Lamination theory in the prediction of failure envelopes for laminated wound materials subjected to biaxial loading. Composites, 1978, 9, 243–246. 10 Partsevskii, V. V. and Kuznetsov, S. F. On the deformation and failure mechanisms of multilayered composite materials. Mekhanika Kompozitaych Materialov, 1981, 16, 1006–1011 (in Russian). 11 Kawata, K., On the yielding-fracture criterion of angleply FW laminates. In Mechanical Behaviour of Materials, Proc. Int. Conf., Vol. 5, Tokyo, 1972, pp. 14–155. 12 Uemura, M. and Yamawaki, In Proc. 9th Int. Conf. Space Technology and Science, Tokyo, 1971, pp. 215–223.

399 13 Guess, T. R., Biaxial testing of composite cylinders: experimental-theoretical comparison. Composites, 1980, 11, 139–149. 14 Highton, J., Adeoye, A. B. and Soden, P. D., Fracture stresses for ± 75 degree filament wound GFP tubes under biaxial loads. J. Strain Anal., 1985, 20, 139–150. 15 Hahn, H. T. and Tsai, S. W., Nonlinear elastic behavior of unidirectional composite laminate. J. Compo. Mater., 1973, 7, 257–271. 16 Alfutov, N. A., Zinov’ev, P. A. and Popov, B. G., Analysis of Multilayered Composite Plates and Shells. Mashinostroenie, Moscow, 1984 (in Russian). 17 Zinoviev, P. A. Thermoelasticity of Multilayered Hybrid Reinforced Plastics, Primenenie Plastmass v Mashinostroenii. Trudy MVTU, Moscow, 1989, N526, pp. 29–40. 18 Zinoviev, P. A. and Lebedeva, O. V., Thermal strength and stability of laminated composites. In Proceedings of the VI National Conference on Mechanics and Technology of Composite Materials, Sofia, 1991, pp. 119–123 (in Russian). 19 Grigoriev, S. V., Zinoviev, P. A., Tarakanov, A. I. and Lebedeva, O. V., Strength Analyzer of Multilayered Hybrid Composites. Software and User’s Manual. Technomic, Lancaster–Basel, 1995. 20 Hinton. M. J. and Soden, P. D., Predicting failure in composite laminates: the background to the exercise. Compos. Sci. Technol., 1998, 58(7), 1001. 21 Soden, P. D., Hinton, M. J. and Kaddour, A. S., Lamina properties, lay-up configurations and loading conditions for a range of fibre reinforced composite laminates. Compos. Sci. Technol., 1998, 58(7), 1011.

Appendix A1. Unidirectional ply in the ‘natural’ coordinate system The model of the composite laminate uses two types of coordinates (Fig. A1). The first coordinate system, X–Y, is applied for the laminate as a whole, this is the ‘global’ coordinate system. The second group of coordinate systems is a family of coordinates related to the plies. Every individual ply has its own coordinate system. Coordinates 1–2 are the ‘natural’ (or local) coordinates of the ply. There is an angle  between axes X and 1,  is the reinforcing (orientation) angle.

Fig. A1. Composite laminate and constitutive unidirectional plies; ‘global’ (X–Y) and ‘natural’ (1–2) coordinate systems.

400 The Hooke’s law relating the stresses and strains in the ith unidirectional ply under a plane stress state looks as follows in the ‘natural’ coordinate system: {12} = [G ]{12], 0

1 g011 2 = g012 12 0

or

g012 g022 0

1 2

12

0 0 g066

(A1)

where [G 0] is the stiffness matrix of the ply, whose components are: E

g011 = 1  v121 v21; E g022 = 1  v122 v21;

Ev

冎

Ev

g012 = 1 1v1221v21 = 1 2v1212v21; g066 = G12.

(A2)

Here E1, E2, G12, v12 are engineering elastic constants of the ply material. Inverting eqn (A1), one obtains {12} = [S ]{12}, or 0

s011 1 2 = s012

12 0

s012 s022 0

0 0 s066

1 2 12

(A3)

where [S 0] is the compliance matrix of the ply. The coefficients of the stiffness matrix, [G 0], are related to the coefficients of the compliance matrix [S 0] (see, for example, Ref. 16).

A2. Transformation of ply characteristics under coordinate system rotation As one changes from ‘natural’ coordinates 1–2 to ‘global’ coordinates X–Y, stress and strain matrices of the ply are transformed. Transformation rules for stress and strain matrices are: {xy}(i) = [T1]{12}

(A4)

{xy}(i) = [T2]{12}

(A5)

where {xy} , {xy} are stress and strain column-matrices of the ply related to (X–Y) axes. The formulas of inverse transformations: (i)

(i)

{12} = [T1]–1{xy}(i) {12} = [T2]–1{xy}(i) where c2 [T1] = s2 sc

s2 c2 –sc

–2sc 2sc , 2 c  s2

c2 [T2] = s2 2sc

s2 c2 –2sc

–sc sc 2 c  s2

(A6)

here s = sin , c = cos . The relationship between mean stresses and strains in the ply related to the axes (X–Y) ¯ ]{xy}(i) {xy}(i) = [G ¯ ] = [T1][G 0][T2]–1 = [T1][G 0][T1]T [G

(A7)

401 ¯ ] is the stiffness matrix of the unidirectional ply related to the axes (X–Y) where [G g¯ 011

g¯ 012 g¯ 022

¯ ]= [G sym

g¯ 016 g¯ 026 g¯ 066

¯ ] and [G 0] matrices (see, for There are the relationships between the coefficients of [G example, Ref. 16).

A3. Stress/strain relationships for the composite laminates under plane stress state Mean stresses arising in the composite laminate under plane stress state are calculated from the formulas:

冘 n

x =

冘 n

 h¯ , (i) (i) x

y =

i=1

冘 n

 h¯ , (i) (i) y

xy =

i=1

¯ (i) (i) xy h

(A8)

i=1

where h¯ (i) = h(i)/H is the relative thickness of the ith ply, and H is the total thickness of the laminate. Assume that the plies in the laminate are bonded ideally, and the strains in all plies are equal to mean strains of the whole laminate:

x = (i) y = (i)

xy = (i) x , y , xy

(A9)

Substituting the Hooke’s law for the ply, eqn (A7), into (A8) and considering eqn (A9), one obtains the relationship between laminate mean stresses and strains under plane state of stresses: {xy} = [G ]{xy} or x g11 y = g12 xy g16

where

g12 g22 g26

冘

g16 g26 g66

x y

xy

(A10)

n

gij =

i=1

¯ (i) g¯ (i) ij h

(A11)

402

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

CHAPTER 3.13

Predicting the nonlinear response and progressive failure of composite laminates Travis A. Bogetti*, Christopher P. R. Hoppel, Vasyl M. Harik, James F. Newill and Bruce P. Burns US Army Research Laboratory; AMSRL-WM-MB, Aberdeen Proving Ground, MD 21005-5066, USA Abstract A comprehensive comparison of laminate failure models was established to assess the state-of-the-art in laminate modeling technologies on an international level (known as the Worldwide Failure Olympics Exercise) [1]. This paper represents one contribution (Part A) to the Exercise, where predictions for laminate response and failure behavior of various laminates under a broad range of loading conditions are made. A method for predicting the nonlinear stress/strain response and failure behavior of composite laminates is described. Predictions are based on an incremental formulation of a well-established threedimensional laminated media analysis [2] coupled with a progressive-ply failure methodology. Nonlinear lamina constitutive relations for the composites are represented using the Ramberg–Osgood equation [3]. Piece-wise linear increments in laminate stress and strain are calculated and superimposed to formulate the overall effective nonlinear response. Individual ply stresses and strains are monitored to calculate instantaneous ply stiffnesses for the incremental solution and to establish ply failure levels. The progressive-ply failure approach allows for stress unloading in a ply and discrimination of the various potential modes of failure. Laminate response and failure predictions for 14 different cases are presented. The cases include prediction of the effective nonlinear stress-vs.-strain responses of laminates, as well as, initial and final ply failure envelope predictions under multi-axial loading. Comparison of predictions with actual experimental data will be made in a companion paper to be published in Part B of the Exercise. Keywords: Composite laminate

1. Introduction 1.1. Background Predicting the mechanical response and failure behavior of laminated composites is vitally important for efficient design in structural applications. Few would argue that predicting the effective laminate strain response to mechanical load is far easier than predicting the failure * Corresponding author.

403 (or post failure) behavior of the laminate. It is not surprising that this subject has received a great wealth of attention since the early days of composite mechanics. Many different approaches exist for laminate failure analysis with varied complexity and successes. Surveys abound on the subject: Chamis [4], Sandhu [5], Soni [6], Tsai [7], and Nahas [8]. Review of the literature reveals the broad spectrum of approaches that are employed in laminate failure prediction. With such a broad range of approaches, predictions for particular laminate configurations and loading are also likely to be widely varied. To address this issue, a recent effort has been launched by Hinton and Soden [1] to assess the state-of-the-art in prediction capabilities for laminate response and failure. This effort requested originators of a variety of laminate failure theories to make performance predictions of specific carbon- and glass-fiber-reinforced epoxy laminates subjected to a range of biaxial loads, using the same given material properties, laminate arrangements, and loading conditions. Each of the contributors has submitted a paper documenting their predictions for 14 different laminate cases including a report on their respective failure theory and approach [9–20]. Specifics of the exercise are published in a separate report [21]. The predictions submitted by the contributors have been compared, and differences in their respective approaches have been identified and discussed [22]. For the most part, the submitted laminate predictions for stresses and strains are based on classical laminated plate theory or a similar derivative while laminate failure behavior is modeled with a wider variety of approaches. This paper represents our laminate response and failure predictions to the 14 laminate cases described in Part A of the Exercise [1]. Subsequently, we plan to participate in Part B of the Exercise and compare our predictions with the experimental data.

1.2. Current approach The approach taken in our investigation is based on the three-dimensional laminated media analysis presented by Chou et al. [2]. This analysis is similar to Classical Laminated Plate Theory (CLPT) except that through thickness stresses and strains are accounted for in the formulation. Additionally, laminate curvatures are not a permitted type of deformation, which makes this theory more relevant to thick laminated composite analysis. As with CLPT, ply stresses and strains are calculated from applied average stress resultant type mechanical loads. In-plane laminate behavior and ply stress and strain predictions for this theory are nearly identical to those made by CLPT for laminates which possess balanced and symmetric lay-ups – those which do not possess bending-twisting-coupling modes of deformation. Specific details of the analysis are highlighted in the Analysis section of this paper. All of the 14 laminates cases studied in this exercise possess balanced and symmetric architectures. The laminated media analysis technique presented by Chou et al. [2] was used for predicting linear-elastic material response and failure in composite laminates [23]. In the present study, we have extended this capability to predict nonlinear material behavior by adopting a piece-wise linear incremental approach. Essentially, the effective nonlinear laminate stress/strain response predictions are determined from the superposition of piecewise linear segments in stress and strain during an incremental loading scheme. The individual ply stresses and strains are computed at each step during the incremental loading history. The effective laminate stiffness matrix is updated at each load increment and is based on straindependent tangent ply properties.

404 Progressive laminate failure is modeled with a maximum strain-based ply failure criteria and a ply modulus discount method. When a strain allowable in any ply is reached during the incremental laminate loading, the associated modulus to the particular failure mode is reduced and the corresponding load is subsequently redistributed within in the laminate. Incremental loading is continued until the laminate cannot sustain load without undergoing excessive deformation or strain. Details of the laminated media analysis, the piece-wise linear incremental loading strategy, and the progressive ply failure methodology are described in the following section.

2. Analysis 2.1. Three-dimensional laminate media analysis In this work, the analytic model developed by Chou et al. [2] is used to predict the effective laminate stress/strain response. It is also used to calculate ply-level stresses and strains during incremental loading for failure and strength prediction [23]. The following section outlines the laminated media model upon which our analysis is based. Chou et al. [2] use a control volume approach to yield a closed-form solution to the problem of effective homogeneous property determination for a laminated media composed of individual layers. Unlike the works of White and Angona [24], Postma [25], Rytov [26], Behrens [27], and Salamon [28], which required the individual layers to be isotropic, Chou et al. [2] permitted general anisotropy of the layers. The analysis is based on the assumptions that all interlaminar stresses are continuous across ply interfaces and that all in-plane strains are continuous through the thickness dimension of a representative volume element (i.e., a repeating sublaminate configuration). The following expression is used to represent the effective (i.e., homogeneous) stress/strain constitutive relationship for an N-layered laminate (see Fig. 1): *i = C *ij *j for (i, j = 1, 2, 3, 4, 5, 6).

(1)

The barred notation is used to denote that the relationship applies in the global x–y–z coordinate system of the laminate. The asterisk superscript is used here to denote the “average” or effective laminate stress and strain quantities. In-plane strains are assumed

Fig. 1. Laminate configuration.

405 uniform (i.e., constant within each ply) and equal to the effective strains of the laminate. Mathematically, this is expressed as

ki = *i for (i = 1, 2, 6; k = 1, 2, . . . , N ),

(2)

where ki represents the strain in the kth ply of the laminate (see ply numbering convention in Fig. 1). To ensure stress continuity across ply interfaces, all ply stress components associated with the out-of-plane direction (i.e., z-direction) are assumed uniform and equal to the corresponding effective stresses in the laminate. Mathematically, this is expressed as ki =  *i for (i = 3, 4, 5; k = 1, 2, . . . , N ),

(3)

where ki represents the stress in the kth ply of the laminate. All remaining effective laminate strains and stresses are assumed to be the volume average of all their corresponding ply strain and stress components, respectively. Mathematically, these assumptions are expressed as

冘

V kki for (i = 3, 4, 5)

(4)

冘

V kki for (i = 1, 2, 6)

(5)

N

*i =

k=1

and N

*i =

k=1

where V k is the ratio of the original (i.e., undeformed) volume of the kth ply over the original volume of the entire laminate. The constitutive equation for each ply in the laminate is written below [Eq. (6)] using the superscript notation. ki = C kij kj for (i, j = 1, 2, 3, 4, 5, 6; k = 1, 2, . . . , N ).

(6)

(For completeness, the ply stiffness matrix coefficients (C kij) are defined in terms of the lamina engineering constants and layer orientations in the Appendix.) Eqs. (1)–(6) represent 12N + 6 linear algebraic equations with 12N + 12 unknowns. Solution to Eqs. (1)–(6) yields the following effective three-dimensional stress/strain constitutive relation, which can be used as an equivalent (i.e., homogeneous) representation for the laminated media where the coefficients in the laminate stiffness matrix, C kij, are given by

冘 冘 N

冘 N

C *ij =

k=1

V k C kij 

C k13C k3j C k33

C

k i3

ᐉ=1 N

+ C

k 33

ᐉ=1

V ᐉ C k3j C k33 V ᐉC kij

for (i, j = 1, 2, 3, 6),

(7)

C k33

C *ij = C *ji = 0 for (i = 1, 2, 3, 6; j = 4, 5)

(8)

406 and

冘 N

C *ij =

冘冘 N

N

k=1

ᐉ=1

k=1

Vk k C ij k

V kV ᐉ k k (C 44C 55  C k45C k54) kᐉ

for (i, j = 4, 5),

(9)

where k =

冏 冏 C k44 C k45 C

k 54

C

k 55

= C k44 C k55  C k45 C k54.

(10)

The effective stress/strain constitutive relation for the laminated media is therefore given by Eqs. (1) and (7)–(10). In determining the individual ply-level stresses and strains, the assumption is made that the applied mechanical loading on the laminated media (*i ) is known, uniform, and represents the ‘average’ or ‘effective’ stress acting on the sublaminate configuration. The associated “effective” or “smeared” laminate strains (*i ) can be obtained directly from the inversion of Eq. (1). From the assumption made in Eq. (2), all in-plane strain values (defined in the global x–y–z coordinate system) for plies 1 through N are therefore known. Similarly, from the assumption made in Eq. (2), all out-of-plane stresses for plies 1 through N are known (also defined in the global x–y–z coordinate system). The out-of-plane ply strains and in-plane ply stresses remain to be determined. Sun and Liao [29] derived the following expression for determination of the remaining outof plane ply strains C k33C k34C k35 k3 k 4 = C k43C k44C k45

k5

–1

C k53C k54C k55

C k31C k32C k36 k3 k 4  C k41C k42C k46

k1 k2

k5

k6

C k51C k52C k56

.

(11)

Once all of the ply strains are known, the remaining in-plane ply stresses can be calculated straightforwardly through the following relation

k1 C k11C k12C k13C k14C k15C k16 k2 = C k21C k22C k23C k24C k25C k26 

k 6

k 61

k 62

k 63

k 64

k 65

C C C C C C

k 66

k1 k2 k3 k4 k5 k6

.

(12)

2.2. Defining nonlinear lamina constitutive relations Material nonlinearity in our laminate analysis is accounted for on the lamina or ply level. The nonlinear lamina constitutive relations (i.e., stress-vs.-strain relations) for each of the principal lamina directions are defined with the Ramberg–Osgood equation [3]. For the treatment of

407 unidirectional lamina in our three-dimensional analysis, this would include the fiber direction (1), in-plane transverse direction (2), transverse normal direction (3), interlaminar shear directions (23 and 13), and the in-plane shear direction (12). The Ramberg–Osgood equation provides an expression for stress written explicitly in terms of strain and three unique parameters, =

Eo

冉 冉 冊冊 Eo 1+ o

n

1 n

(13)

.

here Eo is the initial modulus, o is the asymptotic stress level, and n is a shape parameter for the stress versus strain curve. Fig. 2 graphically illustrates the significance of these parameters with a typical nonlinear stress-vs.-strain relationship. For computational considerations, it is desired to define the instantaneous or tangent lamina stiffness as a continuous function of strain. Taking the derivative of Eq. (13) with respect to strain, the following expression is obtained: Et =

d = d

Eo

冉 冉 冊冊 Eo 1+ o

n

1+

1 n

(14)

where Et is the instantaneous or tangent lamina stiffness modulus expressed explicitly in terms of strain and the three Ramberg–Osgood parameters. A unique set of Ramberg–Osgood parameters for each of the principal directions in the lamina is required. A fitting routine was implemented to find the Ramberg–Osgood parameters which realistically represent the stress/strain response for each of the four materials used in the study. As an example, the data fit to Eq. (13) is illustrated in Fig. 3 for the nonlinear 12-shear direction stress/strain response of the E-glass/MY750 material. A full account of all the Ramberg–Osgood parameters used in our analysis is provided in the Results section of this paper.

Fig. 2. Ramberg–Osgood parameters definitions.

408

Fig. 3. Ramberg–Osgood parameters fit to E-glass/MY750 epoxy data.

2.3. Incremental approach (solution strategy) The nonlinear response of the laminate is generated through the summation of piece-wise linear increments in stress over a pre-established load schedule. An incremental form of Eq. (1) is used to determine the linear increments in laminate stress-and-strain. The laminate stiffness matrix is updated at the end of each stress increment (based on all current ply strain levels) during the incremental loading strategy. The schematic presented in Fig. 4 provides a mathematical representation of the incremental loading strategy for an arbitrary laminate.

Fig. 4. Incremental laminate loading methodology.

409 Assume that at point (a), corresponding to the end of the nth stress increment, the strain and stress state of the laminate is known (*nj, *ni). From this point, the objective is to determine the strain and stress state at point (b) or (nj + 1, *ni + 1) The effective laminate stiffness matrix at the end of stress increment n, C *nij, is computed from an incremental form of the laminated media model constitutive relation, Eq. (1). With the increment in load defined, *ni, the corresponding increment in laminate strain, *nj, is calculated from an inverse form of Eq. (1): *i = [C *ij]–1*i, (i, j = 1, 2, 3, 4, 5, 6).

(15)

Individual ply stress and strain increments are calculated according to the equations presented previously. A cumulative summation is maintained to track the total stress-and-strain levels in each ply of the laminate. The tangent modulus values for each ply and material direction are calculated according to Eq. (14) and used in the determination of the laminate stiffness matrix for the next laminate stress increment calculation. The entire nonlinear response for the laminate is obtained by the cumulative sum of all stress and strain increments throughout the entire stress loading history. The implementation of a progressive ply failure methodology into this incremental loading strategy is described in the next section.

2.4. Lamina failure methodology Failure of individual plies and their effect on the overall laminate response during incremental loading are accounted for in our analysis. Our ply failure predictions are based on the wellestablished Maximum Strain Failure Criterion [8,30]. The Maximum Strain Failure Criterion predicts that a material will fail when the strain in any direction exceeds its corresponding allowable level. The principal ply strains in the six directions (1, 2, 3, 4, 5, and 6) are compared to their corresponding maximum strain allowables: if 1 > 0 and if 1 > Y1T, then the failure mode is fiber tension,

(16a)

if 1 < 0 and if | 1 | > Y1C, then the failure mode is fiber compression,

(16b)

if 2 > 0 and if 2 > Y2T, then the failure mode is transverse tension,

(16c)

if 2 < 0 and if | 2 | > Y2C, then the failure mode is transverse compression,

(16d)

if 3 > 0 and if 3 > Y3T, then the failure mode is transverse tension,

(16e)

if 3 < 0 and if | 3 | > Y3C, then the failure mode is transverse compression,

(16f)

if | 4 | > Y23, then the failure mode is interlaminar shear,

(16g)

if | 5 | > Y13, then the failure mode is interlaminar shear,

(16h)

and if | 6 | > Y12, then the failure mode is in-plane shear.

(16i)

410 In Eqs. (16a)–(16i), Y1T is the maximum tensile strain in the 1-direction (longitudinal), Y1C is the maximum compressive strain in the 1-direction, Y2T is the maximum tensile strain in the 2-direction (transverse), Y2C is the maximum compressive strain in the 2-direction, Y3T is the maximum tensile strain in the 3-direction (out-of-plane), Y3C is the maximum compressive strain in the 3-direction, Y23 is the maximum shear strain in the 23-plane, Y13 is the maximum shear strain in the 13-plane, and Y12 is the maximum shear strain in the 12-plane. As the laminate is loaded and laminate strains develop, the individual ply strains are monitored. When ply failure is predicted in any ply, according to the maximum strain failure criteria, the incremental loading to that point is stopped and the entire laminate stress vs. strain response is recorded. The modulus associated with the particular mode of failure in the failed ply is then reduced to an insignificant value (as well as the associated Poisson’s ratio), and the incremental loading strategy is repeated from the beginning (all stresses and strains are set to zero). The loading procedure is continued until the next failure in a ply is detected. The corresponding modulus value is again discounted, the laminate response is recorded, and the procedure is repeated. This progressive ply failure response is repeated until final failure is determined, which is assumed when the laminate looses sufficient stiffness such that it cannot carry any load without undergoing an arbitrarily excessive amount of deformation (say greater than 5% strain). The entire laminate response is determined by the stress vs. strain response up to the point of failure, and then the load is assumed to drop to the level of the subsequent stress vs. strain curve response. The load path then continues until failure and drops again. This methodology essentially corresponds to progressive ply failure where the load in failure plies is redistributed to adjacent plies under a displacement controlled load path history.

2.5. Thermal residual stresses Thermal residual stresses resulting from thermal expansion mismatch in adjacent plies in the laminates during cool down from the stress-free state at the cure temperature were not accounted for in the predictions. Their actual calculation follows straightforwardly from the analysis derivation described in the previous section. For completeness, however, a full description of their determination is given elsewhere [23]. It is acknowledged that the inclusion of thermal residual stresses will have some effect on the ultimate laminate strength predictions. The exact effect, however, will depend on the specific laminate architecture and loading considered.

2.6. Analysis execution The aforementioned laminate analysis and progressive ply failure methodology has been programmed into a FORTRAN-based software code entitled LAM3DNL. The LAM3DNL code employs a user-friendly database format for input of laminate architectures, lamina properties, and failure parameters [23]. Output from the code includes the effective laminate stress and strain files as well as a failure assessment summary file that identifies all ply failures that occur during a laminate response prediction program run.

411 3. Results and discussion 3.1. Test case summary In this section, we present predictions for the 14 different laminate test cases described by Soden et al. [21]. These cases have been grouped into three classes (a) biaxial failure envelopes of unidirectional lamina, (b) bidirectional failure envelopes of multidirectional laminates, and (c) stress vs. strain curves of laminates under uniaxial and biaxial loading. For completeness, a summary of the test cases is presented in Table 1 [21,22]. It is also noted that four different materials were included in the study: (a) E-glass/MY750 epoxy, (b) E-glass/ LY556 epoxy, (c) T300 graphite/BSL 914C epoxy, and (d) AS4 graphite/3501-6 epoxy. 3.2. Lamina properties and failure allowables Lamina material properties and failure allowables were provided by Soden et al. [21]. Since the required three-dimensional material properties for our analysis were not available, we have made some assumptions in order to represent the through-the-thickness material response. Upon examining the data provided in [21], we have made the reasonable assumption that the longitudinal and transverse lamina responses are linear. We also assume that the material properties are transversely isotropic such that E3 = E2, G13 = G12, and v13 = v12. Accordingly, the interlaminar shear modulus is assumed linear according to G23 = E2/2(1 + v23). Table 1 Details of the laminates and loading cases [22]. Loading case Laminate lay-up Material

Description of loading cases

1

0

E-glass/LY556/HT907/DY063

2

0

T300/BSL914C

3

0

E-glass/MY750/HY917/DY063

4 5 6 7

90/ ± 30/90 90/ ± 30/90 0/ ± 45/90 0/ ± 45/90

E-glass/LY556/HT907/DY063 E-glass/LY556/HT907/DY063 AS4/3501-6 AS4/3501-6

8 9 10

0/ ± 45/90 ± 55 ± 55

AS4/3501-6 E-glass/MY750/HY917/DY063 E-glass/MY750/HY917/DY063

11 12

± 55 0/90

E-glass/MY750/HY917/DY063 E-glass/MY750/HY917/DY063

13 14

± 45 ± 45

E-glass/MY750/HY917/DY063 E-glass/MY750/HY917/DY063

Biaxial failure stress envelope under transverse and shear loading (y vs. xy) Biaxial failure stress envelope under longitudinal and shear loading ( x vs. xy) Biaxial failure stress envelope under long. and transverse loading (y vs. x) Biaxial failure stress envelope (y vs. x) Biaxial failure stress envelope (x vs. xy) Biaxial failure stress envelope (y vs. x) Stress–strain curves under uniaxial tensile loading in y direction (y : x = 1 : 0) Stress–strain curves for (y : x = 2 : 1) Biaxial failure stress envelope (y vs. x) Stress–strain curves under uniaxial tensile loading for (y : x = 0 : 1) Stress–strain curves for (y : x = 2 : 1) Stress–strain curve under uniaxial tensile loading for (y : x = 0 : 1) Stress–strain curves for (y : x = 1 : 1) Stress–strain curves for (y : x = 1 : –1)

412 Table 2 Ramberg–Osgood parameters for nonlinear analysis (and Poisson’s ratio). Material and its parameters

Spacial directions for constitutive modeling 1

2

3

12

13

23

AS4/3501-6 E0 (GPa) 0 (GPa) n v

126 100 10 –

11 100 10 –

11 100 10 –

6.80 0.097 1.96 0.28

6.80 0.097 1.96 0.28

3.79 100 10 0.40

T300/BSL914C E0 (GPa) 0 (GPa) n v

138 100 10 –

11 100 10 –

11 100 10 –

5.94 0.083 2.05 0.28

5.94 0.083 2.05 0.28

3.79 100 10 0.40

E-glass/LY556 E0 (GPa) 0 (GPa) n v

53.5 100 10 –

17.7 100 10 –

17.7 100 10 –

6.36 0.076 1.85 0.278

6.36 0.076 1.85 0.278

6.10 100 10 0.40

E-glass/MY750 E0 (GPa) 0 (GPa) n v

45.6 100 10 –

16.2 100 10 –

16.2 100 10 –

6.42 0.077 1.80 0.278

6.42 0.077 1.80 0.278

5.59 100 10 0.40

Additionally we assume v23 = 0.40 for all materials. It is noted that the predominant source of nonlinearity in our predictions is from the 12 shear response through G12. We fit the lamina material properties to the Ramberg–Osgood equation for input into the analysis. As stated previously, the longitudinal and transverse properties were assumed linear. To capture the linear behavior with the Ramberg–Osgood equation, a linear modulus was assumed as the initial modulus parameter (Eo), an asymptotic stress level (o) was assumed which is much higher than the actual strength of the material and an arbitrarily high shape factor as also used (n = 10). This approach ensures that a linear modulus is used during the entire incremental loading history. The stress-vs.-strain data provided in the exercise for the shear material responses were fit to Eq. (13). A summary of all fitted Ramberg–Osgood parameters for the four materials are summarized in Table 2. Maximum strain failure allowables were also provided by Soden et al. [21] and are summarized in Table 3. 3.3. Results for selected case studies 3.3.1. Bioxialfailure envelopes of unidirectional lamina (cases 1, 2, and 3) The biaxial failure envelope predictions of unidirectional lamina of the E-glass/LY556 epoxy under transverse and shear loading (y vs. xy) are presented in Fig. 5. The typical rectangular-

413

Fig. 5. Loading case 1: biaxial failure envelope (y vs. xy) and failure modes for [0] E-glass/LY556 epoxy.

shaped curve results from the failure strain in each direction being assumed independent of the other directions. For this loading case, the initial and final failure envelopes coincide everywhere except in the second quadrant, where the Poisson’s effects result in early transverse tensile failure occurring in the 3-direction prior to the final transverse compressive failure in the 2-direction. This is similar to the transverse tensile failures that have occurred during axial compression of [0/ + 30/0/–30]2S laminates in other studies [31]. The biaxial failure stress envelope for unidirectional T300/BSL914C under longitudinal and shear loading (x vs. xy) is shown in Fig. 6. Like the previous case, this case again shows a rectangular-shaped curve with shear- and axial-type failures occurring independently of other failure modes. There were no initial failures occurring prior to final fracture for this loading curve. The biaxial failure stress envelope for loading case 3, unidirectional E-glass/MY750 under transverse and longitudinal loading (y vs. x), is shown in Fig. 7. For this failure case, the Table 3 Maximum strain failure allowables. Material

Y1T (%) Y1C (%) Y2T (%) Y2C (%) Y3T (%) Y3C (%) Y23 (%) Y13 (%) Y12 (%)

AS4/3501-6 T300/BSL914C E-glass/LY556 E-glass/MY750

1.38 1.09 2.13 2.81

–1.18 –0.65 –1.07 –1.75

0.44 0.25 0.20 0.25

–2.00 –1.82 –0.64 –1.20

0.44 0.25 0.20 0.25

–2.00 –1.82 –0.64 –1.20

2.00 4.00 3.80 4.00

2.00 4.00 3.80 4.00

2.00 4.00 3.80 4.00

414

Fig. 6. Loading case 2: biaxial failure envelope of (x vs. xy) and failure modes for [0] T300 graphite/ BSL 914C epoxy.

Fig. 7. Loading case 3: biaxial failure envelope of (y vs. x) and failure modes for [0] E-glass/MY750 epoxy.

415

Fig. 8. Loading case 4: biaxial failure envelope (y vs. x) and failure modes for [90/ + 30/–30]s E-glass/ LY556 epoxy. (See Table 4 for ply failure details.)

Fig. 9. Loading case 5: biaxial failure envelope of (x vs. xy) and failure modes for [90/ + 30/–30]s E-glass/LYS56 epoxy. (See Table 5 for ply failure details.)

416 Table 4 Damage modes in the failure envelope for the [90/ ± 30/90]s laminate shown in Fig. 8. Loading case

4 Ratio: 0:1 4 Ratio: 1:1 4 Ratio: 1:0 4 Ratio: 1 : –1 4 Ratio: 0 : –1 4 Ratio: –1 : –1 4 Ratio: –1 : 0 4 Ratio: –4.26 : 1

Damage modes

Y2T Y1C Y2C Y2T Y2T Y1T Y2T Y1T Y2C Y2T Y2C Y1T Y2C Y3T Y2T Y1C Y3T Y2C Y1C Y2T Y3T Y2C Y2T Y3T Y2C

Failed plies

90° 90° ± 30° ± 30° 90° 90° ± 30° 90° 90° ± 30° 90° 90° 90° all ± 30° ± 30° all ± 30° 90° 90° all ± 30° 90° all ± 30°

Stress levels (MPa) y

x

0 0 0 73 91 357 68 254 74 61 120 193 0 0 0 0 –97 –231 –192 –132 –152 –200 –89.5 –170.4 –196.0

61 474 315 73 91 357 0 0 0 –61 –120 –193 –187 –344 –383 –403 –97 –231 –192 0 0 0 21 40 46

final tensile and compressive failures in the fiber direction (x) are almost independent of the transverse stress-state. The tensile and compressive failures in the transverse (y) direction are strongly influenced by the axial (x) stress due to the Poisson’s effects in the material. The initial failures that are predicted in the graph are tensile failures in the 3-direction due to Poisson’s effects, similar to those predicted in case 1. 3.3.2. Bidirectional failure envelopes of multi-directional laminates (cases 4–6 and 9) For loading case 4, the biaxial failure stress envelope (y vs. x) of the [90/ ± 30/90]s E-glass/ LY556 laminate is shown in Fig. 8. For this failure envelope, the failure modes are strongly influenced by the biaxial stress-state in all directions. The laminates also experience initial

417 Table 5 Damage modes in the failure envelope for the [90/ ± 30/90]s laminate shown in Fig. 9. Loading case

5 Ratio: 1:0 5 Ratio: 1:1 5 Ratio: 0:1 5 Ratio: 2.35 : 1 5 Ratio: –1 : 0

Damage modes

Y2T Y1C Y2C Y2T Y2C Y1T Y2T Y2C Y1C Y2C Y2T Y2C Y1C Y2C Y3T Y2T Y1C

Failed plies

90° 90° ± 30° 90°& –30° + 30° + 30° –30° + 30° –30° 90° –30° 90° –30° 90° all ± 30° ± 30°

Stress levels (MPa) x

xy

61 474 311 60 164 278 0 0 0 0 –110.4 –185.6 –253.8 –186 –344 –383 –403

0 0 0 60 164 278 56 186 239 45 47 79 108 0 0 0 0

failures prior to final fracture for all loading directions. Details of the progressive failure for loading case 4 are presented in Table 4. The biaxial failure envelope (x vs. xy) for the same [90/ ± 30/90]s laminate of E-glass/ LY556 epoxy (loading case 5) is shown in Fig. 9. Like loading case 4, the failure modes are strongly influenced by the biaxial stress-state in all directions, and the laminates also experience initial failures prior to final fracture for all of the loading directions. Details of the progressive failure for loading case 5 are presented in Table 5. Fig. 10 shows the biaxial failure envelope (y vs. x) for the quasi-isotropic [0/ ± 45/90]s laminate of AS4/3501-6 (loading case 6). The biaxial failure envelope (y vs. x) for [ + 55/–55]s E-glass/MY750 epoxy (loading case 9) is shown in Fig. 11. Details of the progressive failure for loading cases 6 and 9 are presented in Tables 6 and 7, respectively. 3.3.3. Stress vs. strain curves of laminates under uniaxial and biaxial loading (cases 7, 8, 10–14) Cases 7 and 8 predict the stress–strain response of the quasi-isotropic [0/ ± 45/90]s laminate of AS4/3501-6 used in case 6. The predictions for (y : x = 1 : 0) are shown in Fig. 12, and the predictions for (y : x = 2 : 1) are shown in Fig. 13. In both of these cases, the materials display linear behavior with several initial failures prior to the ultimate failure of the laminate.

418

Fig. 10. Loading case 6: biaxial failure envelope (y vs. x) and failure modes for [0/ + 45/–45/90]s AS4 graphite/3501-6 epoxy. (See Table 6 for ply failure details.)

Fig. 11. Loading case 9: biaxial failure envelope (y vs. x) and failure modes for [ + 55/–55]s E-glass/ MY750 epoxy. (See Table 7 for ply failure details.)

419 Table 6 Damage modes in the failure envelope for the [0/ ± 45/90]s laminate shown in Fig. 10. Loading case

6 Ratio: 0:1 6 Ratio: 1 : 1 6 Ratio: 1:0 6 Ratio: 1 : –1 6 Ratio: 0 : –1 6 Ratio: -1 : –1 6 Ratio: –1 : 0 6 Ratio: –1 : 1

Damage modes

Y2T Y2T Y1T Y2T Y1T Y2T Y2T Y1T Y2T Y12 Y1C Y1C Y2T Y3T Y1C Y1C Y2T Y2T Y12 Y1C

Failed plies

0° ± 45° 90° all all 90° ± 45° 0° 90° ± 45° 90° 90° 90° all all 0° 0° 0° ± 45° 0°

Stress levels (MPa) y

x

0 0 0 318 860 224 592 625 171 362 386 0 0 –395 –849 –582 –254 –171 –362 –386

224 592 625 318 860 0 0 0 –171 –362 –386 –582 –254 –395 –849 0 0 171 362 386

Cases 10 and 11 predict the stress–strain response of a [ + 55/–55]s laminate of E-glass/ MY750 epoxy. The curves for loading applied in the y-direction (y : x = 1 : 0) are shown in Fig. 14. In this case, the loading results in shear loading on the ply-level; thus, the laminate displays nonlinear behavior until failure. For the case where a biaxial load (y : x = 2 : 1) is applied (Fig. 15), the mechanical response is more linear until final fracture, although some nonlinearity (due to in-plane shearing) is evident near the point of ultimate failure. The stress–strain curves for uniaxial tension (y : x = 0 : 1) of a [0/90]s E-glass/MY750 laminate (case 12) is shown in Fig. 16. The final load cases, stress–strain curves for the biaxial loading of [ + 45/–45]s laminates of E-glass/MY750 epoxy, are shown in Figs. 17 and 18. For case 13, where y = y , the strains in the x and y directions are equivalent so one curve is shown in Fig. 17. Fig. 18 shows that for case 14, where y = –x, the strains are equal and opposite in the x and y directions, with both directions displaying significant nonlinear behavior. Details of the progressive ply failures in loading cases 7–14 are summarized in Table 4. In particular, each level in loading where failure in a ply occurs is indicated. The associated ply and mode of failure for each failure load level are also given.

420 Table 7 Damage modes in the failure envelope for the [ ± 55]s laminate shown in Fig. 11. Loading case

Damage modes

9 Ratio: 0 : 1 9 Ratio: 1 : 1 9 Ratio: 1 : 0 9 Ratio: 0 : –1

Failed plies

Y2T Y12 Y2T Y12 Y12 Y2C&3T Y3T Y2C Y12 Y3T Y1C Y2T Y12

9 Ratio: –2 : –1 9 Ratio: –1 : 0

Stress levels (MPa)

± 55° ± 55° ± 55° ± 55° ± 55° ± 55° ± 55° ± 55° ± 55° ± 55° ± 55° ± 55° ± 55°

y

x

0 0 77 202 281 140 0 0 0 –158 –706 –204 –209

70 104 77 202 0 0 –128 –183 –104 –79 –353 0 0

Table 8 Damage modes in the stress–strain curves. Loading case

Laminate lay-up

7

[0/ ± 45/90]s

8

[0/ ± 45/90]s

10 11

[ ± 55]s [ ± 55]s

12

[0/90]s

13

[ ± 45]s

14

[ ± 45]s

Failure modes

Y2T Y2T Y1T Y2T Y2T Y1T Y12 Y2T Y1T Y2T Y1T Y2T Y1T Y12

Failed plies

90° ± 45° 0° 90° ± 45° 0° ± 55° ± 55° ± 55° 90° 0° ± 45° ± 45° ± 45°

Stress levels (MPa) x

y

0 0 0 132 203 367 0 82 414 78 634 92 621 0

224 592 625 264 406 734 281 164 828 0 0 92 621 75

421

Fig. 12. Loading case 7: stress–strain curves (y : x = 1 : 0) and failure modes for [0/ + 45/–45/90]s AS4 graphite/3501-6 epoxy.

Fig. 13. Loading case 8: stress–strain curves (y : x = 2 : 1) and failure modes for [0/ + 45/–45/90]s AS4 graphite/3501-6 epoxy.

422

Fig. 14. Loading case 10. stress–strain curves (y : x = 1 : 0) and the final failure mode for [ + 55/–55]s E-glass/MY750 epoxy.

Fig. 15. Loading Case 11: stress–strain curves (y : x = 2 : 1) for [ + 55/–55]s E-glass/MY750 epoxy with the initial and final failure modes.

423

Fig. 16. Loading case 12: stress–strain curves (y : x = 0 : 1) for [0/90]s E-glass/MY750 epoxy with the initial and final failure modes.

Fig. 17. Loading case 13: stress strain curve (y : x = 1 : 1) for [ + 45/–45]s E-glass/MY750 epoxy with the initial and final failure modes.

424

Fig. 18. Loading case 14: stress–strain curves (y : x = 1 : –1) for [ + 45/–45]s E-glass/MY750 epoxy with the final failure mode.

4. Conclusions A methodology has been presented for predicting the nonlinear response and progressive failure of composite laminates. The predictions are based on an incremental formulation of a well-established three-dimensional laminated media analysis [2] coupled with a progressive ply failure methodology. Nonlinear lamina constitutive relations for the composite are represented using the Ramberg–Osgood equation [3]. Piece-wise linear increments in laminate stress and strain are calculated and superimposed to formulate the overall effective nonlinear response. Individual ply stresses and strains are monitored to calculate instantaneous ply stiffnesses for the incremental solution and to establish ply failure levels. The progressive-ply failure approach is used to allow for stress unloading in a ply and discrimination of the various potential modes of failure. The laminate response predictive capability presented in this paper is unique in relation to other existing capabilities. By adopting the three-dimensional laminated media theory, we are able to capture through-the-thickness effects in laminate response, which is particularly important for thick laminate analysis, where interlaminar loads may be of concern. The theory presented in this work is easily adapted for implementation in the design and failure assessment of composite structures. Employing the three-dimensional laminate analysis presented here and the widely accepted “smearing-unsmearing” approach [2], the authors have developed a computer software code, LAMPAT [32], that is particularly useful for the analysis and design of thick-section composite structures. In this paper, we have presented our prediction for biaxial failure envelopes and stress strain curves for 14 different cases originally proposed by Hinton, Soden and Kaddour, Refs. [1,21].

425 The cases include prediction of the effective nonlinear stress-vs.-strain responses of laminates as well as initial and final ply failure envelop predictions. Comparison of these predictions with the actual experimental data will be made in a companion paper, Ref. [34], in Part C of the Worldwide Failure Olympics Exercise. References [1] Hinton MJ, Soden PD. Predicting failure in composite laminates: background to the exercise. Compos Sci Technol 1998;58(7):1001. [2] Chou PC, Carleone J, Hsu CM. Elastic constants of layered media. J Compos Materials 1972;6:80–93. [3] Richard RM, Blacklock JR. Finite element analysis of inelastic structures. AIAA Journal 1969;7:432. [4] Chamis CC. Failure criteria for filamentary composites. Testing and Design, ASTM STP 1996;490:336–460. [5] Sandhu RS. A Survey of failure theories of isotropic and anisotropic materials. AFFDL-TR72–71, AD756889, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, OH, USA, 1972. [6] Soni SR. A comparative study of failure envelopes in composite laminates. J Reinf Plast Compos 1983;2:34–42. [7] Tsai SW. A survey of macroscopic failure criteria for composite materials. J Reinf Plast Compos 1984;3:40 62. [8] Nahas MN. Survey of failure and post failure theories of laminated fibre reinforced composites. J Compos Technol Res 1986;8:138–53. [9] Gotsis PK, Chamis CC, Minnetyan L. Prediction of composite laminate fracture: micromechanics and progressive fracture. Compos Sci Technol 1998;58(7):1137. [10] Eckold GC. Failure criteria for use in the design environment. Compos Sci Technol 1998;58(7):1095. [11] Edge EC. Stress based grant-sanders method for predicting failure of composite laminates. Compos Sci Technol 1998;58(7): 1033. [12] Mccartney LN. Predicting transverse crack formation in crossply laminate. Compos Sci Technol 1998;58(7):1069. [13] Hart-Smith LJ. Predictions of the original and truncated maximum-strain failure models for certain fibrous composite laminates. Compos Sci Technol 1998;58(7):1151. [14] Hart-Smith LJ. Predictions of a generalized maximum-shear-stress failure criterion for certain fibrous composite laminates. Compos Sci Technol 1998;58(7):1179. [15] Puck A, Schurmann H. Failure analysis of FRP laminates by means of physically based phenomenological models. Compos Sci Technol 1998;58(7):1045. [16] Rotem A. Prediction of laminate failure with the rotem failure criterion. Compos Sci Technol 1998;58(7):1083. [17] Sun CT, Tao JX. Prediction of failure envelopes and stress/strain behavior of composite laminates. Compos Sci Technol 1998; 58(7):1125. [18] Lui KS, Tsai SW. A progressive quadratic failure criterion for a laminate. Compos Sci Technol 1998;58(7):1123. [19] Wolfe WE, Butalia TS. A strain-energy based failure criterion for nonlinear analysis of composite laminates subjected to biaxial loading. Compos Sci Technol 1998;58(7):1107. [20] Zinoview P, Grigoriev SV, Labedeva OV, Tairova LR. Strength of multilayered composites. Compos Sci Technol 1998;58(7):1209. [21] Soden PD, Hinton MJ, Kaddour AS. Lamina properties, lay-up configuration and loading conditions for a range of fibre reinforced composite laminates. Compos Sci Technol 1998;58(7): 1011.

426 [22] Soden PD, Hinton MJ, Kaddour AS. A comparison of the predictive capabilities of current failure theories for composite laminates. Compos Sci Technol 1998;58(7):1225. [23] Bogetti TA, Hoppel CPR, Drysdale WH. Three-dimensional effective property and strength prediction of thick laminated composite media. ARL-TR-911, US Army Research Laboratory, Aberdeen Proving Ground, MD, October 1995. [24] White JE, Angona FA. Elastic wave velocities in laminated media. J Acous Soc Am 1955;27:311. [25] Postma GW. Wave propogation in a stratified medium. Geophysics 1955;20:780. [26] Rytov SM. Acoustical properties of a thinly laminated medium. Soviet Phys Acoustics 1956;2:68. [27] Behrens E. Sound propagation in lamellar composite materials and averaged elastic constants. J Acous Soc Am 1967;42:378. [28] Salamon MDG. Elastic moduli of stratified rock mass. Int J Rock Mech Min Sci 1968:519. [29] Sun CT, Liao WC. Analysis of thick section composite laminates using effective moduli. J Compos Materials 1990;24:977. [30] Vinson JR, Sierakowski RL. The behavior of structures composed of composite materials. Dordrecht, The Netherlands: Martinus Nijhoff Publishers; 1986. [31] Hoppel CPR, DeTeresa SJ. Effect of angle-ply orientation on compression strength of composite laminates. US Army Symposium on Solid Mechanics Proceedings, Myrtle Beach, SC 14 April 1999. [32] Bogetti TA, Hoppel CPR, Burns BP. LAMPAT: a software tool for the analysis and design of thick laminated composite structures. US Army Research Laboratory Technical Report, ARLTR-890, September, 1995. [33] Whitney JM. Structural analysis of laminated anisotropic plates. Lancaster, PA: Technomic Publishing Co; 1987. [34] Bogetti TA, Hoppel CPR, Harik VM, Newill JF, Burns BP. Predicting the nonlinear response and failure of composite laminates: correlation with experimental results. Compos Sci Technol, V64, pp 477–485, 2004, see also Chapter 5.12 in this book.

Appendix Lamina stiffness matrix coefficients

In this Appendix the lamina stiffness matrix coefficients are defined in terms of the lamina engineering constants and ply orientations. In the following descriptions, the primed notation will be used to denote the principal material coordinate system while the barred notation will be used to denote the global material coordinate system. By definition, the three-dimensional Hooke’s Law linear-elastic stress strain constitutive relation for an individual lamina is written in the following contracted form i k = C ijkj k

for (i, j = 1, 2, 3, 4, 5, 6)

(A1)

427 where C ijk represents the lamina stiffness matrix defined in the principal material coordinate system. The orthotropic lamina stiffness matrix is symmetric (i.e., C ijk = C kji for i, j = 1, 2, 3, 4, 5, 6) and takes the following form [33] C 11 C 12 C 13 0 0 0 0 0 C 12 C 22 C 23 0 0 0 C 13 C 23 C 33 0 C ij k = 0 0 0 0 C 44 0 0 0 0 0 C 55 0 0 0 0 0 0 C 66

(A2)

The non-zero stiffness coefficients of the lamina stiffness matrix coefficients are defined in terms of the lamina engineering constants according to C 11 = (1  v223E3 /E2)E1 /V C 12 = (v12 + v13v23E3 /E2)E2 /V C 13 = (v13 + v12v23)E3 /V C 22 = (1  v213E3 /E1)E2 /V C 23 = (v23  v12v13E2 /E1)E3 /V C 33 = (1  v212E2 /E1)E3 /V C 44 = G23 C 55 = G13 C 66 = G12

(A3)

where V = 1  v12(v12E2 /E1 + 2v23v13E3 /E1)  v213E3 /E1  v223E3 /E2 To define lamina stiffness coefficients in the global laminate system, transformation matrices for ply stress and ply strain between the principal (1, 2, 3) and global (x, y, z) coordinate systems is first considered. The global (barred) ply stresses, ¯ ki, can be expressed explicitly in terms of the principal ply stresses, i k, and the ply orientation angle,  (see Fig. 1). Mathematically, this transformation is accomplished with the following second-order tensor transformation ¯ ki[T()]ij j k

(A4)

428 where the stress transformation matrix is given by n2 m2 m2 n2 0 0 [T()]ij = 0 0 0 0 –mn mn

0 0 0 2mn 0 0 0 –2mn 1 0 0 0 0 m –n 0 0 n m 0 0 0 0 (m2  n2)

(A5)

and where m = cos  and n = sin . Similarly, global ply strains are obtained according to

¯ ki = [T()]ijj k

(A6)

where the strain transformation matrix is given by m2 n2 2 m2 n 0 0 [T()]ij = 0 0 0 0 –2mn 2mn

0 0 0 –mn 0 0 0 mn 1 0 0 0 0 m –n 0 0 n m 0 0 0 0 (m2  n2)

(A7)

The lamina stress–strain constitutive relationship, defined in the global (x, y, z) laminate coordinate system, is written explicitly as ¯ kj = C¯ kij ¯ kj for (i, j = 1, 2, 3, 4, 5, 6).

(A8)

Combining Eqs. (A1)–(A8), it can be shown that the lamina stiffness matrix elements, C¯ kij, can be expressed explicitly in terms of the principal lamina stiffness matrix elements and the ply orientation angle, , through the following expression C¯ kij = [T()]ijC ijk{[T()]ij}–1

(A9)

Through Eqs. (A3), (A5), (A7) and (A9), the lamina stiffness matrix elements can be explicitly expressed in terms of the lamina engineering constants and ply orientations.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

429

CHAPTER 3.14

The predictive capability of failure mode conceptbased strength criteria for multidirectional laminates R. G. Cuntze* and A. Freund Main Department ‘Analysis’, MAN Technologie AG, Franz Josef Strauß Str. 5, D-86153 Augsburg, Germany Abstract This contribution is a post-runner to the ‘failure exercise’. It focuses on two aspects of the theoretical prediction of failure in composites [1–3]: the first is the derivation of failure conditions for a unidirectional (UD) lamina with the prediction of initial failure of the embedded lamina. and the second the treatment of non-linear, progressive failure of 3-dimensionally stressed laminates until final failure. The failure conditions are based on the so-called Failure Mode Concept (FMC) which takes into account the material-symmetries (by the application of invariants) of the UD-lamina homogenized to a ‘material’, and on a strict failure mode thinking. The results of the investigation are stress–strain curves for the various given GFRP-/CFRP-UD-laminae, biaxial failure stress envelopes for the UD-laminae, and initial as well as final biaxial failure envelopes for the laminates. In addition a brief comparison between Puck’s and Cuntze’s failure theory is presented by the authors themselves. Keywords: B. Non-linear behaviour; Multi-axial stressing. Multidirectional laminates: C. Failure criterion

Nomenclature In the nomenclature, self-explaining symbols are used if a property is addressed. A lamina (is defined to be the calculation unit) may consist of several physical layers. Unidirectional lamina as, bs b⊥, b⊥储, b⊥储 E1 = E储, E2 = E3 = E⊥

Ramberg/Osgood parameters in softening regime Curve parameters Elastic moduli of a UD lamina in the directions x1, x2, x3

* Corresponding author. Tel.: + 49-821-505-2593; fax: + 49-821-505-2630. E-mail addresses: [email protected] (R. G. Cuntze).

430 E1(tan), E3(sec) Eff (res) Eff (mode) maxEFF(mode) e1t储 , ec储 F 储 , F 储 , F ⊥, F ⊥, F⊥储 f (Mode) Res

fE f (res) Res G21, G21(sec) I1, I2, I3, I4, I5 MS m ˙ Rp0.2 R t储 ⬅ X t, R c储 ⬅ X c R t⊥ ⬅ Y t, R c⊥ ⬅ Y c R⊥储 ⬅ S ¯ R R, vf x1, x2, x3

1, 2, 3 v12

1, 2, 3 c2, t1 储, ⊥ , ˆ ˆ {}(L), {}(R) (mode) eq 12 = 21, 13 = 31, 23 = 32 ⊥储, ⊥⊥

A tangent and a secant elastic modulus Resultant stress effort of all interacting failure modes. Corresponds to Puck’s exposure factor fE Stress effort of a UD-lamina in a failure mode, e.g. 储eq/R c储 = Eff 储 Corresponds to 1/ f 储Res if linear behaviour Stress effort of the maximum stressed failure mode Tensile and compressive failure strain of a UD-lamina in x1 direction Failure functions for FF and IFF Reserve factor = stretching factor for the applied stress state necessary to achieve the failure stress state of the mode, e.g. t ⊥ f ⊥ Res = R ⊥/eq = 1 Stress exposure factor of Puck Resultant reserve factor of all interacting failure modes Shear modulus of a UD lamina in the x2, x1 direction, secant shear modulus Invariants of the transversally-isotropic UD-material Margin of safety = fRes  1 Mode interaction coefficient Stress value at 0.2% plastic strain UD tensile and compressive (basic) strength parallel to the fibre direction UD tensile and compressive strength transverse to the fibre direction Shear strength of a UD lamina transverse/parallel to the fibre direction Mean strength, design allowable Volume fraction Coordinate system of a unidirectional (UD-)lamina (x1 = fibre direction, x2 = direction transverse to the fibre, x3 = thickness direction) Normal strains of a unidirectional lamina Major Poisson’s ratio in the ‘failure exercise’ (corresponds to v⊥储 in the German guideline VDI 2014. There is no rationale for v12 or v21. In the early times the application of v21 was preferred because this denotation makes more sense (location first, cause second) Normal stresses in a unidirectional layer Compressive stress across, tensile stress in fibre direction Stresses parallel and transverse to the fibre direction Laminate mean stresses Load-dependent stresses; residual stresses 储 ⊥ ⊥ ⊥储 Equivalent stresses of a mode (储 eq, eq, eq , eq , eq ), includes load induced mechanical stresses and residual stresses Shear stresses of a unidirectional lamina in the elastic symmetry directions. The first subscript locates the direction normal to the plane on which the shear stress is acting; the second subscript indicates the direction of the shear force Shear stressing transverse/parallel and transverse/transverse to the

431

12 = 21, 13 = 31,

23 = 32

fibre direction Shear strains of a unidirectional layer

Characteristics of the fibres E1f 1f, 2f

Elastic modulus in x1 direction Stress in x1 direction; stress in x2 direction

Potential fracture plane (for the comparison Puck–Cuntze) R (⊥+ )A, R (⊥ )A R A⊥储 R A⊥⊥ x1, xn, xt n, nt, ␶nl  fp

Fracture resistance of the action plane against its fracture due to transverse tensile and compression stressing. They correspond to strength values R1⊥, Rc⊥. Fracture resistance of the action plane against its fracture due to transverse/parallel shear stressing: = R⊥储 Fracture resistance of the action plane against its fracture due to transverse/transverse shear stressing Coordinate system rotated with respect to the fibre direction by an angle  from the x2 direction to the xn direction Normal stress, normal/longitudinal shear stress, normal/transverse shear stress acting on the potential fracture plane (Mohr–Coulomb stresses) Angle between the x2 axis and the xn axis Angle of the fracture plane

Abbreviations CLT CoV DLL F FEA FF FMC FoS FPF FRP IFF MS

Classical laminate theory Coefficient of variation Design limit load Failure function Finite element analysis Fibre failure Failure mode concept Factor of safety First ply failure Fibre-reinforced plastic Inter-fibre failure Margin of safety

Indices, signs A c, t f, m fp (sec) (res)

indicates an Action plane quantity compression, tension (German Guideline VDI 2014) fibre, matrix fibre-parallel fracture plane secant modulus resultant

432 s Res ˆ ( + ), (  )  , 

Symmetric lay-up, softening Reserve laminate mean stress or average stress of laminate mathematical notations for tension and compression statistical mean indicate the failure induced by the normal or shear Mohr stress

1. Introduction For a reliable Strength Proof of Design of a laminate composed of UD-laminae reliable failure criteria and a reliable progressive failure analysis are needed. The non-linear behaviour of laminates composed of brittle laminae (these are the materials in the ‘failure exercise’) originates from the damage development around inherent defects in the constituent matrix and at the interface fibre-matrix (ductile matrix materials would show necking and so-called crazing, which appears in case of glass fibre composites as whitening in a tensile test). These defects grow to micro-cracks and later to cracks under increased stressing. Therefore, the usually in the ‘plasticity theory’ to be applied global yield failure condition [1–4] which has to be anisotropic here (see Hill’s yield condition with associated flow rule), is to be replaced by a single global or several partial fracture conditions. Also, the so-called associated flow rule (Normality criterion: the subsequent failure surface is indicated by a vector normal to the actual global yield failure surface) is replaced by the idea of proportional stressing, that means, the surface increases in the direction of the actual stressing which is seldom the normal direction. Partial mode-related fracture surfaces will confine the subsequent global anisotropic yield surface piecewise. These fracture surfaces are essentially described by those fracture conditions for the UD lamina (defined here to be the material the laminate consists of) which are matrix-dominated. The development of UD failure criteria and of degradation models for the progressive failure analysis gave rise to activities in Germany. These activities concentrated as far as possible on the improvement of failure criteria and their verification by multi-axial testing with the existing specimens and test rigs [5–11]. Since 1980 a group close to Puck has tried to improve Puck’s ‘old’ criteria [8,9] which already distinguished between the two failure types, fibre failure (FF) and inter fibre failure (IFF). In 1992, Puck [12] eventually established his ‘new’ set of IFF-criteria following an idea, proposed 1980 by Hashin [13], which was based on a modified Mohr/Coulomb theory. From 1992 [8] to 1997, Cuntze [14] as well as others [15–22] focussed on the Puck/Hashin IFF-Strength Criteria, which are based on the determination of the fracture plane. The result of this work was incorporated in the final report of a research project [6]. In parallel, since 1994 Cuntze has investigated invariant-based formulations of strength criteria for isotropic and anisotropic materials [23–29]. Cuntze’s main idea is not the basement on invariants but the strict allocation of a strength criterion to one failure mode and to one associated basic strength. While Cuntze was studying the invariant-based v. Mises yield criterion – it describes one (strength) failure mode, the ‘shear yielding’, and allows for the determination of the slip line angles – the question raised to him: Why should it not be possible to formulate for each single failure mode of an anisotropic material an appropriate invariant-based mode failure criterion

433 which might probably (a further condition has to be applied) later allow for postdetermination of the failure angle, if desired? The application of invariants is almost standard for isotropic materials. However there, the main intention is to build up a yield criterion (this means for one single failure mode or one phenomenon) or a global fracture criterion that includes all fracture failure modes occurring in the isotropic case. Such a global criterion has on the one hand numerical advantages because one has to apply only one criterion, but on the other hand, it may lead to erroneous results due to its physical shortcoming because it tries to map several failure modes. Invariant-based failure criteria have been formulated for a large number of isotropic materials. As the first Hashin [13] seems to have postulated (1980) in the same paper, in parallel to his ‘Mohr Coulomb model’-based IFF criteria, invariant-based UD-failure criteria. Based on curve fitting consideration and not on physical reasoning, Hashin chose a quadratic approximation which reads in its general form A1I1 + B1I 21 + A2I2 + B2I 22 + C12I1I2 + A3I3 + A4I4 = 1,

(0)

and which includes six strengths (for the definition of the invariants, see [13]). From the 3D failure criterion above he modelled four distinct failure modes: the tensile and compressive fibre modes and two matrix modes. This results in piecewise smooth failure surfaces which do not fit well (Fig. 3 in [13]). The comparison of Cuntze’s results [29] with Hashin’s formulations show some differences: (1) Hashin’s choice of a single quadratic approximation, (2) two matrix modes, (3) six strengths (Hashin in reality uses R⊥⊥ not Puck’s RA⊥⊥, R⊥⊥ is identical to the strength R⊥t in our actual case of brittle behaviour), and (4) the application of tensile stress 1 combined with longitudinal shear stress 21 (not just the fibre tensile stress alone as with Cuntze or Puck). Only for fibre parallel compression failure Cuntze considers such a contribution of the longitudinal shear stress. But, due to insufficient data for this compression FF also the simple maximum stress criterion is proposed by Cuntze, like Hashin. Since the early 1980s Boehler et al. [30,31] eventually extensively pursued the idea of applying invariant-based criteria which they had partly verified by test. Because this working group did not present their failure criteria in the conventional UD lamina stresses, their valuable results unfortunately did not attract the ‘stress man’. Of course, also Tsai/Wu’s polynomial failure conditions may be transformed into formulations of invariant terms [32]. Later in 1996 Cuntze influenced Jeltsch-Fricker and Meckbach from the University of Kassel to pick up the idea of invariant-based formulations. They approximated the ‘Puck/ Hashin IFF body’ by means of two invariant formulations [33]. The idea of thinking in strength failure modes is not a new idea, but the so-called Failure Mode Concept (FMC) more strictly applies the ‘mode thinking” and more consequently uses the advantage of formulating the failure conditions (interaction of stresses within a mode) by the material symmetries respecting invariants, which contain the lamina stresses of the FEM output. This approach, according to the number of the material symmetries, requires two independent FF modes and three IFF modes. The application to UD material is the most intensive application of Cuntze’s FMC, which is claimed to be applicable to any material [23,26,27]. Each of Cuntze’s five (see also Christensen [34]) failure modes is characterized by one strength and one modulus.

434 The choice of the invariants in Cuntze’s FMC is supported by physical considerations based upon Beltrami. The decision for an individual basic invariant is directed by the fact whether the material element is subjected in the envisaged failure mode to a volume change or a shape change. Cuntze’s previous experience with structural reliability [19,35–38], where failure mode thinking is a basic idea, helped to simply model the interaction of modes within a lamina by the application of a spring model. Cuntze tries to formulate easy-to-handle homogeneous invariant-based criteria with stress terms of the lowest possible order and which make a search of the fracture plane not necessary. The FF criteria are treated as decoupled from the IFF ones. The interaction of FF with IFF is considered probabilistically as within the IFF modes by the spring model mentioned above. Confronted with various questions of the ‘UD failure criteria community’, Cuntze in cooperation with A. Puck tries to outline in this contribution (see Appendix A) the coincidences and main differences of their IFF theories. Puck’s approach uses – as proposed by Hashin – a modified Mohr/Coulomb [39] theory for brittle IFF of unidirectional (transversally-isotropic) laminae. For IFF thereby is an automatic interaction of stresses included due to basing IFF just on the three so-called ‘action plane stresses (n, nt, n1)’. These stresses have a common action plane (Fig. A1). Therefore, these criteria are called ‘action plane strength criteria’. Puck discriminates two fundamental regimes: n > 0 and n < 0. The unknown IFF fracture angle is determined when the action plane of maximum stress effort is ‘found’. The well-known conventional global criteria apply all six stresses of the UD lamina and do not take into consideration whether they might act on the same or on different action planes. It is very simple in the ‘plasticity theory of isotropic materials’ to develop a so-called ‘single yield failure surface’ criterion, that means one global criterion, due to the existence of only one failure phenomenon, the isotropic yielding. A global criterion for fracture may include more than one fracture failure mode potentially occurring under the various stress states. It is sometimes also used instead of a global yield criterion in spite of the facts that a fracture criterion as a mathematical description of the fracture failure surface just confines a global yield failurebody (yield capacity exhausted), and that it usually generates a different failure shape. And for laminae? For them, as already mentioned, instead of a matrix-determined anisotropic global yield criterion a set of fracture criteria on lamina level is applied. These show due to their various failure modes a ‘multi-fold non-linearity’ which requires a much higher effort. A further shortcoming is: A set of failure criteria instead of one global one prevents from a simple implementation into a commercial FEM code in order to take advantage of the code’s solution architecture and pre-/postprocessor capabilities. This point waits to be tackled. Progressive failure analysis [58–60] of laminates or the prediction of laminate behaviour up to fracture is the major challenge compared to the derivation of reliable UD-failure criteria. Cuntze assumes a so-called effective stress–strain curve for the lamina which respects the influence of being embedded [29,40] in the laminate. To be utilized in the non-linear analysis is the secant modulus which alters for a non-linear stress–strain curve. However, data are not only needed for the pure failure mode domains but for the mode interaction domains, too, where the actual stress state affects more than one

435 mode. The influence of the stress state in a mode interaction domain on the secant modulus of each affected mode is considered by a ‘triggering approach’. This approach increases the equivalent stress (which considers all influencing stresses) of the affected mode in the case of hardening (the secant modulus becomes a little smaller) and decreases the equivalent stress in the case of softening (the secant modulus becomes smaller, too). The mode’s equivalent stress–strain curve is identical to the associated uni-axial stress–strain curve measured. A crucial difference between Puck’s [42] approach and Cuntze’s approach is the treatment of degradation in the non-linear-analysis of the laminate. Both theories apply the selfcorrecting secant modulus method, however, describe the successive degradation (the softening) differently as well as the rounding-off in the interaction domains of FF and IFF modes. In the FF–IFF mode interaction domains Puck applies a weakening factor (depending on 1) reasoning that single filament failures have a weakening effect on the resistance against IFF. Cuntze automatically respects this fact by the rounding-off procedure. For more information on the differences and coincidences of Puck’s and Cuntze’s failure theory, see Appendix A. The theoretical background of the following contribution can also be found in the DURACOSYS 99 paper ‘Progressive failure of 3D-stressed laminates: multiple non-linearity treated by the failure mode concept (FMC)’ [29]. The authors hope to add, with this lamina stress-based engineering approach, a ‘physically’-based 3D phenomenological model. 2. Main features of the failure mode concept (FMC) The features of the FMC are briefly summarised in Table 1. Additional aspects are collected in Table 2. These features and some further aspects will be described in the coming sections in more detail. The FMC is a general concept, UD material is one application only. 3. Basics 3.1. State of stress For the unidirectional (UD) material element Fig. 1 depicts the prevailing 3D-state of stress. Additionally, with respect to the symmetries of this transversally-isotropic material (modelled Table 1 Main features of the FMC. • Each mode represents one theoretically independent failure mechanism and one piece of the complete failure surface (surface of the failure body or limit surface) • Each failure mechanism is represented by one failure condition. One failure mechanism is governed by one basic strength and therefore has a clearly defined equivalent stress eq • Curve-fitting of the course of test data is only permitted in the pure failure mode’s regime • Different, however, similar behaving materials obey the same function as failure condition but have different curve parameters • Rounding-off in mode interaction zones is performed by the spring model presented.

436 Table 2 Additional FMC aspects/information. • An invariant formulation of a failure condition in order to achieve a scalar potential considering the material’s symmetries [34] is possible • Each invariant term of the failure function shall be related to a physical mechanism observed in the solid, causing a volume change or a shape change or friction • Hypotheses applied: Hashin/Puck with Beltrami (choice of invariants), Mohr-Coulomb (friction, thinking in Mohr’s stresses) • The rounding-off of adjacent mode failure curves ( partial surfaces) in their interaction zone is leading again to a global failure curve (surface) or to a ‘single surface failure description’ (such as with Tsai/Wu, however without the well-known shortcomings). * Proof of design and strength analysis: – For each mode one reserve factor f (mode) or one stress effort (if nonlinear) is to be determined, Res displaying, where the design key has to be turned – The probabilistics-based ‘rounding-off ’ approach delivers the resultant reserve factor linked to the margin of safety by MS = f (res) Res  1. * Nonlinear stress analysis with Degradation: – Equivalent stresses and stress efforts are used in this (nonlinear) progressive damage description. – Failure mode identification is mandatory for a progressive failure analysis in order to know how the lamina has failed. Criteria which just predict failure do not make a clear degradation of the moduli possible.

an ideal crystal [23,26,34]), the five basic strengths and five elasticity properties are given (Lekhnitskii). A UD-lamina in reality is a low-scale structure with the constituents fibre, matrix and interphase (at the interface). After homogenisation it may be called ‘material’.

Fig. 1. UD lamina (t = tension, c = compression). Stresses, strengths and elasticity properties.

437 3.2. Invariants Strength criteria or failure conditions may be formulated by invariants based on the UDstresses, see [13,30]). Invariants have the advantage that the formulations do not depend on coordinate-system transformations. From the variety of invariants the following forms were chosen to best describe the multiaxial behaviour of the material (the numbering of the invariants is different in the various literature, e.g., I Boehler = I Hashin and I Hashin = –2231  3221 + 2233121). 3 4 5 I1 = 1; (Boehler) I2 = 2 + 3; I3 = 231 + 221; I4 = (2  3)2 + 4223; I5 = (2  3)(231  221)  4233121.

(1)

The sensitivity of I5 to the sign of the shear stresses is suppressed if a ‘main axes transformation’ around the 1-axis is performed (see Fig. 1), leading to 23 = 0. 3.3. Strengths (Cuntze’s view) The characterisation of the strength of transversally isotropic composites requires, according to the FMC – the measurement of five independent basic strengths: R储t, R储c (fibre parallel tensile and compressive strength) as well as R⊥t , R⊥c (tensile, compressive strength transversal to the fibre direction) and R⊥储 (fibre parallel shear strength). R储t is determined by the strength of the constituent fibre and R c储 by ‘shear instability’. The latter includes different micro-failure mechanisms: The matrix may shear under loading and does not stabilise the generally somewhat misaligned fibres embedded within. Hence it comes to bending and ‘kinking’ [41] (structural behaviour). Also, the load grasping fibre as stiffer constituent may shear (this is a constituent’s material behaviour) under c储 and ⊥储. The strength RT⊥ is determined by the relatively low strength properties of the matrix (cohesive failure), by the interphase material in the interface fibre-matrix (adhesive failure caused by a weak fibre-matrix bond), as well as by the fibres acting as embedded stress raisers. 3.4. Rounding-off in the interaction zones Of further interest is the rounding-off of the fracture curve in the mixed failure domain (MiFD) or interaction or transition zone of adjacent failure modes in the envisaged lamina. In [24] a simple probability-based formula – the ‘Series Spring Model’ – as an engineering approach for the resultant reserve factor (which is needed anyway for the proof of design) mode2 1/f (res) + 1/f mode3 +••+ Res = 1/f mode1Res + 1/f Res Res

(2)

was proposed which approximates the results of a time-consuming probabilistic calculation on the safe side. In the case of residual stresses and non-linearity instead of a stress-based reserve factor fRes the stress effort Eff has to be employed.

438 3.5. Classical laminate theory (CLT) (The CLT is addressed here mainly for the reason to depict the definitions and symbols in the German guideline VDI 2014 on ‘Development of FRP components’ sheet 3: Analysis (issued 2003), chosen after many discussions, and which will be employed here. Another reason is given Section 4.2). Assuming transversal isotropy and the state of plane stress (‘in-plane stressing’, 3 = 0, which is the situation of the case studies investigated) the linear stress–strain relations for the kth lamina of a multi-layered laminate are (using the notation; 1 = 储, 2 = ⊥, 12 = 储⊥, [Q ]; [S ]: = stiffness, compliance matrix of the lamina) {}k = (1, 2, 12)Tk = [S ]k{}k and

(3a)

{}k = (1, 2, 12)Tk = [Q ]k{}k.

(3b)

The symmetric elasticity matrix of stiffness (stiffness matrix) of the lamina reads: Q11 [Q ]k = Q21 Q61

Q12 Q22 Q62

Q16 Q26 Q66

E储 1  v储⊥v⊥储 = v⊥储E⊥ 1  v储⊥v⊥储 k 0

v储⊥E储 1  v储⊥v⊥储 E⊥ 1  v储⊥v⊥储 0

0 ,

0 G储⊥

(4)

k

with [Q ]–1 k = [S ]k, v⊥储 · E⊥ = v储⊥ · E储 (Maxwell–Betti law)

(5)

and v⊥储 as the major Poisson’s ratio. Thus, for the application of CLT the knowledge of only four constants is essential: E储, E⊥, G⊥储 and v⊥储 ( ⬅ v12 in ’failure exercise’ [3]). In the case of mechanical loading the following load-strain equations are obtained in the cross section for the load fluxes {n} and the moment fluxes {m°} (moment per unit length of the middle surface)

再 冎 冋 册再 冎 再 冎 n A = m° B

B D

° ° = [K ]  

(6)

with [K ] being the stiffness matrix of the laminate, from which will be utilized the extensional stiffness matrix (see ‘Theory of Laminated Plates’ by Ashton/Whitney [41b])

冘 n

[A] =

[Q ]k · tk, [Q ] = [T][Q ][T]T,

(7)

k=1

and transformation matrices (s = sin , c = cos ) c2 [T] = s2 sc

s2 c2 –sc

–2sc 2sc 2 c  s2

(8a)

439 c2 [T] = s2 2sc

s2 c2 –2sc

–sc sc c2  s2

(8b)

Having determined the strain vector {°} and the curvature vector {} for the middle plane of the laminate, the so-called natural strains {}k (strains in the lamina coordinate system) and stresses {}k in each lamina may be calculated according to {}k = [T]–1 k ({°} + z{}),

(9a)

{}k = [Q ]k{}k.

(9b)

The equations above decouple for a symmetric lay-up to {°} = [A]–1{n}

(10)

If curing stresses have to be considered the equations read {°} = [A]–1({n} + {nT })

(11)

with

冘 n

{nT} =

T [Q ]ktk{ T }k

(12)

k=1

and { T }k = [T ]k{T }k,

(13a)

{T } = (T 储, T⊥, 0)T.

(13b)

In the case of symmetrical lay-ups (test cases of the ‘failure exercise’), for the treatment of material nonlinearity and of degradation, the lamina stresses {}k have to be computed considering {}k = {°}

. . . compatability

{}k = [Q  ]k · ({}k  {T }k)

. . . Hooke

–1 {}k = [T ]–1 k · {}k, {}k = [T]k {}k

(14) (15) (16)

and applying [T]–1 = [T]T,

(17a)

[T]–1 = [T]T.

(17b)

The definitions for the lamina (often called ply if it is a prepreg and layer if it is winding) stresses, angles and thicknesses are illustrated in Fig. 2. The index k of the single lamina will be dropped in the further text.

440

Fig. 2. Laminate and kth lamina subjected to a plane state of stress (mid-plane z = 0).

4. Failure modes and failure conditions of a lamina Failure conditions [49–57] should exhibit – besides a sound physical basis – the numerical advantages: mathematical homogeneity (see F ⊥ in Appendix A4 after the homogenisation) in the stress terms, stress terms of the lowest degree, simplicity, scalar formulations and therefore invariance, numerical robustness and rapid computation. 4.1. Failure modes (types) A designer has to dimension a laminate versus inter-fibre-failure (IFF) and fibre-failure (FF). IFF normally indicates the onset of failure whereas the appearance of FF in a single lamina of the laminate usually marks final failure. In the case of brittle behaving FRP, the failure is a fracture. The IFF incorporates cohesive fracture of the matrix and adhesive fracture of the fibre-matrix interface. Fracture is understood in this article as a separation of material, which was free of damage such as technical cracks and delaminations but not free of tiny defects/flaws (size of microns) prior to loading. Fig. 3 informs about the types of fracture which are recognised in case of ‘dense’ (means: ‘not porous’) transversally-isotropic ideal materials. Whether a failure may be called a shear stress induced shear failure, SF, or a normal stress induced normal failure, NF, depends on the size scale applied. SF⊥储 shows macroscopically shear failure (fracture plane is parallel to 21). However micro-mechanically, it is a 45° normal failure mode of the matrix, caused by tensile matrix stress and visualised by the so-called hackles [41a]. These micro-cracks grow until they touch the next fibre layer where they are turned to later form the basis for the fibre-parallel IFF. The ‘explosive’ effect of a so-called wedge shape failure (a c⊥-caused IFF) of an embedded lamina of the laminate may directly lead [42] to final failure (see a torsion spring) or via local

441

Fig. 3. FMC view of the fracture types ( = failure modes) of brittle transversally-isotropic material. (The physical fracture “planes” are pointed out in the figure [2] fp: = fracture plane angle.) The onset of hackles due to NFm relates IFF2 to IFFI (micro-cracks due to NFm).

delaminations to buckling of the adjacent laminae and therefore to final failure, too. This IFF, may also cause a catastrophic failure like FF.

4.2. Strain energy density basis Beltrami, Schleicher et al. assume at initiation of yield that the strain energy density will consist of two portions. Thus, the strain energy (denoted by W ) in a cubic element of a material reads W=

冕

{}{}d{} = WVol + Wshape.

(18)

Including Hooke’s law in the case of a transversally-isotropic body the expression will take the shape (see Lekhnitskii [42a], sik = compliance coefficients analogue to the 2D formulation of Eq. (3a). See also Ashton/ Whitney [41b]): W = [s1121 + s2222 + s3332 + s4432 + s55(212 + 213)]/2 + s12(12 + 13) + s2323 =

I 21 I 22(1  v⊥⊥) v⊥储I1I2 I3 I4(1 + v⊥⊥) +  + + 2E储 4E⊥ E储 2G储⊥ 4E⊥ volume

volume

volume

shape

shape

(19)

442 Some of the terms above describe the volume change of the cubic material element and others its change of the shape. These changes can be witnessed by the fracture morphology [41a]. In order to formulate a relatively simple failure condition one chooses as basic invariant that term in Eq. (19) which respects whether the cubic material element will experience a volume change in the considered mode or a shape change.

4.3. Failure conditions achieved In engineering application due to property scatter the simplest strength criteria which still describe the physical effects should be applied. This always reduces the number of curve parameters (inherent in the failure criteria) to be determined and, besides this, the numerical effort. Applying the FMC in total three (statistically-based) calibration points at maximum have to be experimentally determined besides the basic strengths serving as anchor points in each mode failure domain. Based on the idea above the following failure conditions, F({}) = 1 have been derived FF1:

F 储 =

I *1 = 1, R¯ t储

FF2:

F 储 =

–I1 = 1, R¯ c储

IFF1:

F ⊥ =

IFF2:

I2I3  I5 ¯3 F⊥储 = I 3/2 = 1, 3 /R ⊥储 + b⊥储 R¯ 3⊥储

IFF3:

F ⊥ = (b⊥  1)

I2 + 兹I4 =1 2R¯ t⊥

(20)

I2 b⊥I4 + b⊥储I3 + =1 R¯ c⊥ R¯ c2 ⊥

with three free curve parameters (b⊥储, b⊥, b⊥储) to be determined from multi-axial test data: (R¯ marks mean strength value. *Mind: 1 → vf · 1f = vf · 1 · E1f = 1 · E t储 with 1f = tensile stress fibre and vf = fibre volume fraction. The very small load-carrying capacity of the matrix is > neglected here in relation to the fibre’s. F =< 1 is called criterion). Each of them has to be calculated from a test point (several measurements) or by curve fitting of the course of test data in the associated pure domain. The (calibration points 䊐 in Figs. 4 and 5 deliver, after inserting them into the equations IFF2 and 3 and a further resolution, the equations ¯ 2 1  (⊥储 21 /R⊥储) from (c2, ⊥储 21 ) c ⊥储2 ¯ 3 22 · 21 /R ⊥储

(21a)

c ¯ c 1 + (c 2 + 3 )/R ⊥ c ¯ c c c 2 ¯ c2 (c 2 + 3 )/R ⊥ + (2  3 ) /R ⊥

(21b)

b⊥储 = b⊥ =

 c ¯ 2 ¯ b⊥储 = 1  (b⊥  1)c 2⊥储/R⊥储  b⊥(2⊥储/R⊥储)

(21c)

443

Fig. 4. Visualization of the reserve factor and computation example. {}(L) = load stress vector, m ˙ = rounding-off exponent [Eq. (45)]. (No curing stresses.)

for the parameter determination. The parameters depend on the material behaviour and on the IFF formulation applied. Bounds on the safe side for GFRP, CFRP and AFRP are assumed to be 0.05 < b⊥储 < 0.15,

1.0 < b⊥ < 1.6,

0 < b⊥储 < 0.4.

The extreme value b⊥储 = 0 means ‘no bulge effect’ and b⊥ = 1 means ‘no friction’ in the ⊥⊥-plane. Above bounds for the parameters and later the mapping of the failure curves are based on multi-axial test data cited in literature [5–7] or carried out at MAN. The author’s experience shows: often, b⊥储 = 0 will map the lamina test data well enough. The skill has to be put into m ˙ as a rounding coefficient on the safe side. Data for the computation of b⊥储 (Fig. 4) are numerous, b⊥储 = 0.1 is a good approach. As calibration points for b⊥ are still missing in the transversal-isotropic domain knowledge from brittle isotropic material is applied which will keep the engineer in the compression domain on the safe side by assuming b⊥ = 1. In the following text the reasons are depicted for the application of Which invariant? and of Which form of invariant? (success check was the mapping of the available multi-axial data): • F 储 : According to the FMC, F 储 originally consists of a quadratic term in stresses. However, being the only (basic) term, the quadratic term can be replaced by a numerically simpler linear term which regards that the fibre tensile stress and not 1 (the UD ‘material’ model does not hold here) has to be applied if formulating a failure condition. Eq. (20a) indicates that for FF not t1 has to reach the value for the UD-strength R t储 but 储 · E储. Why? Poisson’s

444

Fig. 5. Scheme ‘Global Fit’ and ‘Mode Fit’. Example: CFRP-IFF curve of UD-material. (MfFD = mixed failure domain = fracture due to two modes. MfFD = multi-fold failure domain of the same mode “Normal Fracture” NF⊥ working twice. A-curve: 99% reliability, 95% confidence [34]; B-curve: 0%/95%, mean-curve: 50%/50%. In A-, B-design space: R¯ → “design allowable” R.

445

•

•

•

•

effect is not negligible, because a compressive lamina stress state (c2, c3) will cause tensile fibre stress. F  = 1 theoretically may be reached even without a load stress t1! F 储 : Again the basic term is I 21. For reasons of simplicity and due to lacking of test data in the (c2, c3) domain, a shear addressing invariant I3 (reflecting some Wshape) was not considered in F 储 . By this, the I 21 could be reduced to the linear basic term I1. F⊥储: Basic term is I3. The choice of the failure condition is strongly affected by the ‘easy to be used’ desire and by an easy determination fRes, which is simplified if F ({}) is a socalled homogeneous function wherein the stress terms are of the same power (grade). ¯3 Therefore I 3/2 3 /R ⊥储 was applied, instead of a quadratic formulation which was used in the past, thus leading to homogeneity of F⊥储. The term I2I3  I5 is the result of an intensive analytical ‘trial and error search’ of the first author. It respects the different interaction of the stress combinations (2, 21) and (2, 31) a typical material asymmetry at first described by Puck and proven by test [6] (not considerable by Tsai/Wu). A numerical problem existing in F⊥储 has to be mentioned: If b⊥储(I2I3  I5) becomes I 3/2 3 , then the 21(2)-curve in Fig. 4 turns to infinity. In order to generally bypass this difficulty one has to put a query in the program and replace, if applicable, the formulation of the off-turning F⊥储 curve by a limiting ‘horizontal’ line defined by the constant max I 3/2 3 (see Appendix A4). This is very simply done for the 2D test cases. F ⊥: After another intensive search the really straight line in the quasi-isotropic (2, 3)plane could be mapped by employing I2 + 兹I4 in F ⊥. (Is the section line with a hyperbola. Known from isotropy). F ⊥: In F ⊥, besides the basic term I4, the linear term I2 was applied which considers friction. If really necessary, a term I3 may additionally be taken aiming at a better numerical rounding-off in the (F ⊥, F⊥储)-interaction zone.

On the other hand, if I3 is not applied, that means sticking to the basic FMC (b⊥储 = 0), F 1 can c be homogenized, too, by replacing I4 /R c2 ⊥ by 兹I4 /R ⊥. This will lead from a parabola (is in the negative domain already almost straight in Fig. 5 for the parabolic formulation) to a straight line for c2(c3), and shall be the authors’ engineering choice in future (see Appendix A4). With respect to the 3D character of the IFF conditions above they may serve also as criteria for the onset of delamination (F ⊥: wedge failure, F ⊥: transversal tensile failure) generated by the interlaminar stresses 3, 32, 31. Hydrostatic compressive and tensile stressing is automatically considered. One has to keep in mind: one or two modes will be the design driving ones in a local ‘material’ point of a composite’s lamina. The basic strength of the mode-related linear or nonlinear stress–strain curve controls the (size) volume of the mode failure surface (body) being one part of the global failure surface (body). Curve parameters are representing an effect, such as friction (b⊥) in the material. They control the shape of the mode failure surface.

(res) 5. Reserve factors f (mode) Res , f Res of the lamina

5.1. General Reserve factors which have to be determined for the Proof of Design of each lamina in the laminate are defined load-related. These are:

446 • for the initial failure, indicated by the so-called knee in the laminate’s stress–strain curve and originated by F ⊥, F⊥储 in the laminae f initial Res =

initial failure load jp0.2 · DLL

(22a)

• for the final failure, indicated by F 储 , F 储 or F ⊥, f final Res =

final failure load jult · DLL

(22b)

with DLL = Design Limit Load and jp0.2, jult = design factors of safety (FoS). The various failure loads to be inserted into Eqs. (22) are either a result from experiment or from analysis (applying a failure criterion). In linear analysis the reserve factor fRes is normally defined that factor all mechanical loadinduced stresses applied to the laminae have to be multiplied with in other to generate failure. Geometrically it means that the stress vector {}(L) has to be stretched in its original direction by this factor in order to cause failure. This visualisation is valid as far as linear modelling can be applied: If there are no residual stresses and high design FoS, j, then a linear elastic modelling is permissible and a stress-based fRes can be predicted. In case of non-linear behaviour accurate reserve factors have to be referred to loads, which is in accordance to the fact that load FoS are given. Analysis provides via the failure criterion with the modes’ equivalent stresses eq and stress efforts Eff outlining the remaining load capacity for the computation of the resultant reserve factor. The value of the reserve factor then is the ratio fRes =

failure load at Eff (res) = 1. j · DLL

As failure load is often taken, the maximum load achieved when computation stops due to numerical problems. Nonlinear analysis in general means stress redistribution in the structure. This lowers the stress level of the ‘hot spots’ in the laminae (defined material) of the laminate.

5.2. Determination of mode reserve factors If linear analysis is permitted: • Case “No residual stresses”: {}(L) = f ( j · DLL) {}failure = fRes · {}(L) = {}(L) + MS · {}(L) with the margin of safety MS = fRes  1. • Inserting the above definition into the failure condition F = F ({}failure) = F ( fRes · {}(L) = 1 yields an equation for the stress-based fRes fRes · ᐉ(L) + f 2Res · q(L) + f 3Res · c(L) + . . . = 1.

(23)

447 Special example: The failure condition only has linear and quadratic stress terms. Then the reserve factor can be calculated [23] by resolving for fRes as of a polynomial a root which delivers (mind: R¯ → R in the case of Proof of Design) fRes = 1/ᐉ(L)(e.g. = 1/(vf · I1f /R t储)) . . . linear

(24a)

fRes = (– ᐉ(L) + 兹ᐉ2(L) + 4q(L))/2q(L) . . . quadr.

(24b)

with ᐉ(L) = linear terms, q(L) = quadr. terms. • Case “With residual stresses” (linear modelling) {}failure = fRes · {}(L) + {}(R).

(25)

In the case of linear terms, after substitution of the failure causing state of stress one yields F = F ({}failure) = F ( fRes · {}(L) + {}(R)) = 1

(26)

with {}(R) from curing stresses’ computation etc. This procedure can be applied as long as the residual stresses have not caused an essential amount of damage which would lead to stress-redistribution and a reduction of the size of the residual stresses.

5.3. Determination of resultant reserve factor (rounding-off of failure modes) The (resultant) Reserve Factor (superscript res) takes account of the interactions of all modes. In case of linearity it may be estimated (Fig. 4 just includes the relevant interacting modes) by the rounding-off equation or spring model m ˙ (modes) m ˙ ⊥储 m ˙ ⊥ m ˙ 储 m ˙ 储 m ˙ (1/f (res) ) = (1/f ⊥ Res ) = f ( f Res Res) + (1/f Res) + (1/f Res) + (1/f Res) + (1/f Res)

(27)

with m ˙ as the rounding-off exponent, which fits test data. As a simplifying assumption m ˙ is taken the same for each interaction zone. The value of m ˙ has to be set by fitting experience and by respecting the fact that in the interaction zones micro-mechanical and probabilistic effects will commonly occur and cannot be discriminated. From numerical reasons m ˙ should be an odd number between 2.5 and 4. If inserting a unidirectional fracture stress (this is the strength value) into the equation above, then a failure curve or a failure surface described by f (res) Res = 1 is achieved. Fig. 5 refers to the (2, 3)-plane as one failure plane of the various ones. In the upper part it visualizes the evaluation of test data and in the bottom part the rounding-off (by the spring model) in the multi-fold (MfFD) and mixed failure domains (MiFD) as well as the shrunk design space (mean strength R¯ of mapping is replaced by a strength design allowable R) to be used by the designer in the ‘dimensioning’ and in the ‘proof of design’. The rounding shown in the Figs. 4 and 5 seems to exclude the FF modes. These modes, however, have no relevant interaction with the failure curves 21(2) and 2(3). Additional to the FMC-based ‘Mode Fit’ the ‘Global Fit’ (e.g. Tsai/Wu’s ‘single failure surface’ criterion describes a global failure surface or body) is pointed out. The Global Fit interacts between the UD-stresses and includes independent failure modes in one equation, achieving a description of the global (complete) failure surface. This procedure is simple, however error-prone in some domains, due to its physical shortcomings.

448 In order to consider failure probability or the multifold failure chances in the (t2, t3)m ˙ domain (MfFD) the term (1/f ⊥ Res) has to be made ‘2-fold’ effective. A simple numerical way to implement this is by including in Eq. (27) (see Fig. 5), m ˙ MfFD m via (1/f ⊥ )˙, Res) + (1/f ⊥

the multi-fold failure term [43]. f MfFD = 2R t⊥/(t2 + t3) ⊥

(28)

Eq. (28) is applied only, if test data mapping makes it necessary. The experimental behaviour of brittle isotropic materials justifies the MfFD rounding in the quasi-isotropic plane of the UD-lamina. In the following set of formula the so-called equivalent stress of each mode is applied. This stress includes all load stresses and residual stresses which are acting together in a mode equation.

5.4. Application to the UD-lamina (3D-conditions) The Mode Reserve Factors explicitly read f (mode) = R mode/(mode) , Res eq

generally

ˆ R t储/(1 · E t储) = R t储/储eq; • f 储 Res = • f

储 Res

c 储

c 储

储 eq

= –R /1 = –R / ;

• f ⊥ Res = • f ⊥ Res =

2R t⊥ I2 + 兹I4

=

R t⊥ ⊥ eq

R c⊥ (–b⊥  1)I2 + 兹(b⊥  1)2I 22 + 4b⊥储I3) · 2 b⊥I4 + b⊥储I3

3/2 1/3 • f ⊥储 Res = R⊥储/(I 3 + b⊥储(I2I3  I5))

(29) (30a) (30b) (30c)

(30d) (30e)

Remark. If f (mode) becomes negative, caused by the numerically advantageous automatic Res insertion of {} = (1, 2, 3, 23, 31, 21)T as FEM output into Eq. (30), a value of + 100 shall replace the negative value. A negative value e.g. results if a positive 1 (better 11E储) is inserted into Eq. (30b). For an effective design the stress engineer is to be provided with a table which indicates the design driving mode reserve factors (an example: see Appendix A3).

6. Equivalent stress, mode effort and effective secant moduli In the case of small FoS (e.g. in spacecraft) just nonlinear analyses will enable the stress engineer to predict the stress effort and then the load-based fRes. The actual stress effort of a mode, Eff (mode), is the actual portion of the maximum 100% achieved at mode failure. The procedure of determining the resultant stress effort Eff (res) in each lamina of the laminate is similar to that of f (res) Res . The stress effort (Puck calls it stress exposure factor fE) can be related

449 to the reserve factor in case of linear behaviour and zero residual stresses, that means on stress level, by Eff (res) = 1/f (res) Res .

(31)

Also similar to the ‘f (res) Res procedure’ at first the equivalent stress vector 储 储 ⊥ ⊥ ⊥储 T {(modes) equiv. } = ( eq,  eq,  eq ;  eq ;  eq )

(32)

will be computed. It includes the equivalent stress of each mode of the lamina and within the nonlinearly load dependent load stresses {}(L) and the equally nonlinearity-dependent residual stresses {}(R) from curing etc. Consequently the resultant stress effort is represented by

冘 5

Eff

(res)m˙

=

储 ¯ c m ˙ ⊥ ¯ t m ˙ ⊥ c m ˙ ⊥储 ¯ m ˙ ¯ t m˙ Eff (modes) = ( 储 (33) eq /R 储) + ( eq/R 储 ) + ( eq /R ⊥) (eq /R ⊥) + (eq /R⊥储)

1

of Puck (see Appendix A) with Eff (modes) corresponding to some extent to the f (domains) E In case of fracture stresses holds, analogous to f (res) Res = 1, Eff (res) = 1 = 100%.

(34)

Usually in the laminae of a laminate, multi-axial states of stress are acting which have an impact on more than one of the failure modes. Because in the interaction zones adjacent failure modes are commonly affected, a corresponding degradation (displayed by a stiffness reduction) has to be considered by a drop in the secant moduli applied in the non-linear analysis. A ‘triggering’ of the adjacent equivalent stresses takes into account this effect for each of the associated moduli. As ‘triggering approach’ is recommended (see also Appendix A1.2): • for increasing stress (Hardening)  > 0 TeF correq> (mode) = (mode) eq

(35a)

being a modulus decrease • for decreasing stress (Softening)  < 0 /TrF correq< (mode) = (mode) eq

(35b)

being a modulus decrease, with the trigger-factor TrF = Eff (res)/max Eff (mode).

(36)

In these equations the stress effort of the maximum stressed mode governs the ‘triggering’ and TrF is dedicated to all IFF modes affected. As Eq. (36) leads to a sharp decay, a damped m ˙ triggering according to newTrF = 兹TrF is proposed for the future.

450 This approach has to be verified – before general acceptance – for all possible stress combinations, of course (see also Appendix A1.2).

7. Description of non-linearity Non-linear behaviour [58–60] of well-designed composites is most often physically (laminae behaviour) but rarely geometrically (laminate behaviour) caused. A full 3D-input in stress analysis demands for five elastic properties in the case of Fibre Reinforced Plastics (FRP) and in strength analysis for five strengths. In the 2D-case the required input consists in four elastic properties and five strength properties. Further, for the non-linear stress analysis the relevant non-linear stress–strain curves are to be provided, which should discriminate the so-called hardening and the softening (Fig. 6). Material hardening is the domain until the stress reaches its strength value Rm which addresses here an initial failure level of IFF type. From that level on, that means for the progressive failure or damage regime, the term softening is used. Of course, some damaging already begins with material hardening.

7.1. Mapping of hardening The degree of non-linearity essentially depends on the nonlinearly behaving matrix material which affects E c⊥ and G储⊥. For the secant moduli to be applied in the non-linear stress analysis the following values are determined by the Ramberg/Osgood equation which maps the course of non-linear stress–strain data very well (with E(o) the initial tangent modulus)

 = /E(o) + 0.002 (/Rp0.2)n with the Ramberg/Osgood exponent (see [5])

Fig. 6. Mapping of measured stress–strain of an isolated UD-specimens. [Eqs. (37) and (40)]. [Example t21( 21)].

(37)

451 n = ᐉn(pl(Rm))/ᐉn(Rm /Rp0.2)

(38)

estimated from the strength point Rm, pl(Rm). Data for the secant moduli of E⊥, G储⊥ are provided from above Ramberg/Osgood mapping of test data course (denotations see Fig. 6) by E(sec) = E(o) /(1 + 0.002 · E(o) /Rp0.2 · (/Rp0.2)n  1)

(39)

7.2. Mapping of softening Above the Initial Failure level an appropriate progressive failure analysis method has to be employed (or a Successive Degradation Model for the description of post initial failure) by using a failure mode condition that indicates failure type and damage danger (level of stress effort). Final Failure occurs after the laminate (and thereby the structure) has experienced a stiffness reduction and has degraded to a level where it is no longer capable of carrying additional load. Figure 6 depicts hardening with softening. In detail: for an isolated e.g. tensile coupon specimen (a) in the usual load controlled test, (b) in a strain controlled test. A measurement of curve (b) would be possible at the institute BAM in Berlin, which possesses a MTS test rig of a very high frame stiffness (load goes via the stiff frame and not via the relatively weak specimen), however, tests have not yet performed. The curve (b) is assumed here due to the lack of experimental data from there. Modelling of Post Initial Failure behaviour of a laminate requires that assumptions have to be made regarding the decaying elastic properties of the actually degrading embedded lamina [curve (c) in Fig. 7]. E c⊥ and G储⊥ are decreasing gradually rather than being suddenly annihilated. A rapid collapse (often named ‘ply discount method’) of E t⊥ is unrealistic and probably further leads to convergence problems. A simple function was used to map this softening, in order to later derive the secant moduli. It generally reads (the suffix s denotes softening) s = Rm /(1 + exp[(as + )/bs])

(40)

with two curve parameters as, bs to be estimated by the data of two calibration points, e.g. (Rm, (Rm))

and

(Rm · 0.5, (Rm · 0.5)).

(41)

The above softening function [Eq. (40)] practically models the stress–strain curve of a lamina which is embedded in a laminate, and thus, it includes the effect of the altering microcrack density up to the critical damage state (CDS). Curve (c) is therefore an effective curve. 7.3. Constraint effect on an embedded lamina If applying test data from tensile coupons to an embedded lamina in a laminate, one has to consider that tensile coupon tests deliver test results of weakest link type (series model). An embedded [40] or even an only one-sided constraint lamina, however, belongs to the class of redundant type behaviour, to the ‘parallel spring model’ type. Due to being strain-controlled the material flaws in a thin lamina cannot grow freely up to micro-crack size in thickness direction, because the neighbouring laminae will act as micro-crack-stoppers (problem of energy release in fracture mechanics).

452

Table 3 Mechanical and thermal properties of the four UD-laminae of the ‘failure exercise’ [Sod98]d. Fibre type

AS4

T300

E-glass 21xK43Gevetex

Silenka E-Glassl200tex

Matrix

3501-6 ep.

BSL914C ep.

LY556/HT907/DY063 epoxy

MY750/HY917/DY063 epoxy

Specification Manufacturer Fibre volume fraction, Vf Longitudinal modulus (GPa) E储 Transverse modulus (GPa) E⊥ In-plane shear modulus (GPa) G储⊥ Major Poisson’s ratio v Through thickness Poisson’s ratio v⊥⊥ Longitudinal tensile strength (MPa) R t储 Longitudinal compressive strength (MPa) R c储 Transverse tensile strength (MPa) R t⊥ Transverse compressive strength (MPa) R c⊥ In-plane shear strength (MPa) R⊥储 Longitudinal tensile failure strain (%) ec储 Longitudinal compressive failure strain (%) ec储 Transverse tensile failure strain (%) et⊥ Transverse compressive failure strain (%) ec⊥ In-plane shear failure strain (%) ⊥储 Strain energy release rate (J m–2) GIC Longitudinal thermal coefficient (10–6/°C) 储 Transverse thermal coefficient (10–6/°C) ⊥ Curing: Stress free temperature (°C)

Prepeg type Hercules 0.60 126a 11 6.6a 0.28 0.4 1950b 1480 48 200b 79b 1.38 1.175 0.436 2.0 2 220c –1 26 177

Filament wind. DFVLR 0.60 138 11 5.5a 0.28 0.4 1500 900 27 200 80b 1.087 0.652 0.245 1.818 4 220 –1 26 120

(Effective temperature difference (°C)e

–125e

–68

Filament wind. DLR 0.62 53.48 17.7 5.83a 0.278 0.4 1140 570 35 114 72b 2.132 1.065 0.197 0.644 3.8 165 8.6 26.4 120 2 h at 120°C 2 h at 150°C –68

Filament wind. DRA 0.60 45.6 16.2 5.83a 0.278 0.4 1280 800 40 145b 73b 2.807 1.754 0.246 1.2 4 165 8.6 26.4 120 2 h 90°C, 1.5 h 130°C, 2 h 150°C –68 )

a

Initial modulus. Non-linear behaviour, stress/strain curves and data points are provided. c Double cantilever specimen. d Assumption: linearized, reference temperature = RT = 22°C. e –177 + RT + 30 (with moisture effect,which is never to be considered in the exercise) = –125°C. Temperature drop: = Stress free temperature minus RT (→ just curing stresses: are considered). b

453

Fig. 7. The differences in the stress–strain behaviour of isolated and embedded UD-laminae [for the (b) and (c) curve Eq. (40) is applied. The softening parameters for (b) and (c) are different].

Cuntze sees the peak value of so-called effective stress–strain curve (in situ, embedded lamina) slightly higher than the strength point R¯ of the isolated specimen due to the change from the ‘weakest link behaviour’ to the real redundant behaviour (Fig. 7) of a laminate. For the sake of simplicity this ‘peak value’ is lowered down to R¯ in the following analytical description of softening. For the execution of non-linear analysis the application of an effective stress–strain curve is necessary which estimates the behaviour of the lamina in the laminate regarding the stack, its position, and the thickness. In order to provide the non-linear analysis with the input needed, normalized stress–strain curves have been constructed (Fig. 8) with a hardening part measured and a softening part assumed (as long as no test data are available). In the non-linear analysis normally mean values have to be regarded in order to perform a stress analysis that corresponds to an average structural behaviour. Therefore, when executing a non-linear stress analysis of the structure by a mean stress–strain curve, the secant moduli to be utilized are mean values, too. However later, in the strength analysis (Proof of Design) of the ‘hot spots’ so-called ‘A’ or ‘B’ design allowables [44] as minimum strength values R (no bar upon R) have to be regarded. For simply deriving clear data for the secant moduli two regimes have to be distinguished: one below and one above (R¯ m). 7.4. Choice of different m ˙ values In the rounding-off or interaction equation just one constant value for m ˙ is inserted. This might not work if the interaction effects covered by refined conditions (e.g. [23]) are replaced by more practicable simpler formulations, (e.g. setting b⊥储 = 0, Appendix A4). In that case the rounding-off equation may be split into several mode interaction formulae replacing the single equation, because interaction addresses two or at maximum four of the five modes, only. The advantage of this computing intensive procedure would be the possibility of accounting for

454

Fig. 8. Normalized stress–strain curves.

different values with respect to different interaction effects in the various mode interaction zones. If the failure curve is reached, then f (res) Res = 1, and for this level one can stay with the advantage of one single formula. Due to the fact that Ith = 1x · m˙ different interaction effects can be accounted for. A recommendation of the author for an improved treatment of the micromechanically linked modes F ⊥ and F⊥储 is derivable from m ˙1 ⊥ m ˙3 ⊥储 m ˙4 1 = (1/f 储 Res) + . . . + (1/f Res) + (1/f Res) + . . .

(42)

Utilizing different exponents the solution has to be achieved iteratively: (res1) (res2) (res2) (res1) (res3) (resj) {}(L) → f (res1) 1. Res , f Res , {}(L) → f Res , f Res · f Res {}(L) → f Res until f Res (res1) (res2) Hence will be f (res) Res = f Res · f Res · · · . The procedure for the stress effort Eff (res) is analogous.

7.5. Variation of Poisson’s ratio The alteration of the major Poisson’s ratio v⊥储 (notation VDI 2014 [23a]) is linked to the associated failure mode. For example, in the case of shear failure under compressive lateral stresses the value for v⊥储 will be higher than for tensile lateral stresses. Respecting the low effect Poisson’s ratios have – if using FRP with stiff fibres – the following estimation will be a good approach before mode failure occurs: F ⊥: v⊥储 = v⊥储(0) · E⊥(sec) /E⊥(0). Also in the case of F⊥储 the value for v⊥储 is reduced.

455 7.6. Remarks on design and modelling • In composite structures composed of stiff fibres and hopefully well-designed by netting theory the fibre net controls the strain behaviour. • The FMC considers the inter-laminar stresses and classifies the failure modes. Therefore, associated degradation models are inherent and make a gradual degradation of the affected property possible. • In order to design a laminate properly, not only verified failure conditions have to be available, but also proper stresses have to be analytically provided [45]. Therefore, analogous to isotropic materials, the non-linear stress–strain curves have to be taken into account below reaching the initial failure level. • Above the initial failure level an appropriate progressive failure analysis method has to be employed by taking a Successive Degradation Model and by using a failure mode condition that indicates failure type and quantifies damage danger or fracture risk. • Final failure occurs after the structure has degraded to a level where it is no longer capable of carrying additional load. This is most often caused by FF, however in specific cases by IFF, too. An inclined wedge-shaped inter-fibre crack caused by F ⊥ can lead to final failure [42]. • Multidirectional laminates are usually still capable of carrying load beyond initial failure which usually is determined by IFF.

8. Calculation procedure Figure 9 presents a suitable flow chart of the non-linear calculation. The solution procedure of the non-linear analysis aims to establish static equilibrium on each load step after material properties have been changed. For each iteration the procedure is repeated until convergence (equilibrium) is reached or total failure. A correction of the fibre angle in accordance with the

Fig. 9. Non-linear calculation scheme (chosen).

456 change of the specimens geometry has been considered. By employing the equivalent stress reached in each failure mode the associated secant modulus of each mode was determined for the hardening and the softening regime. Considering a consistent stress concept for all (modes) an explicit dependency for Esec((mode) ) eq eq has to be provided. For reasons of achieving such an explicit formulation two separate formulae are discriminated which are linked in the strength point. This automatically respects that the chosen non-linear calculation procedure demands for the dependencies of the secant moduli on the corresponding equivalent stress. These dependencies are: •  > 0 (increasing stress, hardening) E t⊥(sec) = E t⊥(o) ⊥ ⊥c n ⊥  1 E c⊥(sec) = E c⊥(o) /[1 + 0.002 · (E c⊥(o) /R⊥c ] p0.2) · (eq /Rp0.2) c

⊥储 ⊥储 n⊥储  1 G储⊥(sec) = G储⊥(o) /[1 + 0.002(G储⊥(o) /R⊥储 ] p0.2) · (eq /Rp0.2)

(43)

•  < 0 (decreasing stress, softening)

冋冉

⊥ ⊥ ⊥t E t⊥(sec) = ⊥ eq /(eq ) = (eq /bs )/ ᐉn

冊 册

R t⊥  ⊥ a⊥t eq  s⊥t . ⊥ eq bs

(44)

For the further modes the same formula is valid, however, the mode parameters are different. Eq. (44) may be transferred to Puck’s degradation function  (see also Appendix A 1.2). After having reached Eff (res) = 1 this value is kept in the further degradation procedure which causes a stress redistribution towards the fibres as far as the fibre net allows it. Thereby, also the residual stresses are reduced similar to the situation with metallic materials where increasing non-linearity reduces stiffness, and, the residual stresses. If the laminate’s stiffness matrix is recomputed after each step of damage increase the laminate’s damage evolution may be continuously monitored. The approach may be called a self-correcting secant modulus procedure.

9. Application to test cases 9.1. Definition of test cases In Tables 3–5 the mechanical and thermal properties for laminae and its constituents fibres and matrices are presented. Table 6 provides a survey of the required plots and the associated loading conditions. The plots are nonlinearly to be computed.

Table 4 Mechanical and thermal properties of the four fibres utilized [Sod98]. Fibre type

AS4

T300

E-glass 21xK43, Gevetex

Silenka E-glass 1200

Longitudinal modulus (Gpa) Ef储 Transverse modulus (GPa) Ef⊥ In-plane shear modulus (GPa) Gf储⊥ Major Poisson’s ratio vf⊥储 Transverse shear modulus Gf储⊥ Longitudinal tensile strength (MPa) R tf储 Longitudinal compressive strength (MPa) R cf储 Longitudinal tensile failure strain (%) etf储 Longitudinal compressive failure strain (%) ecf储 Longitudinal thermal coefficient (10–6/°C) Mf储 Transverse thermal coefficient (10–6/°C) Mf储

225 15 15 0.2 7 3350 2500 1.488 1.111 –0.5 15

230 15 15 0.2 7 2500 2000 1.086 0.869 –0.7 12

80 80 33.33 0.2 33.33 2150 1450 2.687 1.813 4.9 4.9

74 74 30.8 0.2 30.8 2150 1450 2.905 1.959 4.9 4.9

Table 5 Mechanical and thermal properties of the four matrices utilized. 3501-6 ep.

BSL914C ep.

LY556/HT907/DY063 epoxy

MY750/HY917/DY063 epoxy

Manufacturer Longitudinal modulus (GPa) Em In-plane shear modulus (GPa) Gm Major Poisson’s ratio vm Longitudinal tensile strength (MPa) R tm Longitudinal compressive strength (MPa) R cm In-plane shear strength (MPa) R m Longitudinal tensile failure strain (%) etm Longitudinal thermal coefficient (10–6/°C) Tm

Hercules 4.2 1.567 0.34 69 250 50 1.7 45

DFVLR 4.0 1.481 0.35 75 150 70 4 55

Ciba Geigy 3.35 1.24 0.35 80 120 – 5 58

Ciba Geigy 3.35 1.24 0.35 80 120 – 5 58

457

Matrix type

458

Table 6 Summary of laminate types, material types and plots required from contributors (ˆ x = nx /t; ˆ y = ny /t; t = laminate thickness). Laminate type

Material type

• 0° unidirectional lamina (isolated) E-glass/LY556/HT907/DY063 T300/BSL914C E-glass/MY750/HY917/DY063 E-giass/LY556/HT907/DY063 • [90/ + 30/–30]s laminate t = 2.0 mm, t90 = 0.172, AS4/3501-6 (quasi-isotropic, widely used) • [90/45/–45/0]s laminate t = 1.1 mm, tk = t/8 • [ + 55/–55]s angle-ply laminate t = 1.0 mm, tk = t/4

E-glass/MY750/HY917/DY063 (piping, pressure vessels)

• [0/90]s cross-ply laminate t = 1.04 mm, tk = t/4 • [ + 45/–45]s angle ply laminate t = 1.0 mm, tk = t/4

E-glass/MY750/HY917/DY063 E-glass/MY750/HY917/DY063

Plots required and description of loading conditions 1. 2 vs 21 failure stress envelope 2. 1 vs 21 failure stress envelope 3. 2 vs 1 failure stress envelope 4. ˆ y vs ˆ x failure stress envelope 5. ˆ x vs ˆ xy failure stress envelope 6. ˆ y vs ˆ x failure stress envelope 7. Stress/strain curves under uniaxial tensile loading for ˆ y /ˆ x = 0/1 8. Stress/strain c. for ˆ y /ˆ x = 2/1 9. ˆ y vs ˆ x failure stress envelope 10. Stress/strain curves under uniaxial tensile loading for ˆ y /ˆ x = 1/0 11. Stress/strain c. for ˆ y /ˆ x = 2/1 12. Stress/strain curve under uniaxial tensile loading for ˆ y /ˆ x = 0/1 13. Stress/strain c. for ˆ y /ˆ x = 1/1 14. Stress/strain c. for ˆ y /ˆ x = 1/–1

Note: There is a discrepancy between Soden’s Table 11 and his text [3]. This causes no difference for plot 7 but for plot 10. The latter was corrected to 1/0 according to Soden’s text.

459 9.2. Assumptions and remarks for the plots • Post-initial failure is considered by gradually degraded properties of embedded laminae (no Sudden Death of the failed lamina). The course of the softening (suffix s) is assumed • First FF is final failure. The two FF F 储 (tensile fibre failure) and F 储 (shear instability, local buckling), and sometimes the IFF F 储 , are defined to cause final failure • Failure mode identification according to Cuntze’s definition is inherent to the Failure Mode Concept • Parameters m, ˙ b⊥, b⊥储, and b⊥储 are roughly assumed for the given UD-test cases • Comment: as temperature drop the difference stress free temperature minus room temperature as effective temperature difference (Table 3) is applied (if to be regarded) in order to consider the effect of curing stresses (are thermal residual stresses of the 1st kind). Moisture may be assumed here to have a balancing effect of 30°C. Chemical shrinking [6] and thermal curing stresses do not affect the shear stresses. Micro-mechanical curing stresses (residual stresses of the 2nd kind at filament/matrix level) could not be assessed and are not considered. They are usually assumed to be respected in the values for the UDstrengths, • The given stress–strain curves of the UD-lamina are interpreted mechanical load-based macromechanical stresses. It is assumed that the stress–strain curves are mean curves ¯ (R-values are given), the curve type one needs for test data mapping (see Fig. 5) • An edge effect (3D state of stress) is not considered, because the laminates are assumed to be part of a ‘closed’ composite structure • A progressive behaviour of E t储 (valid for C-fibres, only) was not regarded (see Fig. 15) • The loading is monotonic and proportional. No loading path effects are considered (should be considered some time) • In respect of the few multiaxial lamina test data one single value m ˙ = 3.1 = const will be taken for the various ‘test cases’.

Fig. 10. In-plane shear stress–strain curve 21( 21); UD-lamina (softening parameters assumed, no ¯ 储⊥0 = 5.83 GPa; n⊥储 = 6.6; curing stresses). GFRP: E-glass/MY750/HY917/DY063 [3]. R¯ ⊥储 = 73 MPa, G ⊥储 ⊥储 as = –7,0%, bs = 0.53% (assumed).

460

Fig. 11. Transv. compr. stress–strain curve c2(2); UD-lamina (softening parameters assumed). GFRP: E-glass/MY750/HY917/DY063 [3]. R¯ c⊥ = 145 MPa, E¯ c⊥0 = 16.2 GPa; n⊥c = 6.6; a⊥c s = –3.45%, b⊥c s = 0.47%.

For the computation of the test cases the following failure conditions will be employed (3 is included only in the equations where they are effective): FF1, 2 :

1 · E t储 –1 = 1; =1 储 t Eff · R 储 Eff 储 · R c储

IFF1, 2 : IFF3 :

2 321 + b⊥储22221 = 1; =1 Eff⊥ · R t⊥ (Eff⊥储 · R⊥储)3

(b⊥  1)(2 + 3) b⊥(2  3)2 + b⊥储 · 221 + =1 Eff⊥ · R c⊥ (Eff⊥ · R c⊥)2

(45)

Fig. 12. Transv. tensile stress/strain curve t2(t2) UD-lamina (softening assumend). CFRP: AS4/3501-6 ⊥t epoxy [3]. R¯ t⊥ = 48 MPa, E¯ t⊥0 = 11 GPa; a⊥t s = –1.2%, bs = 0.15%.

461

Fig. 13. In-plane shear stress–strain curve 21( 21); UD-lamina (softening parameters assumed). CFRP: ⊥储 ¯ 储⊥0 = 6.6 GPa; n⊥储 = 5; a⊥储 AS4/3501-6 epoxy [3]. R¯ ⊥储 = 79 MPa, G s = –4.0%, bs = 0.46%.

Herein 3 = –pex is to be inserted in the case of tube specimens loaded by external pressure pex, For flat specimens holds 3 = 0. The consideration of 3 = –pex shifts the biaxial strength capacity to higher values. In the modes IFF1 and 2 the pressure 3 = –pex has no effect. For the computation of the stress effort the particular 2D-state of stress (1, 2, 3) has to be inserted into Eq. (45). This will either not lead to failure, if Eff (mode) < 1, or to failure if Eff (mode) is exceeding the value 1. The modes IFF1 and IFF2 may be called harmless failures whereas IFF3 may cause a catastrophic failure which is respected in the non-linear analysis.

Fig. 14. Transv. compr. stress–strain curve UD-lamina (softening parameters assumed). CFRP: AS4/ ⊥c ⊥c 3501-6 epoxy [3]. R¯ c⊥ = 200 MPa, E¯ c⊥0 = 11 GPa; n⊥c d = 5; ad = –2.7%, bd = 0.12%.

462

Fig. 15. Longit. tensile stress/strain curve T1(1) UD-lamina. CFRP: AS4/3501-6 epoxy [3]. R¯ t⊥储 = 1950 MPa, E¯ t储0 = 126 GPa.

The equivalent stress, building up the denominators, was defined by Eff · R = eq({}),

(46)

including the residual stress by a superposition to the load stress according to {} = {}(L) + {}(R).

(47)

The residual stresses in the lamina of the laminate are decaying with decreasing stiffness caused by the degradation which accompanies increasing non-linearity. In other words: in

Fig. 16. In-plane shear stress–strain curve 21( 21); UD-lamina (degradation assumed). E-glass/LY556/ ⊥储 ¯ 储⊥0 = 5.83 GPa; n⊥储 = 5; a⊥储 HT907/DY063, R¯ ⊥储 = 72 MPa, G s = –7.0%, bs = 0.54%.

463

Fig. 17. In-plane shear stress–strain curve 21( 21); UD-lamina (degradation assumed). CFRP: T300/ ⊥储 ¯ 储⊥0 = 5.5 GPa; n⊥储 = 5; a⊥储 BSL914C epoxy [3]. R¯ ⊥储 = 80 MPa, G s = –7.0%, bs = 0.53%.

parallel to the decay of the stiffness the non-linear analysis sets matrix-dominated stresses free: These include thermal residual stresses (curing stresses), thermal stresses, and mechanical stresses across the fibres. Further, the reduction of matrix-dominated stresses directly is followed by a reduction of the matrix stresses balancing fibre stresses. This fully holds for curing stresses of the 1st kind (upper or material level). A reduction of the not respected curing stresses of the 2nd level (fibre-matrix level) also takes place, however, less pronounced. Curing stresses are respected for laminates, only (on 1st kind level). In order to differentiate between the various stress terms the following definition is given: Stresses: load stresses, hygro-thermal stresses, residual stresses Curing stresses from chemical shrinkage, mismatch of coefficients of thermal elongation Residual stresses: actively or passively built-in stresses from curing, pre-stressing.

Fig. 18. Biaxial failure stress envelope (␶21, 2); and (31, 2) in MPa. UD-lamina (no curing stresses). GFRP: E-glass/LY556 epoxy [3]. Assumed: b⊥储 = 1.5, b⊥储 = 0.13, b⊥储 = 0.4, m ˙ = 3.1 (further data, see [5,6]). [Eq. 45]. No. 1 of Table 11 [3].

464

Fig. 19. Biaxial fail. stress envelope (2, vf 1f ) in MPa. UD-lamina. Eglass/MY750 epoxy [3], [Eq. (45)]. b⊥ = 1.56, b⊥储 = 0.12, b⊥储 = 0.4, m ˙ = 3.1 [further data: see e.g. [46], No. 3 of Table 11 [3].

9.3. Stress–strain curves of the UD-lamina In Figs. 10–17 the course of the test data (solid lines) is displayed as well as the softening curve which is assumed for the embedded UD-lamina (dotted curve). One remark has to be added here: The dotted part of the F ⊥-curve (Figs. 10 and 14) is only active if catastrophic failure of the ‘delamination initiating’ wedge (its oblique micro-cracks are still closed yet deliver some compliance) is prevented by the laminate.

9.4. Biaxial failure envelopes for the UD-lamina In UD failure envelopes the residual stresses are not regarded. Thus, only the so-called load stresses from the mechanical load test are considered. For the non-linear analysis the Ramberg/Osgood exponent and the assumed softening parameters of Eq. (40) are added to each capture. The course of the presented test curves has been verified by tests at MAN and tests cited in literature [6,46]. Figures 5 and 18–22 depict several cross-sections of the five-dimensional IFF-body: • Figure 5: in the graphs (2, 3) and (23, 2), the latter was not shown here, fracture may be excellently described by the homogenized stresses. • Figure 18: the graph (21, 2) represents the IFF-responsible stresses in the plane of the lamina; the graph (31, 2) outlines that 31 does not have the same action plane as 2 (at first investigated by Puck, not derivable in Tsai/Wu’s approach). • Figure 19: the graph (2, 1) shows the limited applicability of the homogenized lamina stresses, because 1 or I1 is not the fracture stress. This is the fibre stress 1f . In order to-

465

Fig. 20. Biaxial failure stress envelope (2 = 3, vf · 1f); t⊥⊥ R t⊥/m兹2 [23]. in MPa. UD-lamina E-glass/MY750 epoxy [3]. b⊥ = 1.56, b⊥储 = 0.12, b⊥储 = 0.4, m ˙ = 3.1 [Eq. (45)].

Fig. 21. Zoom of Fig. 20.

466

Fig. 22. Biaxial fail stress envelope 21(Vf 1f ) in MPa. UD-lamina T300/BSL914C epoxy [3]. b⊥ = 1.53, b⊥储 = 0.15, m ˙ = 3.1. See also [46]. [Eq. (45)] No. 2 of Table 11 [3].

maintain the composite level in the graph the fibre stress is multiplied by the fibre volume fraction (approach: 1f · vf = 1 · E t储). • Figures 20 and 21: in the graph (2 = 3, 1) the peculiarities of a 2D lateral stressing are depicted. In the domain c2 = c3 > –10R c⊥ failure is caused, not by IFF, yet due to Poisson’s effect by fF 储 . The zoom, visualizes the rounding-off in one interaction domain (F ⊥ /F 储 ). • Figure 22: this graph eventually highlights the (21, 1)-interaction.

Fig. 23. Initial and final failure envel. ˆ y(ˆ x) [ + 55/–55]s-laminate, E-glass/MY750 epoxy [3]. b⊥ = 1.5, b⊥储 = 0.13, b⊥储 = 0.4, m ˙ = 3.1 [Eq. (45)] ˆ y = average hoop stress of the laminate, x = 0° direction. Limit of usage (lou) at y = 4%. Curing stresses not included. No. 9 of Table 11.

467

Fig. 24. Initial and final failure envel. ˆ y(ˆ x) in MPa. [90/ + 45/–45/0]s-laminate, AS4/3501-6 [3]. [Eq. (45)] No. 6 of Table 11 [3].

9.5. Initial and final biaxial failure envelopes For the determination of the failure envelopes (see Figs. 23–26) the code Mathcad, non-linear CLT, and an assumed softening behaviour were applied. The symbols used to indicate the mode of failure are the symbols which characterize the failure function, e.g. ⊥储 for F 储 and so on. The angle marks the associated lamina. • Figure 23 incorporates the initial and the final failure envelope of this GFRP-laminate.

Fig. 25. Initial and final biaxial failure envel. ˆ xy(ˆ x) [90/ + 30/–30]s-laminate. E-glass/LY556 epoxy. No. 5 [3]. m ˙ = 3.1. [Eq. (45)] ˆ x is parailel to 0°-direction. No. 5 of Table 11 or of Table 6, at hand.

468

Fig. 26. Initial and fmal biaxial failure envel. ˆ y(ˆ x) [90/ + 30/–30]s laminate. E-glass/LY556 epoxy [3]. [Eq. (45)] No. 4 of Table 11 [3] or of Table 6, at hand.

In the positive quadrant there are no corners. Generally, corners become smoothed due to the effect of high interaction of the failure modes. In the domain A–B both F 储 in the two adjacent laminae are ‘acting together’.

Fig. 27. Stress–strain curves ˆ y : ˆ x = 1.0 [ + 55/–55]s-laminate, E-glass/MY750 [3]; b⊥ = 1.5, b⊥储 = 0.13, b⊥储 = 0.4, m ˙ = 3.1. max = 4%. [Eq. (45)] No. 10 of Table 11, see [3].

469

Fig. 28. Stress–strain curves ˆ y : ˆ x = 2 : 1 [ + 55/–55]s-laminate. E-glass/MY750 No. 11 of Table 11 [3].

In the negative quadrant wedge failure may occur in the compressed laminate specimen. The event of a wedge failure is equal to the onset of delamination damage. In case of a plane laminate specimen, despite the anti-buckling device applied when testing in the compression regime, the wedge will slide and then cause a compressive reaction c3 normal to the lamina’s

Fig. 29. Stress–strain curves ˆ y : ˆ x = 2:1. [90/ + 45/–45/0]s-laminate. AS4/3501-6 epoxy. No. 8 of Table 11 [3].

470 plane onto the adjacent laminae (see Puck’s drive shaft [42]). This will induce delamination or might increase an initial delamination size. However, in case of a pressure loaded tension/ compression-torsion tube specimen (applied at MAN; see also [6]) and in case of high pressure vessels (1000 bar, ARIANE 5 launcher) loaded by external pressure pex the multiaxial strength is increased (3 = –pex is acting in a favourable manner). The sliding friction due to pex is increased similarly until its maximum will be reached. Mind: a correct analysis of boundary conditions and stress state of the test specimen is mandatory before evaluating and applying the data. • Figure 24 depicts the symmetrical failure envelopes of this CFRP laminae. The sharp corners still have to be rounded-off in a refined procedure taking into account the joint failure probability of the laminate [38]. In the negative quadrant IFF covers FF. • The last two failure envelopes (Figs. 25 and 26) are concerning the [90/ + 30/–0]s-laminate subjected to a (ˆ x, ˆ y) state of stress and a (ˆxy, ˆ x) state of stress. Again here, sharp corners

Fig. 30. (a) Stress–strain curves ˆ y : ˆ x = 1 : –1 [ + 45/–45]s-laminate. E-glass/MY750 [3] (without temperature drop). No. 14 of Table 11 [3]. (b) Stress–strain curves ˆ y : ˆ x = 1 : –1 [ + 45/–45]s-laminate. E-glass/MY750 [3] (with temperature drop 120–22° C, as desired in addition by reviewers).

471

Fig. 31. Stress–strain curves ˆ y : ˆ x = 0 : 1 [0/90]s-laminate. E-glass/MY750 [3] No. 12 of Table 11 [3].

should be rounded there where the joint failure probability of the failure modes comes to act.

9.6. Stress–strain curves of the laminates The following stress–strain curves (Figs. 27–33) consider Eq. (45) and the data from Tables 3 and 6 [3]. The loading is monotonic, a temperature drop from curing (causes an off-set) is regarded.

Fig. 32. Stress–strain curves ˆ y : ˆ x = 1 : 1 [ + 45/–45]s-laminate. E-glass/MY750 [3]. No. 13 of Table 11 [3].

472

Fig. 33. Stress–strain curves ˆ y : ˆ x = 1 : 0. [0/ + 45/–45/90]s-laminate. AS4/3501-6 epoxy [3] No. 7 of Table 11 [3].

Figures 27 and 28 outline the deformation behaviour of a pressure vessel, which is usually designed for one special load case ‘inner pressure’ that means for ˆ y /ˆ x = 2 : 1. Load combinations outside of this ratio – such as 1 : 0 (Fig. 27) – will lead to too high shear strains and thereby to a ‘limit of usage’ (lou). This shear strain design limit or limit of usage (lou) was assumed here to be max = 4% shear strain which corresponds more or less to the shear fracture strain of the isolated lamina. As the authors were asked to provide the text with more test data, Fig. 34 was added.

Fig. 34. (2, 21)-lFF curves, tubes [6], GFRP: E-glass/LY556/HT976/DY070, CFRP: T300/LY556/ HT976.

473 10. Some conclusions, outlooks • A general concept was highlighted for the establishment of Failure Conditions (F = 1) for Initial Failure (corresponding to IFF) of dense, brittle laminae and Final Failure of the laminate. • The complete failure surface consists of piecewise smooth regimes (partial failure surfaces). Each regime represents one failure mode and is governed by one basic strength • Sufficient for pre-dimensioning are the basic strengths R. The remaining unknown curve parameters b⊥储, b⊥ b⊥储 can be estimated if test data are not available. The rounding-off exponent m, ˙ after some fitting experience, can be fixed on the safe side by taking a little lower value. • The interaction (rounding-off) of adjacent failure modes is automatically considered when calculating the stress effort Eff res as function of the mode efforts Eff modes. • The concept enables to correctly turn the design key by respecting the most critical mode and the location [25,29] in the Finite Element idealization of the structure (Appendix A3). • Homogenisation of the UD-material comes to its limit if a constituent stress governs the failure. This is the case for F 储 , where the macromechanical stress 1 has to be replaced by the actual fibre stress 1f . A fibre stress may be zero not even for zero 1. Therefore, 1f has to be estimated as 1f = 1 · E1f . In order to remain on composite stress visualization level 1f will be multiplied by the fibre volume fraction vf . • The ‘mode fit’ avoids the shortcomings of the ‘global fit’ which maps the course of test data by mathematically linking failure modes which are in reality not mechanically linked. One typical shortcoming is, that a reduction of the strength of one mode might increase the multi-axial strength in another (independent) mode or part of the global failure surface. • For the prediction of final failure the initial failure approach is not of that high concern, if wedge failure, caused by F ⊥ < 1 and followed by delamination failure, will not occur (see Puck’s drive shaft [42] or torsion spring). • Each failure condition describes the interaction of stresses affecting the same failure mode and assesses the actual state of stress in a ‘material point’. • For (2, 3) states of stress Mohr’s stresses, Mohr’s envelope curve, and the inclined fracture angle fp may be determined. • Damage mechanics is captured in the FMC conditions so far as the stiffness reduction is determinable via the (eq, )-curve, and by the predictability of delamination initiation, applying F ⊥ and F ⊥. • Regarding the investigations in theory and test carried out in Germany on the lamina material level in the last years (still going on) the understanding has improved a lot and seems to be a good basis to tackle laminates stacked-up of UD-laminae and fabric laminae. For other ‘textile preforms’ (3D, stitched etc.) engineering models have to be developed. The transferability to rhombically-orthotropic composites and other materials (fabrics) works [27,28]. • The choice of linear or other terms of stress invariants is based on whether there are volume and/or shape changes of the material element as well as on curve fitting considerations. • In respect of the scatter of the actual test data the parameter set b⊥ = 1, b⊥储 = b⊥储 = 0 will often be an approach good enough for final failure analysis of the laminate.

474 Acknowledgements The authors gratefully express their thanks for the intensive collaboration with Professor A. Puck reviewing this paper in the context with the comparison of the two approaches. The authors also thank the reviewers for their constructive comments.

References [1] Hinton MJ, Soden PD, Kaddour AS. Comparison of failure prediction methods for glass/epoxy and carbon/epoxy laminates under biaxial stress. ICCM11 1997;V:672–82. [2] Hinton MJ, Soden PD, Kaddour AS. Failure criteria in fibre-reinforced-polymer composites (special issue). Composites Science and Technology 1998:58. [3] Soden PD, Hinton MJ, Kaddour AS. Lamina properties, lay-up configurations and loading conditions for a range of fibre-reinforced composite laminates (special issue). Composite Science and Technology 1998;58:1011–22. [4] (a) Soden PD, Hinton MJ, Kaddour AS. A comparison of the predictive capabilities of current failure theories for composite laminates (special issue). Composites Science and Technology 1998;58:1225–54. (b) Kachanov, L.M. Foundations of the Theory of Plasticity. North-Holland (1971). (c) Betten J. Recent advances in mathematical modelling of material behaviour. Math Comput Model 1990;14:37–51. [5] Knappe W, Schneider W. Bruchkriterien fur unidirektionalen Glasfaser-Kunststoff unter ebener Kurzzeit- und Langzeitbeanspruchung. Kunststoffe, Bd 1972;62:864 8. [6] VDI97 Cuntze RG, et al. Neue Bruchkriterien und Festigkeitsnachweise fur unidirektionalen Faser-kunststoffverbund unter mehrachsiger Beanspruchung Modellbildung und Experimente. VDI-Fort-schrittbericht, Reihe 1997;5(506):250. [7] Herrmann G. Zum Bruchverhalten gerichteter Glasfaserverbunde. Dissertation, TU-Stuttgart, Inst. f. Werkstoffe im Bauwesen, 1982. [8] Puck A. Calculating the strength of glass fibre/plastic laminates under combined load. Kunststoffe, German Plastics [German Plastics is the bilingual English and German edition of Kunststoffe] 1969;55:18–19 [in German. p. 780–7]. [9] Puck A, Schneider W. On failure mechanisms and failure criteria of filament-wound glass-fibre/ resin composites. Plast Polym 1969: 33–43. [10] Puck A. Praxisgerechte Bruchkriterien for hochbeanspruchte Faser-Kunststoffverbunde. Kunststoffe 1992;82(2S):149–55 [Fracture criteria for highly stressed fibre plastics composites which meet requirements of design practice. Kunststoffe German Plastics 1992;82(2):36–8]. [11] Puck A. Faser-Kunststoff-Verbunde mit Dehnungs- oder Spannungs-Kriterien auslegen? Kunststoffe 1992;82(5):431–4 [Should fibre-plastics composites be designed with strain or stress criteria? Kunststoffe German Plastics 1992;5:34–36]. [12] Puck A. Ein Bruchkriterium gibt die Richtung an. Kunststoffe 1992;82(7):607–10 [A failure criterion shows the direction further thoughts on the design of laminates. Kunststoffe German Plastics 1992;82(7):29–32]. [13] Hashin Z. Failure criteria for unidirectional fibre composites. J of Appl Mech 1980;47:329–34. [14] Cuntze RG. A physically based 2D/3D-inter-fibre-failure criterion for brittle UD-layers Hashin’s idea and Puck’s realization. Key-note lecture, ISODUR 93, Porto, July 18–21. [15] Jeltsch-Fricker R. Bruchbedingungen vom Mohrschen Typ fur transversal-isotrope Werkstoffe am Beispiel der Faser-Kunststoff-Verbunde. ZAMM 1996;76:505–20. [16] Jeltsch-Fricker R, Meckbach S. Fast solver of a fracture condition according to Mohr for unidirectional fibre-polymer composite. Scripts of the University of Kassel on Appl. Math., No. 1/96 [in German].

475 [17] Michaeli W, Huybrechts D. A new approach for the dimensioning of thick laminates using physically based strength criteria. In: Proc. 39th Conf. of the Society for the Advancement of Material and Process Engineering (SAMPE), vol. 2, 11–14 April 1994, Anaheim, CA p. 2829–40. [18] Huybrechts D. Ein erster Beitrag zur Verifikation des wirkebenebezogenen Zwischenfaserbruchkriteriums nach Puck. Dissertation an der RWTH Aachen; 1996. [19] Rackwitz R, Gollwitzer S. A new model for inter-fibre-failure of high strength uni-directionally reinforced plastics and its reliability implications. NATO-workshop PROBAMAT-21 Century, Perm, Russia, 10–12 September 1997. [20] Kopp J, Michaeli W. Dimensioning of thick laminates using new IFF strength criteria and some experiments for their verification. In: Proceedings Conf. on Spacecraft Structures Materials and Mechanical Testing, ESA, 27–29 March 1996. [21] Kopp J, Michaeli W. The new action plane related strength criterion in comparison with common strength criteria. Proceedings of ICCM-12, Paris, France, July 1999. [22] Hufenbach W, Kroll L. A new failure criterion based on the mechanics of 3-dimensional composite materials. ICCM-10, Whistler, Canada. 1995. [23] Cuntze RG. Fracture-type strength criteria formulated by invariants which consider the materials symmetries of the isotropic/ anisotropic material used. Conf on Spacecraft Structures, Materials and Mechanical Testing. [ESA-CNES-DARA]. Noordwijk, March 1996 [Conf Handbk]. 23a VDI2014: German guideline on development of FRP components analysis; 2003 (in German and English). [24] Cuntze RG. Evaluation of multi-axial test data of UD-laminae by so-cailed fracture type strength criteria and by supporting probabilistic means. ICCM-II, Gold Coast, Australia, 1997. [25] Cuntze RG, Sukarie G. Effective dimensioning of 3D-stressed UD-laminae on basis of fracturetype strength criteria. Conference handbook, Presentation Int. conf. on Mechanics of Composite Materials. Riga, 20–23 April 1998. [26] Cuntze RG. The faiiure mode concept a new comprehensive 3D-strength analysis concept for any brittle and ductile behaving material. Europ. Conf on Spacecraft Structures, Materials and Mechanical Testing. [ESA-CNES-DGLR-DLR] Braunschweig7 November 1998 [ESA SP-428]. p. 269–87. [27] Cuntze RG. Strength prediction for multi-axially loaded CMC-materials. In: 3rd European Workshop on thermal Protection Systems. [ESA-ESTEC] Noordwijk, March 1998 [WP P141]. [28] Cuntze RG. Application of 3D-strength criteria, based on the so-called failure mode concept, to multiaxial test data of sandwich foam, concrete, epoxide, CFRP-UD lamina, CMC-fabric lamina. ICCE/5, Las Vegas, July 1998 [presentation]. [29] Cuntze RG. Progressive failure of 3D-stressed laminates: multiple non-linearity treated by the failure mode concept (FMC). DURACOSYS 99, Brussels, July 1999. [30] Boehler JP. Failure criteria for glass-fiber reinforced composites under confining pressure. J Struct Mechanics 1985;13:371. [31] Boehler JP. Personal note to the first author on fabric invariants; 1995. [32] Jeltsch-Fricker R, Meckbach S. A parabolic Mohr fracture condition in invariant formulation for brittle isotropic materials. ZAMM 1999;79:465–71 [in German]. [33] Meckbach S. Invariants of cloth-reinforced fibre reinforced plastics. Kasseler Schriften zur angewandten Mathematik 1998;1 [in German]. [34] Christensen RM. The numbers of elastic properties and failure parameters for fiber composites. Transactions of the ASME 1998;120:110–3. [35] Cuntze RG. Deterministic and probabilistic prediction of the distribution of inter-fibre failure test data of prestrained CFRP tubes composed of thin layers and loaded by radial pressure. Wollongong. In: Advanced Composites ’93. The Minerals, Metals & Materials Society, 1993. pp. 579–585.

476 [36] Grimmelt M, Cuntze RG. Probabilistic prediction of structural test results as a tool for the performance estimation in composite structures design. Beuth Verlag, VDI-Bericht 1989;771:191–200. [37] Rackwitz R, Cuntze RG. System reliability aspects in composite structures. Eng’ Opt 1987;11: 69–76. [38] Cuntze RG. Failure path analysis of multi-layered fibre reinforced plastic components with the reliability calculation programme FRPREL. Noordwijk, October 1987. [39] Mohr O. Welche Umstande bedingen die Elastizitatsgrenze und den Bruch eines Materials? Civilingenieur 1900;XXXXIV:1524–30, 1572–7. [40] Flaggs DL, Kural MH. Experimental determination of the in situ transverse lamina strength in graphite epoxy laminates. J Comp Mat 1982;16:103–16. [41] (a) Masters J. Fractography of modern engineering materials. Composites and metals, 2nd vol. ASTM STP1203;1994. (b) Ashton JE, Whitney JM. Theory of laminated plates. Progress in Materials Sciences Series, vol. I. Stamford, Technomic Publ. 1970. [42] (a) Puck A. Festigkeitsanalyse von Faser-Matrix-Laminaten Modelle fur die Praxis. Munchen: Carl Hanser Verlag; 1996. (b) Lekhnitskii SG. Theory of elasticity of an anisotropic elastic body. San Francisco: Holden Day; 1963. [43] Awaji H, Sato S. A statistical theory for the fracture of brittle solids under multi-axial stresses. Int J Frac 1978;14:R13–16. [44] MiL hdbk 17. Plastics for aerospace vehicles, vol. I, reinforced plastics; vol. II; vol. III utilization of data. USA: Dep. of Defence (DOD). [45] Rolfes R, Noor AH, Rohwer K. Efficent calculation of transverse stresses in composite plates. MSC-NASTRAN User Conference, 1997. [46] (ZTL80) Dornier, Fokker, MBB, DLR. Investigations of fracture criteria of laminae. Grant from German ministry, BMVg [multiaxial testing] 1975–1980 reports in German. [47] Puck A7 Schurmann H. Failure analysis of FRP laminates by means of physically based phenomenological models [special issue]. Comp Sci Technol 1998;58. [48] Paul B. A modification of the Coulomb-Mohr theory of fracture. J Appl Mech 1961:259–68. [49] Christensen RM. Stress based yield/failure criteria for fiber composites. Int J Solids Struct 1997;34(5):529–43. [50] Christensen RM. Yield functions/failure criteria for isotropic materials. Proc R Soc Lond A 1997;453:1473–91. [51] Hart-Smith LJ. An inherent fallacy in composite interaction failure curves. Designers Corner, Composites 1993;24:523–4. [52] Goldenblat II, Kopnov VA. Strength of glass-reinforced plastics in the complex stress state. Polymer Mechanics of Mechanical Polimerov 1966;1:54–9. [53] Thom H. A review of the biaxial strength of fibre-reinforced plastics. Composites Part A 1998;29:869–86. [54] Tsai SW, Wu EM. A general theory of strength for anisotropic materials. J Comp Mater 1971;5: 58–80. [55] Yeh HY, Kilfoy LT. A simple comparison of macroscopic failure criteria for advanced fiber reinforced composites. J Reinforced Plastics Comp 1998;17:406–45. [56] Wang JZ. Failure strength and mechanism of composite laminates under multiaxial loading conditions. Dissertation, Univ. of Illinois at Urbana, 1993. [57] Rowlands RE. Strength (failure) theories and their experimental correlation. In: Sih GC, Skudra AM, editors. Handbook of composites, band III, Chapter 2. Madison (Wl, USA): Elsevier Science; 1985. pp. 71–125. [58] Matzenmuller A, Lubliner J, Taylor RL. A constitutive model for anisotropic damage in fiber composites. Mechanics of Materials 1995;20:125–52.

477 [59] Sukarie G. Einsatz der FE-Methode bei der Simulation des progressiven Schichtversagens in laminierten Faserverbundstrukturen. Symposium Berechnung von Faserverbundstrukturen unter Anwendung numerischer Verfahren. Munchen, Techn. Univ., 13/14 March 1996. [60] Slight DW, Knight NF, Wang JT. Evaluation of a Progressive failure analysis methodology for laminated composite structures. 38th Structure, Structure Dynamic and Material Conference, April 1997 [AIAA Paper 97-1187].

Appendix A A.1. Comparison of Puck’s and Cuntze’s failure theories A.1.1. Comparison of Puck’s fracture plane based lFFcriteria and Cuntze’s FMC-based invariant formulations The two Sections A1.1 and A1.2 are a common formulation of Puck and Cuntze, because both authors have often been asked for an explanation of the coincidences and differences between their approaches. The following should be mentioned in this context: • With respect to the different effort that has been put by Puck et al. (incl. Cuntze) into the fracture plane based criteria and by Cuntze into the FMC based criteria the ‘Puck criteria’ are approaching the ‘series status’ and the ‘Cuntze criteria’ only the ‘development status’. • The FMC criteria seem to be generally applicable to all materials. Therefore, there are a few shortcomings in their application to UD-material. As early as 1968/1969 Puck concluded from experimental observations that two completely different types of fracture should be distinguished and theoretically treated by separate failure criteria: Fibre Failure (FF) and Interfibre Failure (IFF) [8,9]. In the early seventies the discrimination of these two fracture types became common practice in the German aerospace industry [46]. In all later papers of Puck and Cuntze the separate treatment of FF and IFF has been maintained. For FF both authors use simple maximum stress criteria, based on the consideration, that the composite fails when the fibres reach a certain critical stress. Both authors feel that for the new anisotropic fibres a better approach for FF prediction may be necessary. Since another fundamental paper of 1992 [12], research in Germany has concentrated on the improvement of IFF criteria. This appeared to be of higher importance than assumed in the past after it had been learned from experience on torsion tube springs that the wedge effect of oblique fractures under transverse compression can cause destruction of the whole composite part [47]. Besides this, under alternating loads, microcracks, due to IFF (caused mainly by transverse tensile stress), give rise to high peaks of interlaminar stresses which initiate local delaminations. A.1.1.1. Common foundation of the two approaches The failure theories of Puck and Cuntze are based on the same fundamental assumptions: • The UD-layer is transversally-isotropic and failure occurs by brittle fracture. • Mohr’s statement is valid: The material strengths are determined by the stresses on the fracture plane. • The fracture plane may be inclined with respect to the plane which the external stresses are acting on. This is, for instance, true for uni-axial transverse compression.

478 • For states of stress without longitudinal shear (31, 21), that means plane stress conditions consisting of a stress state (2, 3, 23) which can be replaced by (II, III), the so-called principal stresses of the transversally-isotropic plane, both authors make the same assumption: Paul’s modification of the Coulomb-Mohr theory of fracture [39] is valid. This is based on the assumption, that two different modes of fracture can occur which leads to the following fracture hypothesis (formulation analogue to that for isotropic material): ‘An intrinsically brittle material will fracture in either that plane where the shear stress nt reaches a critical value which is given by the shear fracture resistance RA⊥⊥ of a fibre parallel plane increased by a certain amount of friction. This friction is caused by the simultaneously acting compressive stress n on that plane. Or, it will fracture in that plane, where the maximum principal stress (II or III) reaches the transverse tensile strength R t⊥ ’ (Figs A1/1 and A1/2). For reasons of simplicity the bars over R are skipped in Chaper A1.

Fig. A1/1. Mohr’s fracture envelope and some Mohr circles for fracturing stresses (a) for uniaxial transverse compression II, (b) limiting circle for simultaneous shear fracture (SF) and cleavage fracture (CF) on different action planes. Between SF and CF no circle can touch the fracture envelope, (c) for pure shear 23 = II–III, (d) for uniaxial transverse tension III.

Fig. A1/2. Fracture curve (II, III) resulting from Fig. A1/l with tensile cut-offs and typical fracture angles "fp for uniaxial transverse tension and compression.

479 A1.1.2. Results for plane stress (II, III) For this state of stress without any longitudinal shear (31, 21) there is a complete coincidence of the formulations of Puck and Cuntze. The treatment of this problem by Mohr’s circle (representing the state of stress (II, nt), on any plane, see Fig. A1/1 and Mohr’s fracture envelope [representing the fracture limit for combined (n, nt)-stresses] is well known. • For the domain n < 0 Puck [42] starts with the assumption of a parabolic fracture envelope nt = nt(n), that reads: ) A R⊥⊥n 2nt = (RA⊥⊥)2  2p(⊥⊥

(A1)

wherein RA⊥⊥ is the transverse shear fracture resistance of a fibre parallel plane against its ) fracture caused by a -stressing acting on that plane. The coefficient p(⊥⊥ is a so-called ‘friction coefficient’ for n < 0. At fracture Mohr’s circle and the fracture envelope have a common point of contact, that means the same inclination dnt/dn. From this condition the fracture angle "fp between the action plane of II and the fracture plane can be calculated, which is varying a little with the difference of (II  III), cos2"fp = –cos2"cfp

R c⊥ (for n < 0) II  III

(A2)

with ("cfp = fracture angle under uniaxial transverse compression (angle between the action plane of the uniaxial compressive stress II and the corresponding fracture plane which is 45° < | "cfp | < 60°) and R c⊥ = transverse compression strength. In this equation II and III are stresses at fracture! By means of this result a definite form in II, III for the fracture condition is found which is parabolic and invariant in the transversal plane: F ⊥ =

a⊥ b⊥ (II + III) + c 2 (II  III)2 = 1. c R⊥ (R ⊥)

(A3)

• Cuntze in contrast to Puck starts already with this invariant formulation (A3): The adaptation to experimental uniaxial compression results (strength R c⊥ and fracture angle "Cfp) gives a⊥ = b⊥  1 and b⊥ = 1(2 cos 2"cfp + 1) In Eq. (21b) another adaption of b⊥ to test results is shown. Puck’s and Cuntze’s approach for the domain n < 0 are connected by the relation (A2) for the fracture angle "fp. For the domain n 0 both authors use the ‘tensile cut-offs’ recommended by Paul [48]. That means that the fracture stress is either II = R t⊥ or III = R t⊥, see Fig. A5 or A1/2.

480 A1.3. Results for states of stress with additional longitudinal shear (31, 21) In this field the two authors use rather different approaches: • Puck stays with the ‘physically based’ consideration of the mechanical interaction of the stresses n, nt, n1 on the fracture plane (Fig. A1/3). He uses simple polynomials (parabolic or elliptic) to formulate a master-fracture body in the (n, nt, n1)-stress space. Starting from this (master-) fracture body generally no analytical solutions can be found for the fracture angle "fp (with the exception of (1, 2, 21)-states of stress) and therefore no analytical solutions can be given for the fracture bodies in 1, 2, 3, 23, 31, 21. Therefore, the necessary search for the fracture plane, that means for the plane with the lowest reserve factor minfRes(") or the highest stress exposure factor max fE("), has to be done numerically (using the formulation of the fracture condition in n, nt, n1) in an angle range –90° < " + 90°. By means of the found fracture angle fp, resulting from the numerical procedure, the stresses (1, 2, . . . 21) at facture can be calculated by multiplying the acting stresses (1, 2, 21) by the lowest reserve factor min fRes(") = fRes("fp).

Fig. A1/3. Action planes and fracture causing Mohr stresses.

481 The numerical search for the fracture plane is an inconvenience, but on the other hand the user of this approach automatically gets an information on the fracture angle and on the ‘fracture mode’. Puck defines the ‘fracture mode’ as the stress combination (n, nt, n1) or (, , ⊥储) on the fracture plane. For the calculation of the fracture stresses Cuntze’s invariant formulation is of course the more convenient one. The results can be visualized by fracture bodies in a 3dimensional (II, III, 1)-space, where 1 is the “resultant” of 31 and 21. These fracture bodies are not symmetric with respect to the (II = III)-plane [21]. • Cuntze uses three simple invariant formulations in (1, 2, 21)-one linear, one quadratic and one cubic polynomial- which lead to fracture bodies in the (II, III, 1) space similar to those of Puck. He feels that mechanical and probabilistic interactions cannot be clearly distinguished and therefore he models the ‘mode’ interactions by a simple probabilistic (res) series model (‘rounding-off’ procedure achieved by the determination of f (res) ). Res or Eff • Attention has to be paid to the fact that the expression ‘mode’ has different meanings in the papers of Puck and Cuntze. Puck differentiates between seven interfibre fracture (sub)modes M1–M7 (according to the number of the possible stress combinations acting on the fracture plane) which may be allocated to the three Modes A, B, C (see Fig. A1/4): Group with an n 0 M1 = (t⊥, ⊥⊥, ⊥储) the most general mode M2 = (t⊥, 0, 0) M3 = (t⊥, 0, ⊥储) Domain of Mode A M4 = (0, 0, ⊥储) Group with n < 0 M5 = (c⊥, 0, ⊥储) Domain of Mode B M6 = (c⊥, ⊥⊥, ⊥储) M7 = (c⊥, ⊥⊥, 0)

冎

Domain of Mode C

Fig. A1/4. (2, 21)-fracture curve with IFF-modes A, B, C.

(A5)

482 Cuntze uses the expression ‘mode’ to address his three different invariant IFF conditions, based on the idea that for each of these fracture conditions in their ‘pure’ regimes either the t⊥-, the c⊥, or the ⊥储-stressing is ‘dominant’. Of course, one has to pay for the higher convenience of the invariant approach with a certain loss of ‘physical correctness’ and the inability to predict the fracture angle for states of stress including longitudinal shear 21, 31. However, this may be acceptable in many cases of design practice.

A1.4. Comparison of Puck’s and Cuntze’s failure analysis of laminates This section focuses on a 2D-laminate failure analysis as performed in the ‘failure exercise’, Part A[3]. For fibre failure (FF) of the UD-lamina both authors use the same simple maximum stress failure criterion: f (FF) E =

1 = 1 for 1 0 R t储

and f (FF) E =

– = 1 for 1 < 0. R c储

(A6)

fE is the stress exposure factor used by Puck. It has essentially the same meaning as Cuntze’s resultant stress effort Eff (res). The value of fE or Eff (res), respectively, quantifies the ‘risk of fracture’. Fracture occurs, if fE = 1 = 100%. Both authors also assume that FF in at least one lamina of a laminate means final failure of the laminate. Therefore, the biaxial failure envelopes for final failure of laminates predicted by the two authors do not differ very much, as long as the laminates have three or more fibre directions. The strengths of these laminates are ‘fibre dominated’. Also, the predicted stress/strain curves of such laminates look very similar because the fibres which are much stiffer than the matrix carry the main portion of the loads. Different degradation procedures after the onset of interfibre failure (IFF) do therefore not influence the predicted strains very much. This is especially true for CFRP laminates. Puck’s degradation procedure (known as the -degradation) for the secant moduli E2(sec) and G21(sec) after the onset of IFF is rather simple, since Puck’s IFF-criteria are completely based on the assumption of a mechanistic interaction of 2 and 21. Probabilistic aspects can be dealt with, if necessary, in a separate operation [42]. In Puck’s theory the numerical search for the fracture angle fp, that means the search for the stress action plane with the highest angle dependent stress exposure factor max f (EFF) (), E is not necessary in the special case of a plane state of stress (1, 2, 21). For tensile 2 > 0 and also for moderate compressive stress (| 2 | < 0.4R c⊥) the fracture plane is the same as the action plane of 2 and 21 (Fig. A1/4). That means: fp = 0. For rather high compressive stresses ( | 2 | > 0.4R c⊥) the fracture angle pf 0 can be calculated from a very simple analytical expression:

483

fp = arctan

冑

RA⊥⊥ . – 2

(A7)

Attention! In this equation 2 is the compressive stress at fracture caused by a combined (2, 21) state of stress. Based on the knowledge of the fracture angle there have been found three simple analytical expressions for the stress exposure factor f (IFF) formulated with 2 and 21 instead of n, nt, E n1. Each of the three equations is valid for a certain region of the (2, 21)-fracture curve [42,47]: • Mode A is valid for 2 0 and combines the modes M2, M3, M4 mentioned in Eq. (A5) • Mode B is valid for 0 | 2/21 | RA⊥⊥/ | c21 | and is identical with mode M5. • Mode C is valid for the region with fp 0, i.e. 0 | 21/2 | | c21 |/RA⊥⊥ and combines the modes M6 and M7. One should remember that the expression ‘mode’ has different meanings in the papers of Puck and Cuntze! Puck’s stress exposure factors f (IFF) for his Modes A, B and C are not equivalent E (mode) to Cuntze’s mode efforts Eff but to Cuntze’s Eff (res)! Like f (IFF) also Eff (res) quantifies the E risk of fracture due to the combined action of 2 and 21. Puck’s fracture condition for IFF of a UD-lamina is = 1. f (IFF) E

(A8)

For a UD-lamina in a laminate, this means the onset of progressive IFF (‘matrix cracking’) the three different equations for f (IFF) are (Fig. A1/4): E • For Mode A:

f (IFF) = E

1 · R⊥储

冋冑 冉 221 +

R⊥储  p(⊥储+ ) R t⊥

冊

2

册

22 + p(⊥储+ )2 ,

(A9)

• For Mode B: 1 · (兹221 + p(⊥储 )22 + p(⊥储 )2), R⊥储

(A10)

R c⊥ 221 (– 2) · + . 4(R⊥储 + p(⊥储 )RA⊥⊥)2 (– 2) R c⊥

(A11)

f (IFF) = E • For Mode C: f (IFF) = E

Hence, for Mode A the fracture angle fp iS 0° and, because 2 is a tensile stress, the microcracks tend to open. The resulting decrease of the secant moduli and G21(sec) is modelled by Puck by a simultaneously starting degradation of E2(sec) and G21(sec). That means, secant moduli E2(sec) and G21(sec) with  < 1 are used after the onset of IFF. The degradation factor  is a decaying function decreasing with increasing load, in order to keep f (IFF) = 1. After the onset E

484 of IFF only ‘average stresses’ can be calculated for a micro-cracked lamina. Average stresses are defined as stresses smeared over some length of the cracked lamina (which includes a number of micro-cracks). After the onset of IFF, Puck calculates the average stresses 2() and 21() by using  · E2(sec), G21(sec), and  · v12. He assumes that in the progressive cracking process of a lamina its average stresses 2() and 21() remain approximately constant with increasing load. This is achieved in the calculation by keeping f (IFF) = 1 = constant after the first IFF has occurred. E In contrast to Puck’s completely f (IFF) E -controlled -degradation Cuntze uses his ‘stress exposure factor’ Eff (res) (or fE) only for a certain correction, in order to take into account mode-interactions. His degradation is mainly controlled by using the 2(2)- or the 21( 21)stress/strain curve for finding the valid secant modulus E2(sec) or G21(sec), respectively. The branches with increasing stresses (‘hardening’) of these stress/strain curves are found by the usual experiments with uni-axial 2-stress or pure 21-stress, respectively. The branches with rapidly decreasing stresses (called ‘softening’) are preliminarily assumed (see Figs. 10–17). Like Puck, Cuntze calculates the stresses 2 and 21 in the laminae of the laminate by using secant moduli from the 2(2)- and 21( 21)-stress/strain curves. However, Cuntze has to pay attention to a proper interaction of the interactive modes in the stress and strain analysis in the following manner: In order to take into account the combined probabilistic/mechanistic interaction of the failure modes the secant moduli E2(sec) and G21(sec) are taken from the 2(2)curve or the 21( 21)-curve not just at the stresses 2 or 21, resulting from the stress and strain analysis for the actual level load. Their values are taken at a little higher stress in the ‘hardening branch’ with increasing stress and at a little lower stress in the ‘softening branch’ with decreasing stress. This ‘stress correction’ is controlled by the so-called ‘triggering approach’, which is described in Section 6 [see Eqs. (35) and (36)]. The controlling parameter is the ratio of the resultant stress exposure factor Eff (res) to the maximum mode exposure factor maxEff (mode). By this triggering approach lower secant moduli E2(sec) and G21(sec) are provided for the next calculation loop as those which would result without the correction by the triggering approach. Figs. A1/5 and A1/6 visualize Puck’s -degradation and Cuntze’s ‘triggering approach’. In Cuntze’s theory for the actual load the degradation of E2(sec) and G21(sec) is performed with the same trigger factor TrF. In contrast to Puck’s theory, if one of the corrected equivalent mode stresses has reached its strength level, a rapid decrease of the mode’s average (smeared over the micro-cracks) equivalent stress will follow. There is another difference: Cuntze’s triggering approach is already active before the onset of IFF. This can perhaps be justified by the fact that there is a certain mutual interaction of 2 and 21 on their strains before the fracture stresses have been reached [see Fig. 1] in the literature [47]. Due to the severe lack of experimental experience about the real degradation of laminates after IFF initiation different authors make very divergent assumptions about the average stresses in a lamina after the onset of IFF until final failure of the laminate, as can be seen from the ‘failure exercise’ [4]. Figs. A1/5 and A1/6 demonstrate this for the two authors Puck and Cuntze. In order to demonstrate the different assumptions most drastically the case is considered where compressive 2 does not lead to a premature final failure due to the wedge effect. Because of the assumption that for compressive 2 the cracks stay closed after IFF initiation Puck does not degrade E2s and v12. Therefore a rapid -degradation of 21 alone has to keep f (IFF) = 1. In contrast to this Cuntze treats the compressive stress 2 similar to a tensile stress E

485 2 with a pronounced softening branch of the stress/strain diagram (Fig. 11). He is basing this approach on weakening effects due to the cracking. A.2. Additional biaxial (2, 21)-test data for UD-laminae Fig. 34 provides additional test data for one GFRP and one CFRP material [6]. As test specimen the wound ‘tension/compression- torsion tube specimen’ was used. A.3. Visualization of the reserve factors of a uniformly loaded sandwich plate In this Appendix a still existing example (Table A3/1) is taken in order to visualize the ‘handling’ with the values computed for f (mode) and f (res) Res Res (further see Table A3/2). Failure conditions and corresponding curve parameters are not depicted here, because they belong to a slightly different former set of fracture conditions [26]. But, the following results nevertheless highlight how the designer will work with reserve factors or with efforts. The listing of the reserve factors values in Table A3/2 completely describes the stress situation in all the laminae, with (k) indicating the particular lamina. Numbers in quadratic brackets refer to the finite element (the FE code MARC was employed). Lamina k = 3 contains the minimum resultant reserve factor min f (mode) (0.82), which is due Res to the 90°-angle still a little smaller than that for the 45°-lamina (k = 2) for which Fig. A3/1

Fig. A1/5. Schematic illustration of Puck’s assumptions about the average stresses 2() and 2() after the onset of IFF. The strain repr. is a representative strain, which is proportional to the load on the laminate.

486

Fig. A1/6. Schematic illustration of Cuntze’s assumptions about the stresses 2 and 21 before and after IFF-initiation. Results of the ‘triggering approach’.

Table A3/1 Geometry and loads of the GFRP sandwich plate. Mechanical properties (mean values) and strength properties (vf = 0.60) and data of calibration points (mean values).

487 Table A3/2 Reserve lactors calculated for all laminae [12].

k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k = 10 k = 11

0° 45° 90° – 45° 0° core 0° – 45° 90° 45° 0°

f (res) Res(k)

f 储 Res(k)

f 储Res(k)

f ⊥ Res(k)

f ⊥ Res(k)

f 储Res(k)

1.33 [142] 0.87 [267] 0.82 [686] 0.94 [750] 1.92 [1146] Not relevant here 4.02 [1618] 1.39 [1880] 2.99 [1984] 1.29 [2336] 2.27 [2583]

12.8 [202] 27.6 [349] 38.6 [574] 31.6 [821] 14.9 [996]

100. [. . 1] 53.3 [340] 41.2 [606] 87.0 [762] 100. [961]

1.34 [142] 0.893 [267] 0.820 [686] 0.964 [750] 1.94 [1146]

3.13 [141] 1.75 [267] 4.62 [559] 1.89 [766] 5.21 [1146]

3.51 [142] 2.26 [267] 2.15 [686] 2.44 [750] 5.25 [1146]

100 [1441] 100 [1681] 51.4 [2054] 66.6 [2214] 100 [2401]

11.9 [1567] 25.3 [1681] 30.9 [2029] 22.1 [2243] 10.2 [2640]

4.11 [1618] 1.47 [1880] 3.52 [2069] 1.37 [2336] 2.31 [2583]

6.36 [1667] 2.27 [1880] 5.25 [2029] 2.11 [2298] 3.55 [2583]

6.84 [1618] 2.90 [1880] 2.99 [1984] 2.68 [2336] 4.56 [2582]

is prepared. This plot depicts the distribution of the resultant reserve factor for the lamina ( = + 45°, k = 2) of the sandwich plate. Its smallest value 0.89 is also lower than 1, thus indicating IFF which would cause a redesign if IFF is not permitted. The IFF is caused by t⊥ in the lamina’s plane and is critical over a large domain of the lamina. Similar to fRes Table A/2 can be filled in by the various Eff. A clear determination of the design driving maxEff (mode) would then be pointed out, too.

Fig. A3/1. Reserve factor f (res) Res of failure responsible lamina k = 2 (lower skin) [G. Sukarie, MARC] [25].

488

Fig. A5. The (21, 2, 3)-failure body.

A.4. Simplifications of FMC-based failure conditions • Simplification of F ⊥: As still briefly mentioned, the F ⊥: condition may be homogenized (I4 is not homogeneous to I2) in the form F ⊥ = (b⊥  1)

I2 兹I4 + b⊥ c = 1. R¯ c⊥ R¯ ⊥

(A12)

which means a replacing of I4 by 兹I4 and of setting b⊥储 = 0. Then, the curve parameter b⊥ may be simply determined from the equation b⊥ =

c ¯ c 1 + (c 2 + 3 )/R ⊥ . c ¯ c c C 2 ¯ c2 兹(c 2 + 3 )/R ⊥ + (2  3 ) /R ⊥

(A13)

This value is different to that of the former bc⊥, of course, because F ⊥ is a new one. Now, the reserve factor is linearly computable due to c   f ⊥ Res = R ⊥/(b⊥  1)I2 + b⊥兹I4.

• Determination of max I33/2 for the case discrimination:

(A14)

489 The limit for the applicability of F⊥储 for the given state of stress (marked by a dot) is ˙ ˙ ˙ I˙ 3/2 3 /I2 I3  I5  · b⊥储, with a preliminary to be confirmed  = 1.1. From the ratio above the limiting maximum value on the failure surface can be deduced via 3 I 3/2 3 + b⊥储(I2 I3  I5) = R ⊥储,

I 3/2 3 /(I2 I3  I5) + b⊥储 · 1 =

R 3⊥储 , (I2 I3  I5)

 · b⊥储 + b⊥储 = R 3⊥储 /max(I2 I3  I5).

(A15)

From max(I2 I3  I5) follows 3 maxI 3/2 3 = R ⊥储  b⊥储 max(I2 I3  I5).

(A16)

• Further simplification of F⊥储 (recommendation): 3

2

F⊥储 = (兹I3 + b⊥储兹I2 I3  I5)/R⊥储 2

3

f ⊥储 Res = R⊥储 /(兹I3 + b⊥储兹I2 I3  I5).

(A17)

A.5. The (21, 2, 3)-failure body The most interesting partial IFF body is that for the stress combination (21, 2, 3). By this failure body (a difficult and time consuming work to produce it) the main differences of the new IFF conditions are displayed: This is at first the difference between (21, 2) and (21, 3), see the typical asymmetry outlined in various papers of Puck et al. According to the rounding procedure the (21, t3 = R t⊥) front side is not fully vertical anymore as by Puck documented. Nevertheless it provides with the main information. Secondly the rounding-off in the (t2, t3)domain is depicted. The post-processing of the failure body has caused some smaller irregularities and should be reworked with a better tool.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

490

CHAPTER 3.15

Composite laminate failure analysis using multicontinuum theory J. Steven Mayes*,1a and Andrew C. Hansenb b

a Division of Mechanical Engineering, Alfred University, Alfred, NY 14802, USA Department of Mechanical Engineering, University of Wyoming, Laramie, WY 82071, USA

Abstract Damage in a composite material typically begins at the constituent level and may, in fact, be limited to only one constituent in some situations. An accurate prediction of constituent damage at sampling points throughout a laminate provides a genesis for progressively analyzing failure of a composite structure from start to finish. Multicontinuum Theory is a micromechanics based theory and associated numerical algorithm for extracting, virtually without a time penalty, the stress and strain fields for a composites’ constituents during a routine finite element analysis. A constituent stress-based failure criterion is used to construct a nonlinear progressive failure algorithm for investigating the material failure strengths of composite laminates. The proposed failure analysis methodology was used to simulate the nonlinear laminate behavior and progressive damage of selected laminates under both uniaxial and biaxial load conditions up to their ultimate strength. This effort was part of a broader project to compare the predictive capability of current composite failure criteria. Keywords: Multicontinuum

Nomenclature ± {a} [A] CTE [C ] [C#] Fi, Fij

Indicates the appropriate tensile or compressive value is used depending on the constituent’s stress state. Vector relating constituent to composite thermal strains. Matrix relating constituent to composite mechanical strains. Coefficient of thermal expansion. Composite stiffness matrix. Constituent # stiffness matrix [# = f (fiber), m (matrix)]. Strength parameters (i, j = 1–6).

* Corresponding author. 1 Formerly of the Naval Surface Warfare Center, Carderock Division, Structures and Composites Department, West Bethesda, MD, USA.

491 Ii Ii# ± Ki# ± Sij# ± kl

Sij#

V {} {ii#}

˜ {$} ˜ # {$#} {$o} {$#o} # ˜ {} ij ˜ # {#} ij# T [1]

Composite transversely isotropic stress invariants (i = 1–5). Constituent # transversely isotropic stress invariants (i = 1–5). Constituent # failure parameter [# = f (fiber), m (matrix)]; (i = 1–4). Constituent # strength in the i, j direction [# = f (fiber), m (matrix)]; (i, j = 1–3). Constituent # stress in the i, j direction when stress ± Skl is applied [# = f (fiber), m (matrix)]; (i, j = 1–3); (k, l = 1–3). Volume. Composite coefficients of thermal expansion. Constituent # coefficients of thermal expansion [# + f (fiber), m (matrix)]; (i = 1, 2). Composite strain tensor. Composite total strain tensor in contracted (matrix) notation. Constituent # strain tensor [# = f (fiber), m (matrix)]. Constituent # total strain tensor in contracted notation [# = f (fiber), m (matrix)]. Composite thermal strain tensor in contracted notation. Constituent # thermal strain tensor in contracted notation [# = f (fiber), m (matrix)]. Constituent # volume fraction [# = f (fiber), m (matrix)]. Composite stress tensor. Composite stress tensor in contracted notation. Composite stresses referenced to the lamina (i, j = 1–3) or laminate (i, j = x to z) coordinate system. Constituent # stress tensor [# = f (fiber), m (matrix)]. Constituent # stress tensor in contracted notation [# = f (fiber), m (matrix)]. Constituent # stresses referenced to the lamina coordinate system [# = f (fiber), m (matrix)]; (i, j = 1 to 3). Difference between current and reference temperatures. Identity matrix.

1. Introduction A majority of failure criteria developed for composite materials to date can be classified as macromechanical because the criteria attempt to predict failure using composite stress–strain data. A key element of macromechanics is the combining of constituent’s properties into a homogeneous set of composite lamina properties and possibly combining lamina properties into homogeneous laminate properties. In contrast, micromechanical failure analyses retain the individual identities of each lamina and its constituents, thereby allowing interaction among them. Composite properties are utilized in micromechanics analyses but failure of each constituent and its contribution to lamina and laminate failure is emphasized. All micromechanical models are predicated on a complete set of material constants for each constituent that are consistent with those of the composite they form. This consistency is typically synthesized from a finite element or closed form analytical model of the composite microstructure. Examples of micromechanical

492 approaches can be found in Aboudi [1], Pecknold [2], Rahman [3], and Kwon [4]. A review of these approaches can be found in Mayes [5].

2. Multicontinuum theory Multicontinuum Theory (MCT) is a micromechanics based theory and associated numerical algorithm for extracting, virtually without a time penalty, the stress and strain fields for a composites’ constituents during a routine finite element analysis. MCT development is presented in detail for linear-elastic and linear-viscoelastic composite behavior in papers by Garnich and Hansen [6,7]. The elasticity theory is summarized here to emphasize concepts important to implementing a constituent based failure analysis. The present theory assumes: (1) linear elastic behavior of the fibers and nonlinear elastic behavior of the matrix, (2) perfect bonding between the fibers and matrix, (3) stress concentrations at fiber boundaries are accounted for only as a contribution to the volume average stress, (4) the effect of fiber distribution on the composite stiffness and strength is accounted for in the finite element modeling of a representative volume of microstructure, and (5) ability to fail one constituent while leaving the other intact results in a piecewise continuous composite stress–strain– curve. MCT begins with a continuum definition of stress at a point. The concept of stress in homogeneous materials, such as steel, is a familiar one to most engineers. Yet, if looked at on a microscale, one sees that the “homogeneous” material is hardly homogeneous. It is obvious that stresses will vary significantly from point to point across different phases and inclusions. The homogenized value used to characterize the stress tensor at a point in a single continuum material is derived by taking a volume average of all stresses in the region as ˜ =

1 V

冕

(x) ˜ dV,

(1)

D

where D is the region representing the continuum point. The concept of a multicontinuum simply extends this concept to reflect coexisting materials within a continuum point. In particular, consider a composite material with two clearly identifiable constituents as shown in Fig. 1 [8]. Using Eq. (1) for each constituent we can write: ˜ f =

1 Vf

and ˜ m =

1 Vm

冕

(x) ˜ dV,

(2)

冕

(x) ˜ dV,

(3)

Df

Dm

where D = Df 傼 Dm. Combining Eqs. (1)–(3) leads to

(4)

493

Fig. 1. Composite lamina as a multicontinuum.

˜ = f ˜ f + m˜ m.

(5)

where f and m are the volume fractions of fiber and matrix respectively. Likewise, for strains we have

˜ = f ˜ f + m˜ m.

(6)

It is important to note the averaging process that results in these equations. That is, we are not concerned with stress and strain variations through individual constituents within D but only with their average values. This is an information compromise that separates structural analysis from micromechanical analysis. Accounting for stress variations throughout every fiber at every material point in even a modest structure is simply not possible or desirable. In contrast, providing constituent average stress and strain fields opens a new and manageable information window on a composite material’s response to a load. Changing from direct tensor to contracted matrix notation, the elastic constitutive laws for the composite and the constituents are given by {} = [C ]({}  {o}),

(7)

{f } = [Cf ]({f }  {fo}),

(8)

{m} = [Cm]({m}  {mo}).

(9)

and

Combining Eqs. (5)–(9), constituent fiber and matrix strain fields, {f } and {m} respectively, are derived from the composite strain field {$} using {m} = (m[1] + f [A])–1({}  T {a}),

(10)

and {f } =

1 ({}  m{m}), f

(11)

494 where [A] = 

m ([C ]  [Cf ])–1([C ]  [Cm]), f

and {a} = ([C ]  [Cf ])–1([C ]{}  f [Cf ]{f }  m[Cm]{m}). An isothermal version of Eq. (10) appeared in early work by Hill [9]. Typically [Cf ], [Cm], {f }, and {m}, are developed from known material properties of the constituents, while [C ] and {} of the composite are developed from micromechanical modeling of an assumed fibermatrix distribution incorporating the constituent material properties. Hence, [A] and {a} are known a priori to a structural analysis. A major advantage of a MCT analysis is the increased computational efficiency gained by the theory’s decoupling of micromechanical modeling from structural analysis. MCT’s ability to calculate accurate constituent stress and strain fields is dependent on constituent elastic constants derived from experimentally determined composite values. Further, MCT’s ability to execute realistic failure analysis is dependent on accurate values for constituent strength parameters, also derived from experimentally determined composite values. The link establishing a relationship between composite (macro) and constituent (micro) elastic constants is a finite element micromechanics model for a continuous fiber unidirectional composite. The finite element micromechanics model used in this research was advanced by Garnich [10] which contains discussion of its development. Only major components of the model will be summarized here. The micromechanics model is based on an assumption of uniform hexagonal fiber packing within the lamina’s matrix (Fig. 2). A unit cell, representative of the repeating microstructure, is extracted from a region bounded by symmetry lines. Unit cell geometry, fiber volume fraction, and boundary conditions are used to define the finite element model (Fig. 3). The unit cell is based on a generalized plane strain assumption in the fiber direction but is fully threedimensional. The cell is modeled with a finite element scripting language allowing material properties and fiber volume fraction to be varied as required. Boundary conditions [10,11] necessary to enforce compatibility of unit cell boundaries with adjacent unit cells are generated automatically. Four linear elastic load cases are solved (longitudinal tension, transverse tension, transverse shear, and longitudinal shear) to determine and verify five independent elastic constants for transversely isotropic composite lamina.

Fig. 2. Idealized lamina microstructure.

495

Fig. 3. Finite element model of a unit cell created using the ANSYS software.

All constituent elastic constants (Tables 1 and 2) and strengths (Tables 3 and 4) were backed out via the micromechanics model from experimentally determined composite values

Table 1 Matrix elastic constants calculated from micromechanics. Matrix

Em (GPa)

Gm (GPa)

vm

m (10–6/°C)

3501-6 BSL914C LY556/HT907/DY063 MY750/HY917/DY063

4.50 4.50 4.95 4.85

1.68 1.68 1.83 1.78

0.340 0.340 0.355 0.360

35 35 52 49

Table 2 Fiber elastic constants calculated from micromechanics. Fiber

E11f (GPa)

E22f (GPa)

G12f (GPa)

G23f (GPa)

v12f

v23f

11f (10–6/°C)

22f (10–6/°C)

AS4 T300 E-glass 21xK43 Gevetex Silenka E-glass 1200tex

207.5 227.0 83.2 73.0

25.0 25.0 83.2 73.0

95.0 28.0 33.5 29.6

9.20 9.50 33.5 29.6

0.240 0.245 0.240 0.235

0.359 0.316 0.240 0.235

–1.7 –1.7 6.9 6.6

15 15 6.9 6.6

496 Table 3 Matrix strengths calculated from micromechanics. Matrix

+ S22m (MPa)

 S22m (MPa)

+ 22 S33m (MPa)

 22 S33m (MPa)

S12m (MPa)

3501-6 BSL914C LY556/HT907/DY063 MY750/HY917/DY063

42.3 23.2 27.3 31.5

–176.3 –172.2 –88.9 –114.3

5.52 2.73 4.26 4.63

–23.0 –20.2 –13.9 –16.8

49.54 50.8 44.9 46.3

Table 4 Fiber strengths calculated from micromechanics. Fiber

+ S11f (MPa)

 S11f (MPa)

S12f (MPa)

AS4 T300 E-glass 21xK43 Gevetex Silenka E-glass 1200tex

3202 2466 1772 2040

–2431. –1480. –886. –1275.

101 105 93.0 95.2

provided by the organizers [12]. These in situ constituent values used in the MCT analyses conducted herein were different than those presented for this exercise by the organizers.

3. Failure criterion The Maximum Distortional Energy, or von Mises, criterion is the most widely used criterion for predicting yield points in isotropic metals [l3]. The isotropic von Mises failure criterion is a special case of a general form of quadratic interaction criteria, so named because they include terms to account for interaction between the stress components. Variations of the general criteria have been used to predict brittle failure in orthotropic materials [14]. A generalized quadratic interaction failure criterion, suggested by Gol’denblat and Kopnov [15] and proposed by Tsai and Wu [16], is given as Fi i + Fij i j = 1,

(12)

where Fi and Fij are experimentally determined strength tensors and contracted tensor notation is used (i, j = 1–6). Hoffman [17] has suggested that the linear terms, Fi, are necessary to account for differences in tensile and compressive strengths whereas Tsai and Wu state that they are necessary to account for internal stresses. Tsai and Wu [16] presented a form of Eq. (12) for transversely isotropic composites as F111 + F2(22 + 33) + F11211 + F22(222 + 233 + 2223) + F66(212 + 213) + 2F12(1122 + 1133) + 2F23(2233  223) = 1.

(13)

497 Hashin [18] developed a three-dimensional, stress interactive, failure criterion for unidirectional lamina that recognized two distinct and uncoupled failure modes. While the failure criterion itself was based on composite stresses, it constructs a piecewise continuous failure form based on constituent failure modes. The failure criterion assumes transverse isotropy for a unidirectional composite. A local orthogonal coordinate system is defined in which the fiber axis serves as the principal, x1, material direction, and x2, x3 the transverse and throughthickness directions. The failure state of the material is expressed in terms of transversely isotropic stress invariants. Although Hashin derived these invariants, Hansen [19], in development of an anisotropic flow rule for plastic behavior in composite materials, presented a different form used within this paper. The five transversely isotropic stress invariants are: I1 = 11, I2 = 22 + 33, I3 = 222 + 233 + 2223, I4 = 212 + 213, I5 = 22212 + 33213 + 2121323.

(14)

Hashin’s choice of a quadratic form eliminates I5 from appearing in the failure criterion. Therefore the most general form for a quadratic criterion [18] is K1I1 + L1I 21 + K2I2 + L2I 22 + M12I1I2 + K3I3 + K4I4 = 1,

(15)

where Ki, Li, and M12 are experimentally determined failure coefficients. At this point, it is instructive to compare the criterion of Tsai and Wu with that of Hashin. Rewriting Eq. (13) in terms of the transversely isotropic stress invariants gives F1I1 + F2I2 + F11I 21 + F22I3 + F66I4 + 2F12I1I2 + 2F23(I 22  I3) = 1, or rearranging, F1I1 + F11I 21 + F2I2 + 2F23I 22 + 2F12I1I2 + (F22  2F23)I3 + F66I4 = 1.

(16)

Comparing Eq. (16) to Eq. (15), shows that the Tsai–Wu criterion for transversely isotropic materials and the Hashin failure criterion have the same functional form. Their difference is in defining the coefficients of the stress terms. The Tsai–Wu equation is used to define a smooth and continuous failure surface in both the tension and compression regions of space.

Table 5 Nonlinear shear curve fit parameters. Composite

B0 (Pa)

B1 (Pa)

B2 (Pa)

h1

h2

AS4/3501-6 T300/BSL914C E-glass/LY556/HT907/DY063 E-glassMY750/HY917/DY063

3.31E + 14 1.64E + 11 5.76E + 10 2.69E + 10

–1.09E + 14 –1.51E + 8 –9.51E + 7 –9.961E + 7

4.39E + 14 –1.63E + 11 –5.75E + 10 –2.68E + 10

–0.0536 –43.7 –71.9 –63.1

–0.0132 0.00654 0.00706 0.0161

498 As a result the coefficients are functions of both tensile and compressive composite strengths. In contrast, Hashin identified two composite failure modes; fiber versus matrix influenced, and developed separate equations based on the failure mode to determine a failure state. Hashin further recognized that a composite typically has different ultimate strengths in tension and compression, so both fiber and matrix failure criteria have tensile and compressive subforms. Hence the coefficients of the stress terms are functions of only tension or compression strengths resulting in a piecewise continuous stress-space failure surface. In what follows, we adopt the view of Hashin and develop separate failure criteria for the fiber and matrix failure modes. However, in a major departure from Hashin’s work, we develop failure criteria in the form of Eq. (15) for each constituent as opposed to the composite by utilizing constituent stress information produced by MCT. As a consequence, the transversely isotropic stress invariants, defined in Eq. (14), were used for each constituent of the composite material under consideration. Furthermore, recognizing that constituents typically have different ultimate strengths in tension and compression, each constituent failure criterion has a tensile and compressive subform. A unique aspect of the MCT failure theory is that an anisotropic failure theory is used on an isotropic matrix material. This complexity is necessitated by the fact that the matrix failure behavior will be anisotropic due to microstructural geometry. The root of this phenomenon can be conceptualized by considering a transversely isotropic unidirectional composite. If all fibers were removed but their holes retained only a matrix of “Swiss Cheese” would remain. Because of the remaining microstructure, macroscopic failure of the material will be fundamentally different in axial versus transverse directions resulting in a transversely isotropic failure envelope. As a first approximation, we would like to simplify Eq. (15) for each of the constituents. Pipes and Cole [20] demonstrated some of the difficulties in experimentally determining stress interaction terms such as M12, analogous to F12 in the Tsai–Wu theory. Further, Narayanaswami [21] demonstrated numerically that setting the stress interaction term F12 to zero in the Tsai–Wu quadratic failure criterion in plane stress analyses resulted in less than 10% error for all the load cases and materials considered. Hence, we set M12 equal to zero. Tsai and Wu identify the linear terms in Eq. (13) as necessary to account for internal stresses. Internal stresses refer to self equilibrating stresses within each constituent which, when added together according to Eq. (5), produce no composite stress. Internal stresses may arise in composites operating at a temperature other than the reference temperature due to a mismatch in constituent coefficients of thermal expansion. These internal stresses are accounted for in the formulation of Multicontinuum Theory through the {a} vector. Thus we eliminate the linear terms from Eq. (15). If analytical comparisons against experimental results do not provide a satisfactory correlation, these terms, along with the term M12, could be reexamined for their potential contributions. Noting the above, the general form for a stress interactive failure criterion, after changing to a consistent coefficient notation, is given by K1I 21 + K2I 22 + K3I3 + K4I4 = 1.

(17)

Developing a form of Eq. (17) for fiber failure we note that the majority of fibers used for composite reinforcement have greater transverse strengths than the matrices commonly used in conjunction with them. Hence, we assume that transverse failure of these composites is matrix dominated. Based on this assumption, we set K2 and K3 equal to zero in Eq. (17) as

499 their associated stress invariants involve transverse normal stresses. The fiber failure criterion reduces to K1f I 21f + K4f I4f = 1.

(18)

To determine coefficients for each stress term, we solve Eq. (18) considering individual load cases of pure in-plane shear, tension, and compression applied to unidirectional lamina. For the case of in-plane shear load only (12f 0, ≠ 11f = 0), we find K4f =

1 , S 212f

where, S12f denotes fiber shear strength. For the case of tensile load only (11f > 0; 12f = 0), we find +

K1f = +

1 . S 211f

For the case of compression load only (11f < 0; 12f = 0), we find 

K1f = 

1 . S 211f

The criterion for fiber failure can now be expressed as ± K1f I 21f + K4f I4f = 1.

(19)

The ± symbol indicates that the appropriate tensile or compressive ultimate strength value is used depending on the constituent’s stress state. To determine the coefficients of Eq. (17) for matrix failure we first solve the equation considering load cases of pure in-plane and transverse shear. For the case of transverse shear only (23m ≠ 0, 11m = 22m = 33m = 12m = 0), we find K3m =

1 . 2S 223m

For the case of in-plane shear only (12m ≠ 0, 11m = 23m = 33m = 12m = 0), we find K4m =

1 . S 212m

Noting that a majority of fibers used for composite reinforcement have greater longitudinal strengths than the matrices commonly used in conjunction with them, we assume that the longitudinal failure of these composites is fiber dominated. Based on this assumption and some numerical sensitivity studies we set K1m equal to zero. The approach to our ‘sensitivity analysis’ was to conduct failure analyses, with and without parameter K1m in the proposed failure criteria, on all available test cases. We determined that the presence of K1m did not significantly affect failure predictions results for those cases. Incorporating these results into (17) gives K2m I 22m +

1 1 I3m + 2 I4m = 1. 2 2S 23m S 12m

(20)

500 To determine K2m, we consider the case of transverse tensile load only [(22m + 33m) > 0, 23m = 12m = 0] and find

+

K2m =

1 ( S22m + + + 22S32m)2 +

冉

+

1

冊

S 222m + + 22S 233m . 2S 223m

The numeric superscripts (“22”) in the above failure parameters are used to denote the direction of the applied load. Note that while a pure transverse (one-dimensional) load, 22, on a composite lamina results in 11 = 33 = 0, the constituents experience a fully three-dimensional stress state [6,7]. Likewise, for the case of a pure transverse compressive load [(22m + 33m) < 0, 23m = 12m = 0] –

K2m =

1 ( S22m + –22S33m)2 –

冉

–

1

冊

S 222m + –22S 233m . 2S 223m

The criterion for matrix failure can now be expressed as ±

K2mI 22m + K3mI3m + K4mI4m = 1.

(21)

Transverse shear strength values were not provided as part of the material characterizations provided by the organizers [12]. Parameter ± K2m is highly sensitive to these values and rather than risk using inaccurate values, the matrix failure criterion was modified. Expanding Eq. (21) in terms of local stress components gives ±

K2m(22m + 33m)2 + K3m(222m + 233m + 2223m) + K4m(212m + 213m) = 1.

For the load cases considered in this paper, no transverse shear stresses arise in the constituents so we set 23m = 0 and rearrange the above as ( ± K2m + K3m)222m + ( ± K2m + K3m)233m + ± K2m(222m33m) + K4m(2m12 + 213m) = 1.

(22)

±

K2m scales a stress interaction in the third term of Eq. (22). We set this scale factor to zero as was done previously in the simplification of Eq. (15). This results in ±

K3m(222m + 233m) + K4m(212m + 213m) = 1.

Therefore, in terms of the transversely isotropic stress invariants, the modified matrix failure criterion becomes ±

K3mI3m + K4mI4m = 1.

(23)

where ±

K3m = ±

S

2 22m

1 . + ± 22S 233m

The mode of failure, fiber or matrix, is determined by monitoring their failure criteria given by Eqs. (19) and (23), respectively. The relative contribution of the various stress components to initial, intermediate, and final failure states can be determined by examining the product of

501 Table 6 Initial E-glass/LY55/HT907/DY063 failure envelope summary for a [90°/ ± 30]S laminate under biaxial, y /x, load. Point

Lamina

Primary term

Secondary term

Failure mode

a b c d e

± 30 90 90 ± 30 90

K3m I3m = 1.0 K3m I3m = 1.0 K3m I3m = 1.0 K3m I3m = 0.66 K3m I3m = 1.0

K4m I4m = 0.0 K4m I4m = 0.0 K4m I4m = 0.0 K4m I4m = 0.34 K4m I4m = 0.0

Matrix – tension Matrix – tension Matrix – tension Matrix – comp/shear Matrix – compression

Table 7 Final E-glass/LY55/HT907/DY063 failure envelope summary for a [90°/ ± 30°]S laminate under biaxial, y /x, load. Point

Lamina

Primary term

Secondary term

Failure mode

A B C D E

30 90 ± 30 90 90 ± 30 ± 30 90 ± 30 90 90 ± 30

K4f I4f = 0.77 K1f I1f = 1.0 K1f I1f = 0.78 K1f I1f = 1.0 K1f I1f = 1.0 K4f I4f = 0.83 K3m I3m = 0.66 K1f I1f = 1.0 K3m I3m = 0.97 K1f I1f = 1.0 K3m I3m = 1.0 K4f I4f = 0.71

K1f I1f = 0.33 K4f I4f = 0.0 K4f I4f = 0.22 K4f I4f = 0.0 K4f I4f = 0.0 K1f I1f = 0.17 K4m I4m = 0.34 K4f I4f = 0.0 K4m I4m = 0.03 K4f I4f = 0.0 K4m I4m = 0.0 K4f I4f = 0.29

Fiber – comp/shear Fiber – tension Fiber – tension/shear Fiber – compression Fiber – compression Fiber – shear/tension Matrix – comp/shear Fiber – compression Matrix – compression Fiber – compression Matrix – compression Fiber – shear/comp

F G H

Table 8 Initial E-glass/LY55/HT907/DY063 failure envelope summary for a [90°/ ± 30°]S laminate under biaxial, x /y, load. Point

Lamina

Primary term

Secondary term

Failure mode

a b

90 90 –30 –30 90

K3m I3m = 1.0 K3m I3m = 0.72 K3m I3m = 0.95 K3m I3m = 1.0 K3m I3m = 1.0

K4m I4m = 0.0 K4m I4m = 0.28 K4m I4m = 0.05 K4m I4m = 0.0 K4m I4m = 0.0

Matrix – compression Matrix – comp/shear Matrix – tension/shear Matrix – tension Matrix – tension

c d

502 Table 9 Final E-glass/LY55/HT907/DY063 failure envelope summary for a [90°/ ± 30°]S laminate under biaxial, x /xy, load. Point

Lamina

Primary term

Secondary term

Failure mode

A B

+ 30 90 – 30 – 30 + 30 – 30 90 + 30 – 30 + 30 + 30 – 30 + 30 + 30 90

K4f I4f = 0.54 K3m I3m = 0.72 K3m I3m = 0.95 K1f I1f = 0.86 K4f I4f = 0.98 K1f I1f = 1.0 K4f I4f = 0.91 K4f I4f = 1.0 K1f I1f = 1.0 K1f I1f = 0.95 K4m I4m = 0.72 K4m I4m = 0.61 K4f I4f = 0.77 K4f I4f = 0.77 K1f I1f = 1.0

K1f K1f = 0.46 K4m K4m = 0.28 K4m K4m = 0.05 K4f K4f = 0.14 K1f K1f = 0.02 K4f K4f = 0.0 K1f K1f = 0.08 K1f K1f = 0.0 K4f K4f = 0.0 K4f K4f = 0.05 K3m K3m = 0.28 K3m K3m = 0.39 K1f K1f = 0.23 K1f K1f = 0.23 K4f K4f = 0.0

Fiber – shear/comp Matrix – comp/shear Matrix – tension Fiber – comp/shear Fiber – shear Fiber – compression Fiber – shear/tension Fiber – shear Fiber – tension Fiber – tension Fiber – shear/tension Matrix – tension/shear Fiber – shear/tension Fiber – shear/tension Fiber – tension

C

D E F G H

Table 10 Initial AS4/3501-6 failure envelope summary for a [0°/90°/ ± 45°]S laminate under biaxial, y /x, load. Point

Lamina

Primary term

Secondary term

Failure mode

a b c

All 90 90

K3m I3m = 1.0 K3m I3m = 1.0 K3m I3m = 1.0

K4m I4m = 0.0 K4m I4m = 0.0 K4m I4m = 0.0

Matrix – tension Matrix – tension Matrix – tension

Table 11 Final AS4/3501-6 failure envelope summary for a [0°/90°/ ± 45°]S laminate under biaxial, y /x, load. Point

Lamina

Primary term

Secondary term

Failure mode

A B

All 0 90 ± 45 ± 45 0 90 ± 45 90 90 ± 45 90 ± 45 All

K1f I1f = 1.0 K1f I1f = 1.0 K1f I1f = 1.0 K4f I4f = 0.88 K4f I4f = 0.98 K1f I1f = 1.0 K1f I1f = 1.0 K4f I4f = 1.0 K1f I1f = 1.0 K1f I1f = 1.0 K4f I4f = 0.75 K1f I1f = 1 0 K4f I4f = 0.66 K1f I1f = 1.0

K4f I4f = 0.0 K4f I4f = 0.0 K4f I4f = 0.0 K1f I1f = 0.22 K1f I1f = 0.02 K4f I4f = 0.0 K4f I4f = 0.0 K1f I1f = 0.0 K4f I4f = 0.0 K4f I4f = 0 0 K1f I1f = 0.25 K4f I4f = 0.0 K1f I1f = 0.34 K4f I4f = 0.0

Fiber – tension Fiber – tension Fiber – tension Fiber – shear/tension Fiber – shear/tension Fiber – tension Fiber – compression Fiber – shear Fiber – compression Fiber – compression Fiber – shear/comp Fiber – compression Fiber – shear/comp Fiber – compression

C D E F G H

503 Table 12 E-glass/MY750/HT917/DY063 failure envelope summary for a [ ± 55°]S laminate under biaxial, y /x, load. Point

Lamina

Primary term

Secondary term

Failure mode

A B C D

± 55 ± 55 ± 55 ± 55 ± 55 ± 55 ± 55 ± 55 ± 55

K3m I3m = 1.0 K3m I3m = 0.90 K3m I3m = 0.62 K4m I4m = 0.79 K4f I4f = 0.86 K4f I4f = 0.61 K3m I3m = 0.87 K3m I3m = 0.69 K4f I4f = 0.96

K4m I4m = 0.0 K4m I4m = 0.10 K4m I4m = 0.38 K3m I3m = 0.21 K1f I1f = 0.14 K1f I1f = 0.39 K4m I4m = 0.13 K4m I4m = 0.31 K1f I1f = 0.04

Matrix – tension Matrix – tension/shear Matrix – tension/shear Matrix – shear/tension Fiber – shear/comp Fiber – shear/comp Matrix – comp/shear Matrix – comp/shear Fiber – shear/tension

E F G H

the failure parameter Ki# and its associated stress invariant Ii# (for examples, see Tables 6–12).

4. Description of analysis method A numerical MCT algorithm, based on Eqs. (10) and (11), was developed and incorporated into an in-house finite element code [22]. While the finite element approach may be more powerful than necessary for the analyses conducted as part of this exercise, the methodology was originally developed for failure analyses of general composite structures. Using the finite element framework provides a high degree of analytical flexibility. A majority of composite materials in use today have organic matrices that produce significant nonlinear shear stress–strain behavior as demonstrated by the shear stress–strain curves presented by the organizers [12]. For the research considered herein, unloading or sustained creep of the composite was not a consideration. Therefore, a nonlinear-elastic constitutive model, as developed by Mayes [22], relating changes in elastic constants due to changing composite shear modulus was used. The model uses a three-term exponential series of the form  = B0 + B1e(h1 ) + B2e(h2 ),

(24)

to fit in-plane experimentally determined composite shear stress–strain curves. Bi and hi are curve fit parameters,  is shear stress (Pa), and is engineering shear strain (dimensionless). Nonlinear regression was used to fit the five equation parameters to experimental shear data (Table 5). A strain dependent, tangent shear modulus was computed from the first derivative of Eq. (24) for use during a finite element analysis. Tension and compression elastic moduli for all lamina were assumed to be constant. In the finite element method, numerical integration samples stress, strain, and material values at Gauss quadrature points. MCT failure analyses store a state variable corresponding to composite material damage for every Gauss point. Three composite material conditions or states, listed in increasing damage severity, are defined as:

504 1. undamaged composite, 2. composite damaged by matrix failure, and 3. composite damaged by fiber failure. When either constituent fails, all its moduli are immediately reduced to a near zero value at that Gauss point. Near zero values are used rather than zero to avoid numerical difficulties. Matrix moduli are reduced to 1% of their original value. Fiber moduli, which are typically one to two orders of magnitude larger than matrix moduli, are reduced by whatever percentage is required to bring damaged fiber values to the same magnitude as damaged matrix so that near zero stiffness values are the same for both constituents. Poisson’s ratios remain constant. Their values are rendered irrelevant by the use of near zero moduli values which scale elements of the stiffness matrix, [C#], to near zero values. Since all constituent properties, both intact and failed, are known a priori, the micromechanics model (Fig. 3) is used to determine two additional sets of composite properties, corresponding to damage states 2 and 3, before conducting a MCT failure analysis. The nonlinear character of a failure analysis requires the load to be incrementally applied and the damage tracked progressively. Initially, composite material properties are set to an undamaged condition. At each load step a damage algorithm, using the failure criteria formulated in Eqs. (19) and (23), checks every Gauss point for constituent failure based on accumulative stresses. When constituent failure is detected at a Gauss point, stresses are recalculated using accumulated strains and updated material properties. Gradual softening of the structure due to composite damage at the Gauss points and a nonlinear-elastic constitutive model causes an equilibrium imbalance between the applied (external) and resisting (internal) load vectors. A standard Modified Newton–Raphson nonlinear iterative procedure within each load step calculates differences between external and internal load vectors and applies it to the structure as a “virtual” load. The net effect is to increase nodal displacements, hence Gauss point strains and stresses, until equilibrium is restored and the next load step is then applied. Structural failure of a laminate is defined as that point in the load history when the structure can no longer support the accumulated load and deflections begin to grow without bound. Unbounded growth is detected during equilibrium iterations by monitoring changes in the Euclidean (L2) norm of the structural displacement vector.

5. MCT simulations of load-response to failure for selected laminates Unidirectional (UD) E-glass/LY556/HT907/DY063 and T300/BSL914C lamina failure envelopes under biaxial normal-shear loads are shown in Figs. 4 and 5. These failure envelopes were symmetric about the abscissa and showed a typical quadratic shape caused by interactions between normal and shear stresses in the failure criteria. The weaker matrix was the primary load carrying constituent in the y  xy, loading of the unidirectional (UD) E-glass/LY556/HT907/DY063 lamina. Thus matrix failure determined the final failure envelope. In contrast, the stronger fiber was the primary load carrying constituent in the x  xy biaxial loading of the T300/BSL914C lamina. As a result the lamina failure envelope is sharply skewed in the x direction. The failure envelope for a UD E-glass/MY750 lamina under biaxial x /y, load is presented in Fig. 6. This envelope was characterized by a distinct transition from fiber to matrix failure resulting in a shape analogous to one that would be

505

Fig. 4. Failure envelope tor a [0°] lamina made from E-glass/LY556/HT907/DY063 under biaxial, y /xy, load.

Fig. 5. Failure envelope for a [0°] lamina made from T300/BSL914C under biaxial, x /xy, load.

Fig. 6. Failure envelope for a [0°] lamina made from E-glass/MY750/HT917/DY063 under biaxial, y /x, load.

506 produced by a simple maximum stress ±  ± ±  ± failure criterion (ijf /Sijf or ijm /Sijm). Initial and final lamina failure envelopes in Figs. 4–6 were identical. Initial and final failure envelopes for an E-glass/LY556/HT907/DY063 [90°/ ± 30°]S laminate under biaxial y /x load are shown in Fig. 7. The final failure envelope exhibits a complex shape because of stress interactions between lamina and changing failure modes between constituents. Results for the initial and final failure envelopes are summarized in Tables 6 and 7. The horizontal edge of the initial failure envelope, points a to b, was caused by matrix tensile failure in the ± 30° lamina. The right edge of the initial failure envelope between points b and c is due to matrix tensile failure 90° lamina. Intermediate damage, in the form of matrix failure, occurred later in the ± 30° lamina due to combined tensile and shear stresses. Note that in this regime, all matrix in the laminate had failed but the laminate continued to sustain load. Between points c and d, initial matrix damage slowly switches to a combined compression and shear failure in the ± 30° lamina. From points d to e, the initial and final failure envelopes coincided with compressive matrix failure in the ± 30° lamina controlling the mode. Initial failure from points e to a was due to matrix compressive failure in the 90° lamina. The upper edge of the final failure envelope, points A to B began with combined fiber compression-shear failure in the ± 30° lamina and shifted to fiber tensile failure in the 90° lamina. Catastrophic laminate failure occurred between points B and C due to combined fiber tensile-shear failure in the ± 30° lamina. Fibers in the 90° lamina were still intact. A change

Fig. 7. Failure envelope for a [90°/ ± 30°]S laminate made from E-glass/LY556/HT907/DY063 under biaxial, y /x, load.

507

Fig. 8. Failure envelope for a [90°/ ± 30°]S laminate made from E-glass/LY556/HT907/DY063 under biaxial, x /xy, load.

in the failure envelope shape occurred between points C and D where the failure mode switched to compressive fiber failure in the 90° lamina (in the tension–tension, quadrant I) leaving fibers in the ± 30° lamina intact. Between points D and E, simultaneous fiber failure occurred in the 90° (compressive) and ± 30° (shear) lamina. From points E to F, catastrophic laminate failure became increasingly dependent on fiber shear failure in the ± 30° lamina. From points F to G, the initial and final failure envelopes coincided with compressive matrix failure in the ± 30° lamina which precipitated fiber compressive failure in the 90° lamina. The mechanism for final failure shifted to fiber shear in the in the 30° lamina for the points G to H. Initial and final failure envelopes for an E-glass/ LY55/HT907/DY063, [90/ ± 30]S laminate under biaxial, x /xy, load are shown in Fig. 8. The failure envelope was symmetric about the x axis. Results for both failure envelopes are summarized in Tables 8 and 9. Initial laminate damage in quadrant II, between points a and b, was due to compressive matrix failure in the 90° lamina. Initial failure between points b and c began with combined compression/shear matrix failure in the 90° lamina and tensile/shear matrix failure in the –30° lamina. The failure mode shifted to matrix tension closer to point c. Between points c and d, initial failure was due to tensile matrix failure in the 90 lamina. The step-like shape of the final failure envelope between points A and B was caused by fiber failure oscillating between the ± 30° lamina under combined compressive and shear stresses. In the region about point B, the initial and final failure envelopes coincided. Simultaneous matrix failure in the 90° (compressive) and –30° (tensile) lamina precipitated fiber failure in both the –30° and + 30° lamina. From point C to D, the final failure mode transitioned from

508 fiber compressive failure at C to fiber tensile fiber failure at D in the –30° lamina. At point E, the final failure mode switched to tensile fiber failure in the ± 30° lamina but became increasingly dependent on the shear contribution as one moved towards point F. At point G, final failure began as matrix failure in the –30° lamina, due to combined tensile and shear stresses, which precipitated fiber shear failure in the + 30° lamina. A shape change in the failure envelope occurred at point H due to a switch in failure mode to simultaneous fiber failure in the + 30° lamina (tensile and shear) and 90° lamina (tensile). Both initial and final failure envelopes for a AS4/3501-6, [0°/ ± 45°/90°]S laminate under biaxial, y /x, load are shown in Fig. 9. The failure envelope was symmetric about a line through points A and H. Results for the failure envelopes are summarized in Tables 10 and 11. The initial failure envelope between points a and c was defined by matrix tensile failure in the 90° lamina. At point a, simultaneous matrix failure occurred in all lamina but the laminate retained the ability to sustain load. Later in the load history intermediate laminate damage, in the form of matrix failure in the ± 45° lamina, occurred to the right of points b to c due to combined tensile and shear stresses. The initial and final failure envelopes coincided at point c. Failure there was due initially to matrix failure in the 90° (tension) and ± 45 (shear) lamina precipitating compressive fiber failure in the 90° lamina. Final failure between points A and B was due to fiber tensile failure in the 0° lamina. An abrupt change in shape of the failure envelope occurred at point B as the failure mode shifted to fiber shear stress in the ± 45 lamina. Between points C and D tensile fiber failure in the 0° lamina and compressive fiber failure in the 90° lamina determined final laminate failure. Fiber

Fig. 9. Failure envelope for a [0°/ ± 45°/90°]S laminate made from AS4/3501-6 under biaxial, y /x, load.

509

Fig. 10. Stress/strain curves for a [0°/ ± 45°/90°]S laminate made from AS4/3501-6 under uniaxial tension load y /x = 1/0.

shear stresses precipitated failure in the ± 45° lamina between points D and E. From points F to H final failure was determined by compressive failure of the fiber in the 90° lamina. The stress–strain curves for a AS4/3501-6, [0/ ± 45/0]S laminate under uniaxial tension y /x = 1/0 and y /x = 2/1 are shown in Figs. 10 and 11, respectively. Strain jumps in both plots indicated that initial laminate damage occurred due to transverse matrix tensile failures in the 0° lamina. Intermediate damage in the form of matrix failure in the ± 45° lamina was caused by combined shear and tensile stresses. Under the y /x = 2/1 load, additional intermediate damage occurred through tensile matrix failure in the 90° lamina. Final failure in both laminates was caused by tensile fiber failure in the 90° lamina. The E-glass/MY750/HY917/DY063 failure envelope for a [ ± 55°]S laminate under biaxial, y /x, load is shown in Fig. 12. Initial and final failure envelopes were identical. Results for the failure envelope are summarized in Table 12. Tensile matrix failure in all lamina determined failure from points A to B. It is interesting to note that the E-glass/LY55/HT907/DY063, [90°/ ± 30*]S and the AS4/3501-6, [0°/ ± 45°/90°]S laminates under y /x loading also experienced complete matrix failure in quadrant I but continued to load. From points B to D, rising shear stresses combined with tensile stresses to cause matrix failure. The rough envelope edge around point C was due to the manner in which the load was applied, i.e., load step size and y /x ratio, and does not have physical significance. From points D to E fiber failure under combined shear and compressive stress caused laminate failure. Matrix failure due primarily to compressive stresses determined the failure envelope from points E to F. Rising shear stresses combined with the compressive stresses caused matrix failure between points F and G. Failure due to fiber shear stress began at point G and slowly shifted to fiber tensile failure at point H.

510 Nonlinear shear behavior characterized the stress–strain curves of the E-glass/MY750/ HY917/DY063, [ ± 55°]S laminate under uniaxial load, y /x = 1/0, as shown in Fig. 13. Catastrophic laminate failure was caused principally by shear failure of the fibers. Nonlinear shear effects did not become significant for the [ ± 55°]S laminate under biaxial loading, y /x = 2/1 (Fig. 14), because catastrophic matrix tensile failure occurred at relatively low strain levels. The E-glass/MY750/HY917/DY063 stress–strain curves for a [0°/90°/0] laminate under uniaxial load y /x = 1/0 are shown in Fig. 15. Initial laminate damage due to tensile matrix failure in the 0° lamina occurred at approximately one-third of the ultimate laminate load. This damage is a consequence of the load being applied transversely to the fiber direction in the 0° lamina. Intermediate laminate damage, which was also in the form of matrix tensile failure, occurred in the 90° lamina. This matrix damage was interesting because it occurred in the principal load bearing (11f ) direction which was aligned with the load. Note that there is no term in the matrix failure criterion, Eq. (19), involving stress 11m. Therefore this matrix failure was caused by transverse, 22m and 33m, stresses arising from Poisson’s effect. Tensile fiber failure in the 90° lamina resulted in final laminate failure. Stress–strain curves for a E-glass/MY750/HY917/DY063, [ ± 45]S laminate exhibited near linear response under y /x = 1/1 biaxial load as shown in Fig. 16. Catastrophic tensile failure of the matrix occurred before any significant lamina shear stresses developed. In contrast, Fig. 17 shows a highly nonlinear stress–strain response in a [ ± 45°]S laminate under y /x = 1/–1 biaxial load. These shear induced, laminate strains did not become significantly nonlinear until about the 0.5% level which was approximately twice the ultimate laminate strain in the previous y /x = 1/1 load case. Laminate failure under biaxial load y /x = 1/–1 is due to fiber shear failure.

Fig. 11. Stress–strain curves for a [0°/ ± 45°/90°]S laminate made from AS4/3501-6 under biaxial tension load y /x = 2/1.

511

Fig. 12. Failure envelope for a [ ± 55°]S laminate made from E-glass/MY750/HT917/DY063 under biaxial, y /x, load.

Fig. 13. Stress–strain curves for a [ ± 55°]S laminate made from E-glass/MY750/HT917/DY063 under uniaxial tension load y /x = 1/0.

Fig. 14. Stress–strain curves for a [ ± 55°]S laminate made from E-glass/MY750/HT917/DY063 under uniaxial tension load y /x = 2/1.

512

Fig. 15. Stress/strain curves for a [0°/90°/0°] laminate made from E-glass/MY750/HT917/DY063 under uniaxial tension load y /x = 1/0.

Fig. 16. Stress–strain curves for a [ ± 45°]S laminate made from E-glass/MY750/HT917/DY063 under biaxial load x /y = 1/1.

Fig. 17. Stress/strain curves for a [ ± 45°]S laminate made from E-glass/MY750/HT917/DY063 under biaxial load y /x = 1/–1.

513 6. Comments on the MCT failure results The failure load (stress) for individual points on a failure envelope was taken as the value at the beginning of the load step in which failure occurred. The failure value is therefore dependent on the size of the load step but will monotonically converge with decreasing load step size. Generally the lack of smooth failure envelope edges, (e.g., Fig. 8 in the II and III quadrants) is a result of discrete loading ratios and load step size and has no physical significance. Generating a single 2-dimensional failure envelope took on the order of a hundred finite element runs so time constraints prevented detailed convergence of the failure surfaces. Other rough edges, e.g. Fig. 8 on the positive x axis, are due to changes in failure modes from matrix to fiber and may have physical significance. Establishing initial, intermediate, and final failure envelopes serves to highlight the importance of assessing constituent damage in a structural analysis. The practical implications of the different failure surfaces are in establishing allowable stress levels in a composite design, e.g., at what degree of laminate damage is composite ‘failure’ deemed to have occurred? The MCT approach to failure analysis requires identifying constituent failure modes from composite test data. Identifying the constituent that precipitates failure in longitudinal and transverse lamina tension and compression tests is intuitive and straightforward, i.e., fiber failure for longitudinal loads and matrix failure for transverse loads. Identifying the constituent leading to shear failure is more problematic, as non-catastrophic matrix and fiber damage begins well before ultimate composite strength is achieved [23]. Ultimate constituent shear strengths have previously been determined by utilizing nonlinear regression analysis of load cases involving varying amounts of combined normal and shear stresses [5]. Specifically, we make an educated guess as to each constituent’s shear strength and then use that data to predict lamina failure in off-axis tension tests. Using the experimentally determined lamina failures and our initial guess, we iterate with additional guesses until failure predictions based on constituent shear failure produce composite failures that more or less agree with the experimental data. Armed with these semi-empirical constituent shear strengths we have increased confidence in analysis of more complex problems involving shear. Data from off-angle, balanced, symmetric laminates, [ ± ]S, provide an excellent basis for determining a best fit determination of failure parameters S12m, S23m, and S12f . Hence, some of the laminates analyzed as part of this exercise would, in a normal case, be used as inputs to the failure prediction process. Thermal effects due to curing, were neglected in all analyses conducted as part of the failure exercise. However, as noted previously, in situ material properties, as determined from finite element micromechanics, were utilized in this failure analysis. Differences between the in situ properties used herein and those provided by the organizers may be explained in part by residual thermal stresses. MCT can account for post-cure thermal effects through the thermal vector, {a}, in Eq. (10). MCT’s handling of thermal effects can be demonstrated using the E-glass/MY750/HT917/ DY063 composite as an example. The organizers provide a stress-free reference temperature of 120°C for this material. We assume that uniaxial testing used to determine lamina composite tensile strengths occurred at 20°C. Conducting an MCT analysis of the uniaxial strength test, with a T = –100°C, we backed out the temperature adjusted normal constituent tensile strengths shown in Table 13. A negative T produces internal matrix tensile stresses.

514 Table 13 Effect of T on constituent normal strengths for E-glass /MY750/HY917/DY063. Laminate

+

S11f S11f + S22m  S22m + 22 S33m –22 S33m 

Strength (MPa) T = 0°C

T = –100°C

2040 –1275 31.5 –114.3 4.63 –16.8

2040 –1275 42.0 –103.8 15.09 –6.3

Fig. 18. Stress–strain curves for a [0°/90°/0°] laminate made from E-glass/MY750/HT917/DY063 under uniaxial tension load y /x = 1/0 and T = –100°C.

Accounting for this internal tensile load has the net effect of increasing matrix tensile strength and reducing matrix compressive strength. T = –100°C has no significant effect on the Eglass fiber normal strengths. Next we reanalyzed test case numbers 12 and 13, i.e., y /x = 1/0 loading of the [ ± 45°]S and the [0°/90°/0°] laminates, again assuming a T = –100°C. The MCT program applies T in its entirety as a uniform temperature in the first load step. In the thermal analysis of the [0°/90°/0°] laminate, shown in Fig. 18, the higher matrix CTE (compared to the fiber CTE ) causes the 0° lamina matrix to attempt to contract more, in the

515 Table 14 Effect of T on ultimate laminate strength. Laminate

[ ± 45°]S [0°/90°/0]

Strength (MPa) T = 0°C

T = –100°C

68.8 624

38.4 624

Table 15 Thermally induced matrix stresses in each lamina for T = –100°C. Laminate

[0°]N [ ± 15°]S [ ± 30°]S [ ± 45°]S [ ± 60°]S [0°/90°]S

Laminate stress (MPa) 11m

22m

33m

27.1 26.9 28.4 30.5 28.4 30.5

10.5 13.0 19.5 24.7 19.5 24.7

10.5 11.2 12.5 12.5 12.5 12.5

transverse (global y) direction, than the 90° lamina fibers allow. This lamina interaction induces matrix tensile stresses that partially offset the higher matrix tensile, temperature adjusted, strength. The combined thermal and mechanical matrix tensile stresses cause a 0° lamina matrix tensile failure to occur earlier than the case of T = 0. Later in the load history, the higher matrix tensile, temperature adjusted, strength causes a 90° lamina matrix tensile failure to occur at a higher laminate load than in the case of T = 0. The [0°/90°/0°] laminate ultimate strength, shown in Table 14, is fiber dominated and thus does not change with T = – 100°C. The data listed in Table 14 show that the matrix dominate ultimate strength of the [ ± 45°]S laminate is significantly reduced due to the combination of thermal and mechanical induced matrix tensile stresses. As in the case of the [0°/90°]S laminate, the orthogonal orientation of the ± 45 lamina fibers restrains the matrix thermal contraction inducing high matrix tensile stresses. For comparison purposes, Table 15 presents the magnitudes of thermally induced matrix tensile stresses in several laminate cases. Clearly thermally induced residual cure stresses can be important but in the absence of precise knowledge these stresses induced during the cure process, accounting for thermal processing effects is a questionable endeavor. 7. Concluding remarks MCT is a fully 3-dimensional failure prediction methodology intended to efficiently bring constituent information to bear on the analysis of general composite structures. Because

516 failure of composite laminates begins at the constituent level, the constituent information provided by MCT has tremendous value. Accurate predictions of constituent level failure, within the framework of the finite element method, enables development of a progressive failure analysis for general structures. Permitting only one constituent to fail while keeping the others intact allows load redistribution to other parts of the structure as well as to the remaining constituents. Material failure can be tracked as it occurs region by region. The stiffness and strength of damaged areas can be reduced without necessarily declaring total structural failure. This approach has not been incorporated in general design practice in the past because constituent information was generally unavailable in standard finite element analysis.

Acknowledgements The research reported herein was supported by the Inhouse Laboratory, Independent Research program at the Naval Surface Warfare Center, Carderock Division and by the Office of Naval Research under Grant N00014-97-1-1081.

References [1] Aboudi J. Micromechanical analysis of the strength of unidirectional fiber composites. Compos Sci Technol 1988;33:79. [2] Pecknold DA, Rahman S. Application of a new micromechanics-based homogenization technique for nonlinear compression of thick-section laminates. In: Groves SE, Highsmith AL, editors. Compression response of composite structures, ASTM STP 1185. Philadelphia: American Society for Testing and Materials; 1994. p. 34. [3] Rahman S, Pecknold DA. Micromechanics-based analysis of fiber-reinforced laminated composites. Civil Engineering Studies, UILU-ENG-92-2012, Department of Civil Engineering, University of Illinois, Urbana-Champaign, September 1992. [4] Kwon YW, Berner JM. Micromechanics model for damage and failure analyses of laminated fibrous composites. Eng Fracture Mech 1995;52(2):231. [5] Mayes JS. Micromechanics based failure analysis of composite structural laminates. Naval Surface Warfare Center, Carderock Division Report, NSWCCD-65-TR–1999/15, September 1999. [6] Garnich MR, Hansen AC. A multicontinuum theory for thermalelastic finite element analysis of composite materials. J Compos Mater 1997;31(1). [7] Garnich MR, Hansen AC. A multicontinuum approach to structural anaiysis of linear viscoelastic composite materials. J Appl Mech 1997;64:795. [8] Agarwal BD, Broutman LJ. Analysis and performance of fiber composites, 2nd ed. New York: John Wiley & Sons; 1990. [9] Hill R. Theory of mechanical properties of fibre reinforced materials I. Elastic behaviour. J Mech Phy Solids 1964;12:199–212. [10] Garnich, M. R. 1996. A multicontinuum theory for structural analysis of composite materials. PhD dissertation, University of Wyoming. [11] Brockenbrough JR, Suresh S, Wienecke HA. Deformation of metal matrix composites with continuous fibers: geometrical effects of fiber distribution and shape. Acta Metall Mater 1992. [12] Soden PD, Hinton MJ, Kaddour AS. Lamina properties, lay-up configuration and loading conditions for a range of fibre-reinforced composite laminates. Compos Sci Technol 1998;58(7): 1011.

517 [13] Gibson RF. Strength of a continuous fiber-reinforced lamina. In: Principles of composite material mechanics. New York: McGraw-Hill; 1994. p. 108. [14] Nahas MN. Survey of failure and post-failure theories of laminated fiber-reinforced composites. J Compos Technol Res 1986;8: 138 53. [15] Gol’denblat I, Kopnov VA. Strength of glass reinforced plastics in the complex stress state. Mekhanika Polimerov 1965;1:70 [English translation: Polymer Mechanics, 1966, 1, 54]. [16] Tsai SW, Wu EM. A general theory of strength for anisotropic materials. J Compos Mater 1971;5:58. [17] Hoffman O. The brittle strength of orthotropic materials. J Compos Mater 1967;1:200. [18] Hashin Z. Failure criteria for unidirectional fiber composites. J Appl Mech 1980;47:329. [19] Hansen AC, Blackketter DM, Walrath DE. An invariant-based flow rule for anisotropic plasticity applied to composite materials. J Appl Mech 1991;58:881. [20] Pipes RB, Cole BW. On the off-axis strength test for anisotropic materials. J Compos Mater 1973;7:246. [21] Narayanaswami R, Adelman HM. Evaluation of the tensor polynomial and hoffman strength theories for composite materials. J Compos Mater 1977;11:366. [22] Mayes JS. Multicontinuum failure analysis of composite structural laminates. PhD dissertation, University of Wyoming, 1999. [23] Gipple K, Camponeschi ET. The influence of material nonlinearity and microstructural damage on inplane shear response of carbon/epoxy composites. Advanced Composites Lett 1992;1(1) 9.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

518

CHAPTER 3.16

A bridging model prediction of the ultimate strength of composite laminates subjected to biaxial loads Zheng-Ming Huang*,1 Department of Mechanics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, People’s Republic of China Abstract A micromechanical prediction procedure is described in this paper to simulate the progressive failure strength of a composite laminate subjected to multiaxial loading. Casting the loading in an incremental form, the stress increments exerted on each lamina in the laminate can be determined based on the instantaneous stiffness matrix of the lamina. A recently developed micromechanics model, the bridging model, is used to define this instantaneous stiffness matrix and to relate the stress increments in the constituent fiber and resin materials of the lamina with those stress increments exerted on it. The thermal residual stresses generated in the constituents due to mismatch between coefficients of thermal expansion in the fiber and resin materials are clearly addressed and have been incorporated in the analysis. As long as one of the constituents attains its ultimate stress state, the lamina is considered to have failed and its contribution to the overall stiffness matrix of the laminate is reduced. A total reduction strategy is employed in the paper. In this way, the progressive failure process in the laminate can be understood, and the corresponding failure mode can be identified automatically. A generalized maximum normal stress criterion is adopted to detect the constituent failure. The prediction procedure has been applied to 14 different composite laminates where failure envelopes or stress–strain curves up to final failure subjected to biaxial loading are required. The predicted results have been reported in figures and data tables. Keywords: Unidirectional lamina; B. Mechanical properties; B. Stress–strain curve; C. Failure criterion

Nomenclature 1 2

local coordinate direction along fiber axis, i.e. longitudinal direction local coordinate direction transverse to the fiber axis

* Corresponding author. Tel.: + 65-874-4693;fax: + 65-779-1459. E-mail address: [email protected] (Z.-M. Huang). 1 Present address: Polymer & Textile Composites Laboratory, Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore.

519 3 E11 E22 G12 G23 v12 v23 E G v [S ] [S ]e [S ] p [C ] [I ] 1 V V f m ET Y e %ij MT 11, 22, 33, 23, 13, 12 (d11, d22, d33, d23, d13, d12) 11, 22, 33, 23, 13, 12 (d11, d22, d33, d23, d13, d12) ij [A] [Aij ] [B] #,  u u,c eq (1), (2), (3)

local coordinate direction, also transverse to the fiber axis Young’s modulus in longitudinal, or fiber axial, direction Young’s modulus in direction transverse to fibers in-plane shear modulus transverse shear modulus longitudinal Poisson’s ratio transverse Poisson’s ratio Young’s or effective modulus shear or effective shear modulus Poisson’s ratio (for resin material) compliance matrix, with Sij being compliance elements elastic component of the compliance matrix, [S ] plastic component of the compliance matrix, [S ] stiffness matrix, with Cij being stiffness elements unit matrix superscript denoting inverse to a matrix volume fraction volume of a geometric occupation suffix (superscript or subscript) denoting a quantity of fibers suffix (superscript or subscript) denoting a quantity of resin (matrix material) hardening modulus (tangent to uniaxial stress–strain curve in a plastic region) yield strength von Mises equivalent stress, only for resin material in the text Kronecker delta ( = 0, if I ≠ j; = 1, if i = j) material parameter for resin, see Eq. (4.4) for definition

stresses (stress increments) in the lamina local coordinate system

strains (strain increments) in the lamina local coordinate system deviatoric stress component, see Eq. (4.5) for definition two-dimensional bridging matrix used in the main text, with aij (i ≤ 3 and j ≤ 3) being bridging elements three-dimensional bridging matrix used in Appendixes only, with elements of Aij (i ≤ 6 and j ≤ 6) (Vf [I ] + Vm[A])–1, with bij being its elements bridging parameters (without any suffix) ultimate tensile strength (in longitudinal direction, for fiber material) ultimate compressive strength (in longitudinal direction, for fiber material) equivalent maximum normal stress, defined in Eq. (21.2) principal stresses of a material

520 q u11, u22, u12 T T (T )  {b} N h x, y, z G  [T ]c, [T ]s u0, v0, w0 d0xx, d0yy, d0xy d0xx, d0yy, d0xy dNxx, dNyy, dNxy dMxx, dMyy, dMxy d1st, d2nd

power-index used in the generalized maximum normal stress criterion longitudinal, transverse, in-plane shear strength of a unidirectional composite (lamina) temperature superscript denoting transposition to a matrix or vector superscript denoting a temperature dependent quantity thermal expansion coefficient (with a subscript) thermal stress concentration factor, with bi being its elements number of plies in a laminate thickness of a laminate global coordinates for a laminate, with their origin on the middle surface of the laminate and z normal to the middle surface superscript denoting global coordinate system inclined angle of fibers with the global x-direction coordinate transformation matrices laminate middle surface displacements in x-, y-, and z-directions, respectively laminate middle surface strain increments laminate middle surface curvature increments in-plane force increments on laminate per unit length in-plane moment increments on laminate per unit length in-plane bi-axial stress increments applied to the laminate

1. Introduction Most structural components of composite materials in engineering applications consist of multidirectional plies of laminates to meet the specific structural requirement. In order to make a judgment on the structural safety, knowledge of strength of the composite laminates subjected to multi-axial loads is necessary. Prediction of this strength is thus of great importance in adequate design of the laminates and in development of material systems. The failure mechanisms in a laminate are much more complicated than the failure mechanisms associated with a unidirectional lamina. Most laminates reveal a progressive failure process when subjected to an external load. In general, the failure of the laminate in any one ply does not imply failure of the other plies, nor does it mean the failure of the whole laminate. The final failure of the laminate is usually followed by the weaker ply failure in the form of resin2 cracking or, sometimes, laminate delamination initiated at the free edges [1], and occurs at a load level that can be considerably higher than the load level at which the first ply has failed. Efforts have been made over the past several decades to the development of reliable methods to analyze the failure strength of multidirectional composite laminates [2–7]. Recently, Hinton and Soden [6] and Soden, Hinton and Kaddour [7] organized a “failure

2

Throughout this paper, we refer to “resin” rather than “matrix” for the matrix material in a composite in order to avoid any confusing with the matrix representing a mathematical array.

521 exercise” to compare predictive capabilities of a variety of the most important strength theories for laminated composites in current usage. The majority of these theories has been developed based on phenomenological failure criteria for unidirectional laminae [6]. In general, extensive experiments on the composite laminae including, occasionally, biaxial tests are necessary to determine the critical strength parameters involved in these phenomenological failure criteria [2]. Such experiments may be difficult or expensive in some circumstances [8]. Even with the same constituent materials, a different composite only having a different fiber volume fraction generally requires repeated tests. Further, the measured strength parameters of the laminae may still not be sufficient for prediction of the failure behaviors of the composite laminates if the stress–strain responses of the laminae in the laminates have not been fully understood [2]. It has been recognized that non-linear behaviors are important for understanding the failure mechanisms of the composite laminates. While a lamina generally is statically determinate in the measurement of its strength parameters, it becomes always statically indeterminate in a laminate. The load shared by the lamina in the laminate cannot be accurately determined unless all the instantaneous stiffness matrices of the laminae involved have been correctly specified. Therefore, when the laminae can actually undergo inelastic deformations, the use of their linear elastic stiffness matrices up to failure in the laminate strength analysis will result in prediction error no matter whichever phenomenological failure criterion, developed based on lamina strength parameters, has been incorporated. In most cases, the non-linearity of the stress–strain response of a laminate results mainly from two sources: (a) material non-linearity and (b) progressive failure in the laminate. While a unidirectional lamina generally reveals a linear stress–strain response to a longitudinal load, the composite laminate of angle-plied laminae may display a non-linear stress–strain curve even under a uniaxial load. This is because some laminae in the laminate are subjected to the resulting transverse and shear loads. It is thus necessary to take the material non-linearity into account in developing a laminate failure theory. A sophisticated micromechanics theory, which recognizes the non-linear behaviors of the constituent fiber and resin materials and identifies the progressive failure process in the laminate, would show more promising for the prediction of the laminate failure strength. The aim of this paper is to present such a micromechanics failure theory to analyze the biaxial strengths of a number of multidirectional composite laminates. The theory is based on a new micromechanical model, the bridging model, which has been recently developed by the author [9–18] and has been successfully applied to simulate the nonlinear and strength behaviors of a variety of textile fabric reinforced composites [19–24]. This model can predict the overall instantaneous compliance matrix of the lamina made from various constituent fiber and resin materials at each incremental load level and give the internal stresses of the constituents upon the overall applied load. The lamina failure is assumed whenever one of the constituent materials attains its ultimate stress state. Using the classical laminate theory, the overall instantaneous stiffness matrix of the laminate is obtained and the stress components applied to each lamina are determined. If any ply in the laminate fails, its contribution to the remaining instantaneous stiffness matrix of the laminate will no longer occur. In this way, the progressive failure process in the laminate can be identified and the laminate total strength is determined accordingly. The theory has been applied to all the exercise problems, specified by Soden, Hinton and Kaddour in Ref. [25], in which the composite laminates are subjected to various biaxial loads. The predicted results are reported both in figures and in data tables.

522

Fig. 1. A flow chart to show analysis procedure for a multidirectional laminate.

2. Simulation procedure A detailed flow chart (Fig. 1) to show simulation procedure based on an incremental solution strategy is indicated above. Essentially, two steps are observed in the simulation of a multidirectional laminate. In the first step, a laminate analysis is performed. This is accomplished in the present paper by using the classical lamination theory. The purpose of the laminate analysis is to obtain the in-plane strain and curvature increments of the laminate, and further to determine the stresses sheared by each lamina in the laminate. The most important quantity involved in this step analysis is the instantaneous stiffness matrix of the lamina, which will be used to construct the laminate overall stiffness matrix. Initially, all the laminae in the laminate are in linear and elastic deformation, and their stiffness matrices can be defined in a usual way. However, with the increase of load level, the laminae may undergo inelastic deformation. Their initial (elastic) stiffness matrices are no longer applicable. Thus, the second step analysis, i.e., lamina analysis, is concerned with determination of the lamina instantaneous stiffness matrix even if an inelastic deformation has occurred. This instantaneous stiffness matrix cannot be determined without knowledge of the internal stress states in the fiber and resin materials of the lamina. The reason is that the overall instantaneous stiffness matrix of the lamina depends on the instantaneous stiffness matrices of its constituent materials, which are functions of the constituent internal stresses. On the other hand, having explicitly known the internal stresses of the lamina, its failure status can be detected by checking whether the fiber or the resin has attained its ultimate stress state or not. If fails, a laminate progressive failure strength results. The remaining laminate is analyzed by discounting the stiffness contribution from the failed lamina. In this way, the ultimate strength of the laminate can be incrementally determined. All the three quantities, i.e., the lamina instantaneous stiffness matrix, the internal stresses in the

523 fiber, and the internal stresses in the resin, pertaining to the lamina analysis at any load level can be informed by using the bridging micromechanics model, as described in detail in the following section.

3. Lamina theory The present laminate strength prediction is based on a new micromechanics theory for unidirectional fiber reinforced laminae, which has been reported elsewhere [9–18]. However, for the purpose of completeness, the theory is briefly summarised below.

3.1. Isothermal analysis Consider a unidirectional composite lamina, which is made using transversely isotropic fibers and an isotropic resin. Let E, v, and G represent Young’s modulus, Poisson’s ratio, and shear modulus of a material. Suffixes (either superscripts or subscripts) f and m will refer to the fiber and resin (matrix), with a quantity without any suffix referring to the composite. In order to deal with any nonlinear problem in a uniform manner, let us use an incremental approach, although this is not necessary for a linear elastic material. Suppose that the fibers used are linearly elastic until rupture, having an incremental stress–strain relationship given by

d f11 d f22 = 2d f12

1 E f11



v f12 E f11

0

1 E f22

0

symmetric

1 G f12

d f11 d f11 d f22 = [S f ] d f22 , d f12 d f12

(1)

where 1 and 2 are, respectively, along and transverse to the fiber axis. The resin material, however, is assumed to be elastic-plastic, of which the incremental constitutive equations can be formally expressed as dm11 d m11 m m d22 = [S ] d m22 2dm12 d m12

(2)

The instantaneous compliance matrix of the resin material, [Sm], must be determined based on a plastic flow theory adopted. Using the Prandtl–Reuss flow theory, this compliance matrix is defined as [26]

[S m ] =

再

[S m ]e, when  me ≤  mY , [S m ]e + [S m ] p, when  me >  mY

(3)

524 where 1 Em



[S m ]e =

9 [S ] = · m 4M T ( me)2 m

vm Em

0

1 Em

0

symmetric

1 Gm

1111

p

2211 2222 symmetry

,

(4.1)

21211 212. 22 41212

,

 me = 兹( m11)2 + ( m22)2  ( m11)( m22) + 3( m12)2, M mT =

E mE mT , E m  E mT

再

1 0, if i ≠ j ij = ij  (11 + 22)%ij, %ijn = , 3 1, if i = j

(4.2)

ij =  mij

(4.3) (4.4)

i, j = 1, 2.

(4.5)

In the above,  me is the von Mises equivalent stress,  mY and E mT are the yield strength and hardening (tangential) modulus (tangent to the stress–strain curve in the plastic region) of the resin under uniaxial loading, respectively. At any load level, let us assume that the incremental stresses in the constituent fiber and resin of the composite lamina are correlated through a bridging matrix via [9,10,14] (see, also, Appendix B) a11 d m11 d m22 = a21 d m12 0

a12 a22 0

a13 a23 a33

d f11 d f11 f d 22 = [A] d f22 d f12 d f12

(5)

It should be noted that the bridging matrix, [A], is not symmetric. As the composite lamina is transversely isotropic, there are only four independent elements in the bridging matrix, [A], which are assumed to be a11, a22, a33, and a21. The other dependent elements can be determined by requiring the resulting overall compliance matrix of the lamina, namely (Appendix B), [S ] = (Vf [S f ] + Vm[S m][A])(Vf [I ] + Vm[A])–1,

(6)

to be symmetric, where Vf and Vm are the volume fractions of the fiber and the resin respectively, and [I ] is a unit matrix. Eq. (6) represents the instantaneous compliance matrix of the lamina at any load level with any constituent materials, see Appendixes A and B. The only difference with different constituent materials is in the definition of the bridging elements, which should be varied accordingly. With a perfect bonding assumption for the fiber/resin interface, the independent elements of the bridging matrix can only depend on two types of parameters, i.e., the constituent properties and the fiber packing geometry. As long as these independent elements have been determined using an elastic deformation condition, only the first type of parameters, i.e., the material properties involved, needs to be changed

525 accordingly when the resin undergoes a plastic deformation. This is because the second type of parameters, i.e., the fiber packing geometry such as the fiber volume fraction, the fiber arrangement in the resin, the fiber cross-sectional shape, etc., does not change or only varies very little during the resin inelastic deformation. By making use of some well-known elasticity results for the unidirectional lamina, a set of explicit formulae for defining the independent elements of the bridging matrix are given in the following [9,10,14] (see, also, Appendix B) a11 = Em /E f11,

(7.1)

a22 = 0.5(1 + Em /E f22),

(7.2)

a33 = 0.5(1 + Gm /G f12),

(7.3)

a21 = 0,

(7.4)

where Em and Gm are called effective moduli of the resin material and are defined as

Gm =

再

再

E m, when  me ≤  mY , E mT, when  me >  mY

(7.5)

0.5E m/(1 + vm), when  me ≤  mY E mT/3, when  me >  mY

(7.6)

Em =

The dependent elements thus determined are a12 = (S f12  S m12)(a11  a22)/(S f11  S m11)

(8.1)

a13 =

d2#11  d1#21 , #11#22  #12#21

(8.2)

a23 =

d1#22  d2#12 , #11#22  #12#21

(8.3)

d1 = S m13(a11  a33),

(8.4)

d2 = S m23(Vf + Vma11)(a22  a33) + S m13(Vf + Vma33)a12,

(8.5)

#11 = S m12  S f12, #12 = S m11  S f11, #22 = (Vf + Vma22)(S m12  S f12),

(8.6)

#21 = Vm(S f12  S m12)a12  (Vf + Vma11)(S f22  S m22).

(8.7)

When the resin is also in the elastic deformation, we have S m13 = S m23 = 0. The four engineering elastic moduli of the lamina thus obtained from (6) are E11 = Vf E f11 + Vm E m,

(9.1)

v12 = Vf v f12 + Vmv m,

(9.2)

526 E22 =

(Vf + Vma11)(Vf + Vma22) , (Vf + Vma11)(Vf S f22 + a22VmS m22) + Vf Vm(S m21  S f21)a12

G12 = G m

(G f12 + G m) + Vf (G f12  G m) . (G f12 + G m)  Vf (G f12  G m)

(9.3)

(9.4)

It is seen that Eqs. (9.1) and (9.2) are exactly the same as rule of mixture longitudinal Young’s modulus and Poisson’s ratio formulae, which are accurate enough [27]. Eq. (9.4) is a precise elastic solution for the in-plane shear modulus based on a concentric cylinder model, see Ref. [28]. Only formula (9.3) is different from any of the existing models, the accuracy of which can be seen by comparing its prediction with experiments [27] and with the predictions of other two established micromechanics models, the Chamis model given in Ref. [29] and the Hill–Hashin–Christensen–Lo model presented in Ref. [30], for the transverse modulus of a UD (unidirectional) composite, as shown in Fig. Bl in Appendix B (with “beta = 0.5”). It is noted that only the elastic parameters of the constituents have been used in obtaining all the predictions shown in the figure. It must be pointed out that the formulae (9.1) and (9.2) are obtained based on the independent elements, a11 and a21, given by Eqs. (7.1) and (7.4), whereas the formulae (9.3) and (9.4) are obtained based on a22 and a33, given by Eqs. (7.2) and (7.3), respectively. In reality, the composite longitudinal property is the least “case sensitive” and the formulae (9.1) and (9.2) are applicable to nearly any unidirectional lamina. Therefore, the corresponding independent elements, a11 and a21, given by Eqs. (7.1) and (7.4), need no amendment. On the other hand, the composite transverse and in-plane shear properties are much more dependent on the in situ conditions involved, such as fabrication defects, processing condition, fiber-resin interface bonding, fiber arrangement, fiber cross-sectional shape, etc. In order to account for these variations, the corresponding independent bridging elements, i.e., a22 and a33, must also be able to vary. This can be simply achieved by using, e.g., the following formulae (see Appendix B), a22 = # + (1  #)

Em , E f22

0 ≤ # ≤ 1,

(10.1)

a33 =  + (1  )

Gm , G f12

0 ≤  ≤ 1,

(10.2)

instead of Eqs. (7.2) and (7.3), which correspond to  = # = 0.5. For convenience of expression, we call both # and  as bridging parameters. Physically, the bridging parameters represent the influence of the fiber packing geometry on the mechanical properties of a specific composite [9,16,18]. They do not depend on the constituent properties and can be calibrated/determined using measured transverse Young’s modulus and in-plane shear modulus of the composite, respectively. The transverse modulus formula, Eq. (9.3), is valid for any type of a22 used. The in-plane shear modulus, based on Eq. (10.2), is given accordingly by G12 =

Vf + Vma33 Vf /G f12 + Vma33/G m

(11)

527 (which is obtained by using conditions: d12 + Vf d f12 + Vmdm12, dm12 = a33d f12, d12 = G12d12, d f12 = G f12d f12, dm12 = G mdm12, and d12 = Vf d f12 + Vmdm12). Comparison of the predicted results by using different  (denoted by “Alf)” to obtain the in-plane shear elastic modulus of a Glass/Epoxy UD lamina with available experimental data [27] and with the Chamis model and the Hill–Hashin–Christensen–Lo model predictions has been shown in Fig. B2 in Appendix B. Similarly as in Fig. B1, only the elastic shear moduli of the constituents have been used in Fig. B2. It is seen that with the decrease in  a stiffer in-plane shear modulus has been obtained. The same conclusion can be made for the transverse modulus either, by varying # in Eq. (10.1), see Fig. B1. It is important to realise that once the composite elastic properties have been used in the calibration, the same bridging parameters  and # thus obtained can be employed in the inelastic and strength analysis of the composite. Suppose that the overall applied stress increments are {d} = {d21, d22, d12}T. From relation {d} = Vf {d f } + Vm{dm}, the incremental stresses generated in the fiber and resin materials are determined through b11 d f11 d f22 = 0 d f12 0

b12 b22 0

b13 b23 b33

d11 d11 d22 = [B ] d22 , d12 d12

(12.1)

and dm11 a11 dm22 = 0 dm12 0

a12 a22 0

a13 a23 a33

b11 0 0

b12 b22 0

b13 b23 b33

d11 d11 d22 = [A][B ] d22 , d12 d12

(12.2)

where b11 = (Vf + Vma22)(Vf + Vma33)/c, b12 =  (Vma12)(Vf + Vma33)/c,

(13.1)

b13 = [(Vma12)(Vma23)  (Vf + Vma22)(Vma13)]/c, b22 = (Vf + Vma11)(Vf + Vma33)/c,

(13.2)

b23 =  (Vma23)(Vf + Vma11)/c, b33 = (Vf + Vma22)(Vf + Vma11)/c, c = (Vf + Vma11)(Vf Vma22)(Vf + Vma33).

(13.3) (13.4)

The total stresses in the various materials are simply updated through [ij](K + 1) = [ij ](K ) + [dij ], [ fij ](K + 1) = [ fij ](K ) + [d fij ], [mij ](K + 1) = [mij ](K ) + [dmij ], K = 0, 1, . . . , where [ij ](0)( = [0]), p fij ](0), and [mij ](0) are initial (residual) stresses in the composite lamina, the fiber, and the resin materials, respectively. For instance, [ fij ](0) and [mij ](0) may be the thermal residual stresses generated in the constituent materials, which can be determined through a thermal analysis described below.

528 3.2. Thermal analysis Let T1 represent the working temperature of the composite, and T0 the reference temperature at which the internal stresses of the fiber and the resin are both known (e.g. stress-free). Due to mismatch between the thermal expansion coefficients of the fibers and the resin, thermal stresses will be generated in the constituent materials during the temperature variation, dT = T1  T0. The general constitutive equations of the fiber, the resin, and the composite are modified to {d f } = [S f ]{d f } + { f}dT,

(15.1)

{dm} = [S m]{dm} + {m}dT,

(15.2)

{d} = [Sij ]{d} + {}dT,

(15.3)

and

where  fi, mi, and i, respectively, denote the thermal expansion coefficients of the fiber, the resin, and the composite at the initial temperature T0 with m2 = m1 = m and  f3 = m3 = 0. On the other hand, we have [16–18] {d f } = (Vf [I ] + Vm[A])–1{d} + {b f }dT = [B f ]{d} + {b f }dT,

(16.1)

{dm} = [A](Vf [I ] + Vm[A])–1{d} + {bm}dT = [[B m]{d} + {bm]dT,

(16.2)

where {b f } and {bm} are called thermal stress concentration factors of the fiber and the resin, satisfying Vf {b f } + Vm{bm} = {0}.

(17)

In order to completely address the problem, we can apply either Levin [31] or Benveniste and Dvorak [32] rigorous expression [16,17]. Choosing {bm} as independent, the Benveniste and Dvorak’s expression is given as {bm} = ([I ]  [B m])([S f ]  [S m])–1({m}  { f }). By means of the bridging matrix, the last equation becomes {bm} = ([I ]  [A](Vf [I ] + Vm[A])–1)([S f ]  [S m])–1([m}  { f }).

(18)

The overall thermal expansion coefficients of the composite are determined from {} = Vf { f } + Vm{m} + Vm([S m]  [S f ]){bm}.

(19)

If there is no overall load applied to the lamina, namely, if {d} = {0}, which is the case when the lamina cools down from the fabrication temperature to room temperature, the thermal residual stress increments in the constituents are simply given by {dm}(0) = {bm}dT,

and

{d f }(0) = 

Vm m {b }dT. Vf

(20)

529 3.3. Strength criterion As the stress states in the constituent materials are explicitly known, a straightforward choice is to apply a stress failure criterion to detect the failure status of the constituents. The composite is considered to have failed as long as any of its constituent materials attains a failure stress state. Accordingly, the overall applied the stress is defined as an ultimate strength for the composite. In this paper, a generalized maximum normal stress criterion firstly developed in Ref. [13] is employed to detect the constituent failure. Thus, a tensile failure in either the fibers or the matrix material is attained if eq u,

(21.1)

where eq =

再

(1), when (2) ≤ 0, [((1))q + ((2))q]1/q, when (2) > 0, 1 < q ≤ &.

(21.2)

In the above, q is a power index, which is introduced to account for a reduction in the material load carrying capacity due to a bi-axial tension in comparison with a uniaxial tension. (1) and (2) are the first /maximum and the second /minimum principal stresses of the constituent, which are calculated according to (1) =

11 + 22 1 + 兹(11  22)2 + 4(12)2, 2 2

(2) =

11 + 22 1  兹(11  22)2 + 4(12)2. 2 2

u is the ultimate tensile strength of the material under uniaxial tension. For the fibers, u is measured along the fiber axial direction. When the power-index q = &, (21.1) together with (21.2) is equivalent to the classical maximum normal stress criterion. Throughout this paper, except for a comparative study purpose where effect of different power-indices q on predicted composite strengths is investigated (Fig. 18), the power-index of q = 3 is used. Thus, the difference between the generalized and the classical maximum normal stress criteria is distinct only when the two principal stresses are close to each other. Furthermore, a compressive failure of the material is considered to occur if (2) ≤ (–u,c),

(22)

where u,c is the uniaxial compression strength of the material (taking along its axial direction for the fiber material) and no material buckling is assumed.

3.4. Uniaxial strengths Closed form formulae [34] for composite strengths can be obtained when a UD lamina is only subjected to uniaxial loads (i.e., longitudinal, transverse, and in-plane shear loads), respectively. In such cases, no coupling between normal and shear stresses exists in the resin, and hence S m13 = S m23 = 0 which gives a13 = a23 = 0 in Eq. (5). These strength formulae can be

530 determined based on (12.1), (12.2), (21.1) and (21.2). However, they are more conveniently derived according to the stress relationships connecting the averaged stresses in the fiber, the resin, and the composite (see Appendix A). Thus, let us consider the stress balance in the transverse direction, i.e. (Appendix A), d22 = Vf d f22 + Vmdm22.

(23)

Substituting the second line of Eq. (5) into Eq. (23) and solving for d f22 with respect to d22, we obtain d f22 =

d22 . Vf + Vma22

(24.1)

dm22 =

a22d22 . Vf + Vma22

(24.2)

d f12 =

d12 , Vf + Vma33

(24.3)

dm121 =

a33d12 , Vf + Vma33

(24.4)

d f11 =

d11 Vma12d22  , Vf + Vma11 (Vf + Vma11)(Vf + Vma22)

(24.5)

dm11 =

a11d11 Vf a12d22 + . Vf + Vma11 (Vf + Vma11)(Vf + Vma22)

(24.6)

Combining Eqs. (23) and (24.1) gives

Similarly, we have

and

Suppose that the resin used is bilinearly elastic-plastic and the composite is free of thermal residual stresses. Using the classical maximum normal stress criterion, the lamina uniaxial strength formulae can be derived based on (24.1)–(24.6) as follows [34]. 3.4.1. Ultimate strength due to a longitudinal load (11) only From (24.5) and (24.6), we obtain the total stresses in the constituents (supposing that 11 is sufficiently large) as  f11 =

011 11  011 +  fe1  fp1

and

m11 =

011 11  011 + . me1 mp1

531 Thus, the maximum value of 11, i.e., the lamina longitudinal strength, is given by u11 = min

再

冎

 fu  ( fe1   fp1)011 mu  (me1  mp1)011 , ,  fp1 mp1

(25.1)

where

011 = min

再 冎

mY  fu , , me1  fe1

(25.2)

 fe1 =

E f11 , Vf E f11 + (1  Vf )E m

(25.3)

me1 =

Em , Vf E f11 + (1  Vf )E m

(25.4)

 fp1 =

E f11 , Vf E + (1  Vf )E mT

(25.5)

mp1 =

E mT . Vf E f11 + (1  Vf )E mT

(25.6)

f 11

3.4.2. Ultimate strength due to a transverse load (22) only

u22 = min

再

冎

 fu  ( fe2   fp2)022 mu  (me2  mp2)022 , ,  fp2 mp2

(26.1)

where

022 = min

再 冎

mY  fu , , me2  fe2

(26.2)

 fe2 =

E f22 , Vf E f22 + 0.5(1  Vf )(E m + E f22)

(26.3)

me2 =

0.5(E f22 + E m) , Vf E f22 + 0.5(1  Vf )(E m + E f22)

(26.4)

 fp2 =

E f22 , Vf E + 0.5(1  Vf )(E mT + E f22)

(26.5)

mp2 =

0.5(E f22 + E mT) . Vf E f22 + 0.5(1  Vf )(E mT + E f22)

(26.6)

f 22

532 3.4.3. Ultimate strength due to an in-plane shear load (12) only

u12 = min

再

冎

 fu  ( fe3   fp3)012 mu  (me3  mp3)012 , ,  fp3 mp3

(27.1)

where 012 = min

再

mY

,

兹3me3

冎

 fu ,  fe3

(27.2)

 fe3 =

G f12 , Vf G f12 + 0.5(1  Vf )(G m + G f12)

(27.3)

me3 =

0.5(G f12 + G m) , Vf G f12 + 0.5(1  Vf )(G m + G f12)

(27.4)

 fp3 =

3G f12 , 3Vf G f12 + 0.5(1  Vf )(E mT + 3G f12)

(27.5)

mp3 =

0.5(3G f12 + E mT) 3Vf G + 0.5(1  Vf )(E MT + 3G f12)

(27.6)

f 12

It is noted that in deriving the transverse strength, the constituent longitudinal stress components as per (24.5) and (24.6) due to the only applied transverse load have been assumed to be negligibly small. Further, the resin yield due to in-plane shear stress has been detected using the von Mises equivalent stress [Eq. (4.2)].

4. Laminate theory

4.1. Isothermal analysis Suppose that the laminated composite consists of a sequence of unidirectional laminae, stacking in different ply-angles. Each lamina can have different constituent materials and a different fiber volume fraction. We take a global coordinate system (x, y, z) with x and y in the laminate plane and z along the thickness direction. Let the fiber direction of the kth lamina have an inclined ply-angle k with the global x direction. A schematic diagram of such a lamina with respect to the global system is shown in Fig. 2. According to the classical laminate theory, only in-plane stress and strain increments, i.e., (d}G = {dxx, dyy, dxy}T and {d}G = {dxx, dyy, 2dxy}T, are retained, where G refers to the global system. The out-off plane strain components, i.e., d13, d23, and d33 are assumed to be zero, whereas the out-off plane stress components can be determined using a three-dimensional constitutive relationship if necessary. The global in-plane strain increments of the laminate at a material point (x, y, z) are expressed as [35] dxx = d0xx + zd0xx, dyy = d0yy + zd0yy, 2dxy = 2d0xy + 2d0xy,

533

Fig. 2. A lamina lay out in the global coordinate system.

where d0xx etc. and d0xx etc. are the strain and the curvature increments of the middle surface, respectively. The global stress increments at the considered material point are obtained from {d}G = [C ]Gk{d}G = [(C Gij)k]{d}G = ([T ]c)k([S ]k)–1([T ]Tc ){d}G,

(28)

where [S ]k is the compliance matrix of the kth lamina in its local coordinate system given by Eq. (6) and l 21 [T ]c = m21 l1m1

l 22 m22 l2m2

2l1l2 , 2m1m2 l1m2 + l2m1

(29)

with l1 = m2 = cos  and l2 = –m1 = sin , is a coordinate transformation matrix [36] from the lamina local system, (x1, x2), to the laminate global system, (x, y) (see Fig. 2). It is noted that the local coordinate x3 coincides with/is parallel to the global coordinate z. Hence, the averaged stress increments on the kth lamina can be determined from {d}Gk = ([T ]c)k([S ]k)–1([T ]Tc){d}Gk,

(30)

where

再

{d}Gk = d0xx +

冎

zk + zk  1 0 zk + zk  1 0 dxx, d0yy + dyy, 2d0xy + (zk + zk  1_d0xy 2 2

T

.

(31)

zk and zk  1 are the z coordinates of the top and the bottom surfaces of the lamina. These stresses can be transformed into the local coordinates through {d}k = ([T ]Ts )k{d}Gk,

(32)

where another coordinate transformation matrix, [T ]s, is defined as [36] l 21 [T ]s = m21 2l1m1

l 22 m22 2l2m2

l1l2 . m1m2 l1m2 + l2m1

(33)

Substituting Eq. (32) into the right hand sides of Eqs. (12.1) and (12.2), the averaged stress increments in the fiber and resin phases of this lamina can be calculated. It is thus only

534 necessary to determine the middle surface strains and curvature increments, which can be achieved using the following equations of the classical laminate theory [35], i.e., dNxx dNyy dNxy dMxx dMyy dMxy

冘

Q I11 Q I12 Q I13 Q II11 Q II12 Q II13

=

Q I12 Q I22 Q I23 Q II12 Q II22 Q II23

N

I ij

Q =

k=1

Q I13 Q I23 Q I33 Q II13 Q II23 Q II33

冘

Q II11 Q II12 Q II13 Q III 11 Q III 12 Q III 13

Q II12 Q II22 Q II23 Q III 12 Q III 22 Q III 23

Q II13 Q II23 Q II33 Q III 13 Q III 23 Q III 33




N

1 (C ) (zk  zk  1), Q = 2 G ij k

II ij

d0xx d0yy 2d0xy d0xx d0yy 2d0xy

冘

,

(34)

N

(C ) (z  z G ij k

k=1

2 k

1 ), Q = 3

2 k1

III ij

(C Gij )k(z3k  z3k  1). (35)

k=1

N is the total number of lamina plies in the laminate. (C Gij )k are the stiffness elements of the kth lamina in the global coordinate system, see Eq. (28). In Eq. (34), dNxx, dNyy , and dNxy and dMxx, dMyy , and dMxy are, respectively, the overall incremental in-plane forces and moments per unit length exerted on the laminate. Suppose that the totally applied in-plane stresses are (0xx, 0yy, 0xy). The incremental in-plane forces and moments are calculated from

dNxx =

冕

h/2

(d0xx)dz, dNyy =

–h/2

dMxx =

冕

(d0yy)dz, dNxy =

–h/2

h/2

–h/2

冕

h/2

(d0xx)zdz, dMyy =

冕

冕

h/2

(d0xy)dz,

(36.1)

–h/2

h/2

冕

h/2

(d0yy)zdz, dMxy =

–h/2

(d0xy)zdz,

(36.2)

–h/2

where h = Nk= 1 (zk  zk  1) is the whole thickness of the laminate. 4.2. Post failure analysis Substituting from Eq. (30) into the right hand sides of Eq. (32) and noting that [T ]Ts = [T ]–1 c , we obtain {d}k = ([S ]k)–1([T ]Tc ){d}Gk.

(37)

Eq. (37) indicates that each lamina in the laminate is subjected to a different load share as long as it has a different coordinate transformation matrix, [T ]c. The conclusion is valid even if the lamina instantaneous compliance matrix, [S ], is the same and there is no bending curvature occurred in the laminate. Therefore, some lamina ply must have failed first before others. Once some k0-th lamina fails, according to the strength criterion given in Section 3.3, the corresponding overall applied load on the laminate is defined as a progressive failure strength. If the k0-th lamina failure occurs firstly in the laminate, the corresponding strength is named as the first-ply failure strength. Similarly, we have the second-ply failure strength, etc. As the ply failure has already been defined upon the failure of one constituent material, the

535 corresponding failure mode is automatically identified. It is either the resin failure, or the fiber fracture, or the failures of the both that causes the ply failure. Furthermore, the contribution from the failed lamina to the remaining instantaneous stiffness matrix of the laminate must be reduced. Using a total reduction strategy, the stiffness elements in Eq. (32) are redefined as

冘 N

I ij

Q =

k=1 kⰻ(k0)

冘 N

1 (C ) (zk  zk  1), Q = 2 G ij k

II ij

k=1 kⰻ(k0)

冘 N

(C ) (z  z G ij k

2 k

1 ), Q = 3

2 k1

III ij

(C Gij )k(z3k  z3k  1), (38)

k=1 kⰻ(k0)

where {K0} represents all those laminae which have already failed and the symbol “ⰻ” stands for “not belonging to.” Eqs. (38) imply that the failed lamina cannot sustain any more load, but remains its geometric position. Namely, the failed lamina is replaced using an imagining material layer which has the same geometric dimensions as the original lamina but has no stiffness. Note that the incremental forces and moments are calculated still using the same formulae, Eqs. (36.1) and (36.2). Continued in this way, the whole progressive failure process in the laminate can be characterized and the ultimate failure strength of the laminate is determined using the overall applied load at which all the plies have failed. It must be pointed out that the stiffness reduction, as per Eq. (38), up to the last ply failure is applicable only when the laminate is subjected to an in-plane load. If, however, the laminate is involved with a flexural load, the stiffness reduction must be performed more carefully. For example, let the laminate be subjected to only a bending load. The above stiffness reduction process should be stopped before reaching the last or the second last ply failure [37,38]. This is because under the bending condition, the middle plane strain increments, d0XX etc., are negligibly small. The remaining bending curvature will have very small, if not zero, stress contribution to the last ply failure or the last two-ply failures, according to Eqs. (30) and (31). For instance, if the laminate consists of odd-number (e.g. 5, 7, 9, . . .) of plies each of which has the same global property and the same thickness, the central ply will not carry any load no matter how much a pure bending will be applied to the laminate, according to Eqs. (30) and (31). Thus, the last ply will not fail at all, but the deflection (curvature) can be increased unlimitedly. In bending tests of some laminated beams, we have observed that some ply/plies of the beams cannot be forced to failure due to the limitation of the testing apparatus to excessive beam deflections [37,38]. This is consistent with our theory. Therefore, if a flexural load is involved, the only use of stress failure criterion is no longer sufficient; an additional critical deflection/curvature condition has to be employed in order that the ultimate strength of the laminate can be determined [37,38]. However, only in-plane loads have been specified for all the exercise problems and no bending deflection will be concerned in the present paper. 4.3. Thermal analysis As aforementioned, when a unidirectional lamina is subjected to a free temperature variation, there will be no overall stress (i.e., no thermal load) on the lamina [see Eq. (17)], although its constituents generally sustain thermal stresses. In such case, the constituent thermal stresses are calculated using Eqs. (20). However, if the laminate is only subjected to a temperature variation, each lamina in the laminate will be subjected to overall thermal stresses (called thermal loads) in general. In this latter case, the thermal stresses in the constituents of the

536 lamina will consist of two parts: one part is still given by Eqs. (20) and another part should be evaluated based on Eqs. (12.1) and (12.2). The second part is due to the composite lamination or stacking constraint. Namely, each lamina has different global coefficients of thermal expansion (although the local ones may be the same). Hence, it is only necessary to calculate the thermal loads shared by the lamina. For the kth lamina, the thermal stress and strain increments in the global coordinate system satisfy [refer to Eq. (15.3)] ) ) {d}G,(T = [C ]Gk{d}G,(T  {#}Gk dT, k k

(39.1)

{#}Gk = {(#1)Gk, (#2)Gk, (#3)Gk }T = ([T ]c)k([S ]k)–1{}k.

(39.2)

where

On the other hand, the thermal stress increments applied on the kth lamina in its local coordinate system are obtained from Eq. (32). These stress increments must be substituted into the right-hand sides of Eqs. (16.1) and (16.2), rather than Eqs. (12.1) and (12.2), to determine the thermal stress increments in the constituent fiber and resin materials. Namely, the two parts of thermal stresses are calculated simultaneously. In this way, the thermal residual stresses in each lamina in the laminate can be obtained. ) Note that the thermal strain increments, {d}G,(T , in Eq. (39.1) are defined as [35] k

再

) ) = d0,(T {d}G,(T k xx +

zk + zk  1 0,(T ) zk + zk  1 0,(T ) ) ) 0,(T ) dxx , d0,(T dyy , 2d0,(T yy + xy + (zk + zk  1) dxy 2 2

冎

T

(40)

where the middle surface strain and curvature increments due to the temperature variation are obtained from [35] d' I1 d' I2 d' I3 d' II1 d' II2 d' II3

Q I11 Q I12 Q I13 Q II11 Q II12 Q II13

=

冘

Q I12 Q I22 Q I23 Q II12 Q II22 Q II23

Q I13 Q I23 Q I33 Q II13 Q II23 Q II33

Q II11 Q II12 Q II13 Q III 11 Q III 12 Q III 13

Q II12 Q II22 Q II23 Q III 12 Q III 22 Q III 23

N

I i

d' =

k=1

Q II13 Q II23 Q II33 Q III 13 Q III 23 Q III 33

冘




) d0,(T xx 0,(T ) dyy ) 2d0,(T xy , 0,(T ) dxx ) d0,(T yy 0,(T ) 2dxy

(41)

N

1 (# ) (zk  zk  1) dT, d' = 2 G i k

II i

(#i)Gk(z2k  z2k  1) dT

(42)

k=1

5. Application to the exercise problems 5.1. Constituent properties Measured properties of the fibers, resins, and four UD (unidirectional) laminae together with their fiber volume fractions used for the exercise problems have been specified in Ref. [25]. They are summarized in Tables 1–4. In the present calculations, all the fibers are assumed to be linearly elastic until rupture. The resin of each UD lamina, however, must be considered

537 Table 1 Properties of E-glass 21
K43 Gevetex and LY556/HT907/DY063 epoxy UD lamina (fiber volume fraction: Vf = 0.62; stress-free temperature: 120°C; working temperature: 25°C; bridging parameters used: # = 0.45 and  = 0.35). Properties

E11 (GPa) E22 (GPa) G12 (GPa) v12 (Y )1 (MPa)a (Y )2 (MPa)a (Y )3 (MPa)a (Y )4 (MPa)a (Y )5 (MPa)a (Y )6 (MPa)a (Y )7 (MPa)a (Y )8 (MPa)a (ET )1 (MPa)a (ET )2 (MPa)a (ET )3 (MPa)a (ET )4 (MPa)a (ET )5 (MPa)a (ET )6 (MPa)a (ET )7 (MPa)a Lu (MPa)b Lu,c (MPa)c Tu (MPa)d Tu,c (MPa)e Su (MPa)f 11 (%)b 11,c (%)c 22 (%)d 22,c (%)e 212 (%)f 1 (
10–6/°C) 2 (
10–6/°C) 11 (MPa)g 22 (MPa)g 12 (MPa)g a

Lamina

Fiber

Resin

Provided

Predicted

Provided

Used

Provided

Used

53.48 17.7 5.83 0.278 – – – – – – – – – – – – – – – 1140 570 35 114 72 2.132 1.065 0.197 0.644 3.8 8.6 26.4 – – –

50.87 14.38 5.72 0.257 – – – – – – – – – – – – – – – 1140 570 72 114 84.8 2.227 1.123 0.692 1.087 7.638 6.23 20.62 0 0 0

80 80 33.33 0.2 – – – – – – – – – – – – – – – 2150 1450 – – – – – – – – 4.9 4.9 – – –

80 80 33.33 0.2 – – – – – – – – – – – – – – – 1804.1 908.9 – – – – – – – – 4.9 4.9 –12.55h –6.86h 0

3.35 3.35 1.24 0.35 – – – – – – – – – – – – – – – 80 120 – – – 5 – – – – 58 58 – – –

3.35 3.35 1.24 0.35 31.9 38.4 44.7 49.9 53.6 56.1 58.1 60.0 1566 1337 944 584 338 245 197 56.5 55.7 – – – 5.5434h – – – – 58 58 20.47h 11.2h 0

Assumed to be the same in both tension and compression. Longitudinal tension. c Longitudinal compression. d Transverse tension. e Transverse compression. f In-plane shearing. g Predicted thermal residual stress, from stress-free temperature to working temperature. h Not for use in subsequent predictions. b

538 Table 2 Properties of silenka E-glass 1200tex and MY750/HY917/DY063 epoxy UD lamina (fiber volume fraction: Vf = 0.60; stress-free temperature: 120°C; working temperature: 25°C; bridging parameters used: # = 0.45 and  = 0.35). Properties

E11 (GPa) E22 (GPa) G12 (GPa) v12 (Y )1 (MPa)a (Y )2 (MPa)a (Y )3 (MPa)a (Y )4 (MPa)a (Y )5 (MPa)a (Y )6 (MPa)a (Y )7 (MPa)a (Y )8 (MPa)a (ET )1 (MPa)a (ET )2 (MPa)a (ET )3 (MPa)a (ET )4 (MPa)a (ET )5 (MPa)a (ET )6 (MPa)a (ET )7 (MPa)a Lu (MPa)b Lu,c (MPa)c Tu (MPa)d Tu,c (MPa)e Su (MPa)f 11 (%)b 11,c (%)c 22 (%)d 22,c (%)e 212 (%)f 1 (
10–6/°C) 2 (
10–6/°C) 11 (MPa)g 22 (MPa)g 12 (MPa)g a

Lamina

Fiber

Resin

Provided

Predicted

Provided

Used

Provided

Used

45.6 16.2 5.83 0.278 – – – – – – – – – – – – – – – 1280 800 40 145 73 2.807 1.754 0.246 1.2 4 8.6 26.4 – – –

45.74 13.45 5.31 0.26 – – – – – – – – – – – – – – – 1280 800 78.8 145 90 2.820 1.749 0.835 2.918 9.875 6.46 21.67 0 0 0

74 74 30.8 0.2 – – – – – – – – – – – – – – – 2150 1450 – – – – – – – – 4.9 4.9 – – –

74 74 30.8 0.2 – – – – – – – – – – – – – – – 2092.8 1311.8 – – – – – – – – 4.9 4.9 –13.55h –7.32h 0

3.35 3.35 1.24 0.35 – – – – – – – – – – – – – – – 80 120 – – – 5 – – – – 58 58 – – –

3.35 3.35 1.24 0.35 32.6 39.9 46.8 52 55.6 58 60.1 62.0 1698 1387 918 542 317 244 186 60.9 74.8 – – – 5.7704h – – – – 58 58 20.33h 10.98h 0

Assumed to be the same in both tension and compression. Longitudinal tension. c Longitudinal compression. d Transverse tension. e Transverse compression. f In-plane shearing. g Predicted thermal residual stress, from stress-free temperature to working temperature. h Not for use in subsequent predictions. b

539 Table 3 Properties of graphite AS4 and 3501-6 epoxy UD lamina (fiber volume fraction: Vf = 0.60; stress-free temperature: 177°C; working temperature: 25°C; bridging parameters used: # = 0.45 and  = 0.3). Properties

E11 (GPa) E22 (GPa) G12 (GPa) v12 (Y )1 (MPa)a (Y )2 (MPa)a (Y )3 (MPa)a (Y )4 (MPa)a (Y )5 (MPa)a (Y )6 (MPa)a (Y )7 (MPa)a (Y )8 (MPa)a (ET )1 (MPa)a (ET )2 (MPa)a (ET )3 (MPa)a (ET )4 (MPa)a (ET )5 (MPa)a (ET )6 (MPa)a (ET )7 (MPa)a Lu (MPa)b Lu,c (MPa)c Tu (MPa)d Tu,c (MPa)e Su (MPa)f 11 (%)b 11,c (%)c 22 (%)d 22,c (%)e 212 (%)f 1 (
10–6/°C) 2 (
10–6/°C) 11 (MPa)g 22 (MPa)g 12 (MPa)g a

Lamina

Fiber

Resin

Provided

Predicted

Provided

Used

Provided

Used

126 11 6.6 0.28 – – – – – – – – – – – – – – – 1950 1480 48 200 79 1.38 1.175 0.436 2.0 2 –1 26 – – –

136.7 9.23 5.54 0.256 – – – – – – – – – – – – – – – 1950 1480 65.9 200 98.8 1.420 1.084 0.792 3.404 3.187 0.06 27.9 0 0 0

225 15 15 0.2 – – – – – – – – – – – – – – – 3350 2500 – – – – – – – – –0.5 15 – – –

225 15 15 0.2 – – – – – – – – – – – – – – – 3206.4 2458.6 – – – – – – – – –0.5 15 –22.23h –9.06h 0

4.2 4.2 1.567 0.34 – – – – – – – – – – – – – – – 69 250 – – 50 1.7 – – – – 45 45 – – –

4.2 4.2 1.567 0.34 38.1 41.8 46.1 50.1 54.0 57.6 61.2 64.6 2507 2530 2072 1721 1409 1202 991 65.6 116.4 – – – 2.5424h – – – – 45 45 33.35h 13.6h 0

Assumed to be the same in both tension and compression. Longitudinal tension. c Longitudinal compression. d Transverse tension. e Transverse compression. f In-plane shearing. g Predicted thermal residual stress, from stress-free temperature to working temperature. h Not for use in subsequent predictions. b

540 Table 4 Properties of graphite T300 and BSL914C epoxy UD lamina (fiber volume fraction: Vf = 0.60; stressfree temperature: 120°C; working temperature: 25°C; bridging parameters used: # = 0.45 and  = 0.35). Properties

E11 (GPa) E22 (GPa) G12 (GPa) v12 (Y )1 (MPa)a (Y )2 (MPa)a (Y )3 (MPa)a (Y )4 (MPa)a (Y )5 (MPa)a (Y )6 (MPa)a (Y )7 (MPa)a (Y )8 (MPa)a (ET )1 (MPa)a (ET )2 (MPa)a (ET )3 (MPa)a (ET )4 (MPa)a (ET )5 (MPa)a (ET )6 (MPa)a (ET )7 (MPa)a Lu (MPa)b Lu,c (MPa)c Tu (MPa)d Tu,c (MPa)e Su (MPa)f 11 (%)b 11,c (%)c 22 (%)d 22,c (%)e 212 (%)f 1 (
10–6/°C) 2 (
10–6/°C) 11 (MPa)g 22 (MPa)g 12 (MPa)g a

Lamina

Fiber

Resin

Provided

Predicted

Provided

Used

Provided

Used

138 11 5.5 0.28 – – – – – – – – – – – – – – – 1500 900 27 200 80 1.087 0.652 0.245 1.818 4 –1 26 – – –

139.6 9.09 5.04 0.26 – – – – – – – – – – – – – – – 1500 900 55.8 200 81.2 1.073 0.645 0.641 6.891 4.578 –0.06 29.6 0 0 0

230 15 15 0.2 – – – – – – – – – – – – – – – 2500 2000 – – – – – – – – –0.7 12 – – –

230 15 15 0.2 – – – – – – – – – – – – – – – 2462.5 1499.4 – – – – – – – – –0.7 12 –16.59h –7.49h 0

4.0 4.0 1.481 0.35 – – – – – – – – – – – – – – – 75 150 – – 70 4 – – – – 55 55 – – –

4.0 4.0 1.481 0.35 41.6 49.6 55.8 59.9 63.1 66.3 68.9 71.4 2015 1384 769 548 457 324 275 56.4 116.8 – – – 5.4139h – – – – 55 55 24.89h 11.24h 0

Assumed to be the same in both tension and compression. Longitudinal tension. c Longitudinal compression. d Transverse tension. e Transverse compression. f In-plane shearing. g Predicted thermal residual stress, from stress-free temperature to working temperature. h Not for use in subsequent predictions. b

541 as elastic-plastic. This is because the in-plane shear stress-shear strain curve of every lamina supplied by the exercise organizers is nonlinear [25]. From Eq. (6), we can clearly see that the composite can display a nonlinear deformation if and only if one of its constituent materials undergoes such a deformation. All the thermo-elastic properties of the constituent materials of the four laminae as well as their fiber volume fractions are kept the same as those provided by the organizers. However, the bridging parameters  and # used in defining the independent bridging elements a33 and a22 have to be calibrated against the transverse and in-plane shear moduli of the laminae in such a way that the predicted moduli agree reasonably well with the measured values. The parameter # has been chosen as 0.45 for all the four laminae, whereas the parameters  has been taken to be 0.3 for one lamina and 0.35 for the others, as indicated in Tables 1–4. Both the bridging parameters of each lamina remain unchanged in the subsequent predictions for all the exercise problems. Thus, the formula used to calculate the in-plane shear (elastic) modulus is Eq. (11). It is expected that the resin plastic parameters (yield strength and hardening modulus) measured using monolithic material specimens can be directly employed, just as the employment of the constituent thermo-elastic properties. However, no detailed information about the resin plasticity was reported [25]. On the other hand, the in-plane shear stress-shear strain curves of the unidirectional laminae fabricated from all the four material systems were provided [25]. They are used to retrieve the plastic parameters of the respective resin materials. Compared with a transverse stress–strain response, the in-plane shear stress–strain curve generally displays more distinct non-linear behaviour. It is noted that whenever possibly the resin plastic parameters should not be back calculated based on the overall longitudinal response of the composite, since in some cases the resin material may not sustain a full load share (i.e. to its maximum load carrying capacity) when the composite is longitudinally loaded up to failure. The retrieved uniaxial stress–strain curve of each resin was assumed to consist of eight linear segments. Hence, hardening modulus of the resin at a specific loading range was specified as E mT = (E mT )i, when, (mY)i ≤ me ≤ (mY)i + 1, i = 0, 1, . . . 7, (E mT)0 = E m, (mY)0 = 0,

(43.1)

E mT = (E mT)7, when me ≥ (mY)8.

(43.2)

and

The recovering was performed as follows. Starting from the given elastic modulus, E m, the remaining hardening moduli of the resin were adjusted in such a way that the predicted inplane shear stress-shear strain curve was as close to the provided one as possible. The yield strength, mY ⬅ (mY)1, was determined using a condition that the predicted plastic strain at this stress level was equal to 0.02mY /E m. The retrieved resin plastic parameters of the four laminae are summarized in Tables 1–4, respectively. Using these parameters, together with the given elastic properties, the simulated in-plane shear stress-shear strain curves of the four laminae are plotted in Figs. 3–6, in which the provided shear stress-shear strain curves are also shown for comparison. Further, the stress–strain curves of the corresponding resin materials under uniaxial tensile and in-plane shear loads respectively, obtained using the retrieved plastic parameters, are also graphed in the respective figures. It should be pointed out that as in the present simulation all the lamina nonlinearity has been attributed to resin plastic deformation,

542

Fig. 3. Stress–strain curves of lamina I and its resin material, subjected to either in-plane shear or uniaxial tensile load.

Fig. 4. Stress–strain curves of lamina 2 and its resin material, subjected to either in-plane shear or uniaxial tensile load.

543

Fig. 5. Stress–strain curves of lamina 3 and its resin material, subjected to either in-plane shear or uniaxial tensile load.

Fig. 6. Stress–strain curves of lamina 4 and its resin material, subjected to either in-plane shear or uniaxial tensile load.

544 the retrieved resin plastic parameters may be somewhat different from those measured using a monolithic resin specimen. In reality, some other defects such as damage evolution and the more or less nonlinear behavior of the fiber material can also contribute to the lamina nonlinearity. However, more critical correlation and hence judgment can be made once the experimental data for all the exercise problems as well as complete constituent behaviors including resin plastic parameters measured using monolithic material specimens are available. It is evident that the most important parameters in the present modeling approach to the composite strength are the ultimate strength data of the constituent materials. Although measured values of the constituent strengths using monolithic constituent specimens were provided in Ref. [25], some amendments have been made for these constituent strengths so that some predicted uniaxial strengths of the corresponding UD laminae are equal to the measured values. The tensile strengths of the constituent fiber and resin materials of all the four laminae were back calculated only using respective lamina longitudinal tensile strengths. To do this, a longitudinal load was incrementally applied to the lamina until a level exactly equal to the lamina longitudinal tensile strength. Then, the corresponding equivalent maximum normal stresses [Eq. (21.2)] both in the fiber and in the resin were considered as their tensile strengths. It is noted that the thermal residual stress effect has been incorporated in the constituent strength determination. Regarding the retrieval of the constituent tensile strengths, some additional comments deserve mentioning. Generally speaking, the longitudinal tensile strength of a UD lamina can be used to retrieve the fiber tensile strength, whereas the lamina transverse tensile strength may be employed to back calculate the resin tensile strength. However, using the transverse tensile strengths of the present four UD laminae, the retrieved resin strengths would be lower than expected values. Namely, based on these strengths, the UD laminae cannot sustain longitudinal loads as high as their ultimate values, provided by the organizers [25] (also see Tables 1–4). As in general the measurement of a longitudinal tensile strength of a UD lamina is more accurate than that of the lamina transverse tensile strength, the constituent strengths should be adjusted, if any, in such a way that a longitudinal tensile load as high as the measured longitudinal tensile strength can be safely applied to the lamina. Based on so defined resin tensile strengths, the predicted transverse and in-plane shear strengths of the four UD laminae are all higher than the corresponding measured values, as indicated in Tables 1 4. It is reminded that when an applied shear load is higher than the highest point on a shear stress-shear strain curve (either of Figs. 3–6), the resin hardening modulus should be specified according to (43.2). On the other hand, the compressive strengths of the fiber and the resin were retrieved using the lamina longitudinal and transverse compressive strengths, respectively. The calibrated constituent strength parameters are listed in Tables 1–4. Then, the laminae transverse and inplane shear strengths together with the laminae ultimate strains under uniaxial loads were predicted. Results are also summarized in the tables. It is seen that discrepancies exist between the predicted and the measured transverse tensile and in-plane shear strengths. The predicted values are larger than the measured ones. This is because higher resin tensile strengths have been employed in the predictions. The failure modes of all the four laminae subjected to either transverse tensile or in-plane shear load were the same, i.e., due to the tensile fracture of the resin materials used. It should be noted that the resin plasticity, already retrieved from the lamina in-plane shear response, and the thermal residual stress effect have

545 been incorporated in the predictions of all the ultimate stress and strain data listed in Tables 1–4. In all the following predictions, the constituent parameters listed in the columns of Tables 1–4 designated as “used”, together with the fiber volume fraction Vf , the bridging parameters  and #, and the stress-free and working temperatures are served as input data consistently. The constituent properties are assumed to be temperature-independent during the given temperature variation, i.e., from the stress-free temperature to the working temperature. However, the thermal residual stresses in the constituent fiber and resin materials of the four material systems have been incorporated throughout. Namely, all the composites considered are assumed to have been fabricated at their corresponding temperatures specified in Ref. [25].

5.2. Predictions of the four laminae Before we go to the exercise problems, let us consider the four laminae materials in some more detail. The predicted stress–strain curves of the four laminae subjected to a longitudinal tensile load up to failure are graphed in Fig. 7. In the figure, yield strength of each lamina, which is defined as the overall applied load at which one of the constituent materials (currently, it is only the resin material) initiates yielding, was indicated. However, all the stress–strain curves display a grossly linear behavior. This is because the longitudinal stiffness of the present fibers, which remains constant up to failure, is much higher than the resin stiffness, and the resulting composite longitudinal stiffness is mainly dependent on the fiber stiffness. Thus, any plastic deformation of the resin has essentially no effect on the composite longitudinal stress–strain response.

Fig. 7. Stress–strain responses of the four UD laminae subjected to longitudinal tensile load up to failure.

546 On the other hand, the resin plasticity may have significant influence on the transverse and in-plane shear behaviors of the composite. The amount of the influence depends heavily on the fiber volume fraction. To illustrate, the stress–strain responses of the four laminae made from the same constituent materials at the same fabrication conditions but with different fiber volume fractions were simulated. The laminae were subjected to a transverse tensile and an in-plane shear load. The predicted results are plotted in Figs. 8–11 for the transverse tensile, and in Figs. 12–15 for the in-plane shear stress–strain responses, respectively. Except for the composites made from material systems 3 and 4 (i.e. graphite fibers and epoxy systems) which have relatively low fiber volume fraction and which are subjected to transverse tensile load (shown in Figs. 10 and 11), all the other composites subjected to either transverse tensile or in-plane shear load exhibit apparently nonlinear behavior. Because of this, a plastic flow/ nonlinear deformation theory, such as the Prandtl-Reuss theory used in this paper, must be employed to describe the constitutive relationship of the resin materials. Moreover, the predicted results, graphed in Figs. 10 and 11, indicate that there is not much effect of fiber volume fraction on the transverse tensile strengths of the laminae made using transversely isotropic graphite fibers as reinforcement. However, the influence is distinct with glass fiber material systems. This is mainly because the latter material systems result in more ductile stress–strain behavior. Furthermore, for the composite made from isotropic glass fibers reinforcement, the resulting yield strength increases with the increase of its fiber volume content. On the contrary, when the composite is fabricated using transversely isotropic fibers, the composite yield strength decreases with the increase of the fiber volume fraction. Interestingly enough, this situation does not happen when the composite is subjected to an inplane shear load. Namely, under in-plane shear loading the yield strength value in the composite made from any of the material systems is proportional to the fiber volume fraction.

Fig. 8. Stress–strain responses of UD composites with different fiber volume fractions subjected to a transverse load up to failure.

547

Fig. 9. Stress–strain responses of UD composites with different fiber volume fractions subjected to a transverse load up to failure.

Fig. 10. Stress–strain responses of UD composites with different fiber volume fractions subjected to a transverse load up to failure.

548

Fig. 11. Stress–strain responses of UD composites with different fiber volume fractions subjected to a transverse load up to failure.

Fig. 12. Stress–strain responses of UD composites with different fiber volume fractions subjected to an in-plane shear load up to failure.

549

Fig. 13. Stress–strain responses of UD composites with different fiber volume fractions subjected to an in-plane shear load up to failure.

Fig. 14. Stress–strain responses of UD composites with different fiber volume fractions subjected to an in-plane shear load up to failure.

550

Fig. 15. Stress–strain responses of UD composites with different fiber volume fractions subjected to an in-plane shear load up to failure.

Finally, all the ultimate strengths of the composites under whichever (transverse tensile or inplane shear) load condition have a positive dependency on their fiber contents.

5.3. Predictions of the exercise problems There are a total of 14 exercise problems posed in Ref. [25], all of which will be simulated. The laminates involved in the problems are subjected to combined either uniaxial tension/ compression loads in two orthogonal directions (the global x and y directions) or uniaxial tension/compression and in-plane shear loads. There is no bending moment/lateral force applied to the laminates. With a given ratio between the applied two stresses, a failure state of stress for the composite can be determined using the incremental loading scheme described in the previous sections. Varying this ratio, the predicted failure stress states constitute an envelope in the plane of the two stresses. Let the maximum possible values of the first and the 2nd second applied stresses be denoted by 1st max and max. Corresponding to different stress ratios, the maximum possible combinations of the applied stresses are assigned by 2nd (1st, 2nd) = (1st max cos i, max sin i), i =

i 90°, 36

i = 0, 1, . . . , 36

(44)

551 In the present calculations, the overall load increments, corresponding to these combinations, are taken as (d1st, d2nd) =

1 (1st, 2nd) 3000

(45)

For each problem, related issues such as how to specify the constituent materials, what is the maximum load combination used, and where to show the predicted results are summarized in Table 5. It is pointed out once more that the predicted results of all the 14 problems, either for failure envelopes or for stress–strain curves up to failure, have incorporated thermal residual stresses generated in the composites. The problems 1–14 designated in the table correspond to the problem sequence specified in Table 11 of Ref. [25]. The laminate lay-up and stacking sequence have been described in detail in Ref. [25]. 5.3.1. Predictions: failure envelopes Let the composite be subjected to a fixed ratio [i.e.,  be chosen fixed in Eq. (44)] of biaxial load increments. At each load level, the resulting constituent stresses are checked against the generalized maximum normal stress criterion [Eq. (21)], with q = 3 (see Fig. 18). If either exceeds its ultimate strength (tensile or compressive strength), the composite is considered to have failed. Thus, a data point can be figured out in the corresponding biaxial stress plane. Varying the load ratio, a continuous curve can be plotted, which is called a failure envelope

Table 5 Characterizations of the analyzed problems. Problem Material Maximum load specification combination (MPa) 1

Table 1

2

Table 4

3

Table 2

4 5 6 7 8 9 10 11 12 13 14

Table 1 Table 1 Table 3 Table 3 Table 3 Table 2 Table 2 Table 2 Table 2 Table 2 Table 2

2nd max max (1st max, max) = (yy , xy ) = (100, 200) (–300, 200) 2nd max max (1st max, max) = (xx , xy ) = (2500, 120) (–1500, 120) 2nd max max (1st max, max) = (xx , yy ) = (2500, 100) (–2000, 100) (2500, –300) (–2000, –300) 2nd max max (1st max, max) = (yy , xx ) = ( ± 800, ± 1500) 2nd max max (1st max, max) = (xx , xy ) = ( ± 1500, 600) 2nd max max (1st ,  ) = ( ,  max max yy xx ) = ( ± 1200, ± 1200) 2nd max max (1st ,  ) = ( ,  max max xx yy ) = (1000, 0) 1st 2nd max (max, max) = (xx , max yy ) = (500, 1000) 2nd max max (1st max, max) = (yy , xx ) = ( ± 1500, ± 600) 2nd max max (1st max, max) = (xx , yy ) = (1000, 0) 1st 2nd max (max, max) = (xx , max yy ) = (1000, 2000) 2nd max max (1st max, max) = (xx , yy ) = (1000, 0) 1st 2nd max (max, max) = (xx , max yy ) = (500, 500) 2nd max max (1st max, max) = (xx , yy ) = (–500, 500)

Results shown

Fig. 16 and Table 6 Fig. 17 and Table 6 Figs. 18, 19 and Table 7

Fig. 20a and Tables 8a and 8b Fig. 22a and Table 9 Fig. 23a and Tables 10a and 10b Fig. 24 and Table 12 Fig. 25 and Table 12 Fig. 26a and Table 11 Fig. 27 and Table 12 Fig. 28 and Table 13 Fig. 29 and Table 13 Fig. 30 and Table 13 Fig. 31 and Table 13

552 of the composite in the considered stress plane. As long as a load combination of the two stresses is within the envelope, the composite safety can be guaranteed. Otherwise, i.e. outside the envelope, the composite will fail. Presented in the following, pertaining to each problem involved, are: (a) first-ply failure envelope if any, which is represented by a broken-line, (b) last-ply failure envelope, which is represented by a solid line, (c) resin failure mode corresponding to the last-ply failure (i.e., under the corresponding load combination the lastply failure is due to resin failure), which is designated by an empty triangle, (d) fiber failure mode corresponding to the last-ply failure, which is designated by a solid triangle, (e) resin yield load (i.e., the load level at which the resin begins to yield), which is designated by an empty ball, and (f) no resin yield (i.e., until the last-ply failure the resin does not yield at all), which is designated by a star. Note that the resin yielding is defined according to the condition that me ≥ mY, where me is calculated using Eq. (4.3) and mY is given by (mY )1 in each of Tables 1–4. Also note that the plastic parameters, including mY, have been assumed to be the same under both tension and compression for all the resin materials considered. Problem 1 (UD E-glass/LY556/HT907/DY063 lamina subjected to combined ␴yy and ␴xy loads). Only one half of the predicted results are plotted in Fig. 16. It is evident that the predicted envelope is symmetric with respect to the horizontal axis (transverse stress axis) if max we take max xy = –200 MPa rather than use xy = 200 MPa (see Table 5). This is because the applied transverse and in-plane shear loads do not exhibit coupled effect on any of the internal constituent stresses. Therefore, the principal stresses resulted from xy and ( –xy) will be the same and the same maximum normal stress will be attained. It should be pointed out that the conclusion made herein may not be applicable to an angle-plied laminate.

Fig. 16. Predicted failure envelope of xy vs yy for problem 1.

553 As the ultimate strength (the maximum load combination) was determined by checking any constituent failure status, the failure mode can be clearly shown. For each load combination, the corresponding failure mode was indicated on the failure envelope. It can be seen that for the present load combination, i.e., subjected to combined transverse and in-plane shear loads, the failure of the lamina was always caused by the resin failure. In addition, the resin material has always undergone plastic deformation before composite failure. The yield envelope, inside which the resin and hence the composite does not yield at all whereas outside which the resin is always under a plastic deformation, is also plotted in the figure. From Fig. 16, one can see that the lamina shear strength increases due to the application of a moderate compressive transverse stress. This is attributed to the effect of thermal residual stress. A comparative plot based on the present prediction without thermal residual stress is also shown in the figure, from which the maximum shear strength is achieved when the transversely applied load is zero. When the fiber CTE (Coefficient of Thermal Expansion) is lower than the resin CTE and if the composite cools down to a lower temperature, tensile residual stresses will be generated in the resin material. A moderate compressive load will release the resin from this tensile residual stress and hence will increase the lamina transverse strength to some extent. Note that the lamina transverse strength is governed by the resin strength. The conclusion made herein is applicable to all the four material systems used in this exercise. Problem 2 (UD T300/BSL914C lamina subjected to combined ␴xx and ␴xy loads). In contrast to the problem 1, where a failure envelope due to transverse tensile and in-plane shear loads is a gradually varied smooth curve as shown in Fig. 16, the present failure envelope (only the upper half part), indicated in Fig. 17, under longitudinal tensile and in-plane shear

Fig. 17. Predicted failure envelope of yy vs xy for problem 2.

554 loads consists of almost one half of a rectangle. An interesting phenomenon represented by this figure is that at some load combination, the resin material does not yield at all when the composite fails. For the present composite, this happened when a longitudinal compressive load together with a moderate in-plane shear load was applied to the lamina. Further, under a longitudinal tensile load near to the composite longitudinal tensile strength, the composite failure mode varied from fiber failure to resin failure and back to fiber failure when the composite was subjected to an additional increasing in-plane shear load. From the fiber failure mode to the resin failure mode, the composite failure envelope varied smoothly. However, when the failure mode changed back from the resin to the fiber again, there was “a dent” in the failure envelope. Such kind of “dent” may probably not occur in practice. It would be more pertinent if one connects the fiber failure points (i.e., the points on the envelope which were due to fiber failure) in between the “dent” using a straight line, as indicated by a dashand-dot segment in the figure. Problem 3 (UD E-glass/MY750/HY917/DY063 lamina subjected to combined ␴yy and ␴xx loads). For the present problem, the predicted failure envelope is significantly influenced by the power-index parameter q used in Eq. (21.2), as can be seen from Fig. 18. Since, phenomenologically, taking q = & would give a prediction error when both principal stresses are close to each other, a choice of q 4, particularly q = 3, might be suitable. Hence, all the numerical calculations in this paper including those shown in Fig. 19 are based on q = 3. However, experiments are required to correlate this choice. It should be noted that the due to thermal residual stresses the composite longitudinal as well as transverse tensile strength depends on the power-index q used either. Although a longitudinally applied load only generates a longitudinal stress component, as shown in Eqs.

Fig. 18. Comparison of different power-indexes used in prediction of failure envelopes of yy vs xy for problem 3.

555

Fig. 19. Predicted failure envelope of yy vs yy for problem 3.

(24), a transverse thermal residual stress has already been generated in the resin material. This stress may affect the composite longitudinal strength through a finite number of q, because under the longitudinal load condition both the fiber and the resin materials reached their ultimate load carrying capacity (see Section 4.1). The composite transverse strength can be influenced more distinctly by q: in addition to a thermal residual stress component in the longitudinal direction, a transversely applied load also generates stress component in the longitudinal direction. For the present composite, the maximal difference of the predicted strengths based on different q can be as large as 15%. Regarding Fig. 19, the lower part of the failure envelope (i.e. the points on the failure envelope which have negative transverse stress) is essentially half of a rectangle. However, the upper part even in the second quadrant is not so. The transverse stress involved increased gradually from a minimum (zero) to a maximum, which is located in the second quadrant. This means that a proper longitudinal compression will increase the composite transverse load carrying capacity to some extent. Problem 4 [(90°/ ± 30°/90°) E-glass/LY556/HT907/DY063 laminate subjected to combined ␴yy and ␴xx loads]. Simulation indicates that the last-ply failure envelope does not coincide with the first-ply failure envelope for this laminated composite, although the two envelopes are somewhere very close to each other. Results are plotted in Fig. 20a. In the prediction, different failure modes were observed due to different load combinations. Information on the ply that failed either firstly or lastly has been given in Fig. 20b. It is seen that at some load combinations the last-ply failure occurs in the ( ± 30°) laminae, whereas at the other combinations the last-ply failure occurs in the (90°) laminae. It is noted that at the

556

Fig. 20a. Predicted failure envelope of xx vs yy for problem 4.

present biaxial normal load conditions, the (–30°) lamina and the ( ± 30°) lamina assumed the same amount of internal stresses. Corresponding to the first quadrant load combination, i.e., biaxial tensile loads or xx ≥ 0 and yy ≥ 0, the laminate first-ply failure was always caused by the resin tensile fracture, whereas the last-ply failure might result from either the fiber or the resin failure. In the latter case, the failure could be further initiated by either an over-loaded tensile or a compressive load. Detailed information can be gained from Figs. 20c and d, which clearly show the failure modes and the corresponding load levels sustained by the constituent fiber and resin materials when the first-ply and the last-ply failures occurred in the laminate. It is seen from Fig. 20c that the last-ply failure of the composite was due to the fiber tensile fracture as long as the maximum possible load combination, xx = 1500 cos() (MPa) and yy = 500 sin() (MPa)

Fig. 20b. Predicted failure envelopes for problem 4, on which the ply in failure was shown.

557

Fig. 20c. Variation of maximum and minimum normal stresses of the fiber vs partition parameter, i, in Eq. (44) at the first-ply and the last-ply failures for problem 4, when subjected to biaxial tensile loads.

(referring to Table 5), took an angle in between 55°  65°. On the other hand, Fig. 20d indicates that the composite last-ply failure was caused by the resin tensile fracture, when the combination angle, , was smaller than 55°, whereas by the resin compressive failure if this angle was larger than 65°. The last observation implies that when a composite laminate is subjected to biaxial tensile loads, its constituent materials may carry a resulting compressive stress and the composite failure can be caused by this over-loaded compressive stress. It has been shown [14,16] that the resin plasticity does not cause much difference in strength prediction of unidirectional laminae, as long as the fiber used is sufficiently stiffer than the resin. However, this conclusion is no longer applicable to multidirectional laminates, in general. To show this, predictions of the failure envelopes of the present laminate with different resin plasticity have been made. Results are graphed in Fig. 21. In the figure, “Predicted 1” and “Predicted 2” refer to the predictions based on a bilinear stress–strain

Fig. 20d. Variation of maximum and minimum normal stresses of the resin vs partition parameter, i, in Eq. (44) at the first-ply and the last-ply failures for problem 4, when subjected to biaxial tensile loads.

558

Fig. 21. Predicted failure envelopes of xx vs yy for problem 4 with different resin plastic properties.

assumption for the resin used. A yield strength of 45 MPa (mY = 45 MPa) has been assumed for the “Predicted 1”, whereas a yield strength of 25 MPa (mY = 25 MPa) has been employed in the “Predicted 2.” The resin hardening modulus, E mT , was determined in such a way that the ultimate strain of the resin, corresponding to the ultimate strength used, 56.5 MPa (mu = 56.5 MPa, see Table 1), was equal to 5% [25]. Therefore, the hardening modulus of 315 and 741 MPa has been applied for “Predicted 1” and “Predicted 2”, respectively. The “Predicted 3” was accomplished without any resin plasticity (i.e., a linearly elastic deformation until rupture was assumed for the resin), whereas the “Predicted 4” was obtained using the resin plastic parameters given in Table 1. Further, the symbol “(F)” stands for the first-ply failure, whereas “(L)” denotes the last-ply failure. All the other parameters used have been taken from Table 1. It is evident from Fig. 21 that the predicted failure envelopes of the multidirectional laminate, both the first-ply failure and the last-ply failure envelopes, depend significantly on the resin plastic parameters used. Because of this, the exact resin stress–strain curve must be employed in the laminate inelastic and failure analysis. Problem 5 [(90°/ ± 30°/90°) E-glass/LY556/HT907/DY063 laminate subjected to combined ␴xx and ␴xy loads]. There is no coincidental part between the first-ply and the last-ply failure envelopes for this laminate, which is subjected to combined uniaxial tension/ compression and in-plane shear loads, as shown in Fig. 22a. Information of the failed ply which caused either the first-ply or the last-ply failure is indicated in Fig. 22b. Comparing Fig. 22b with Fig. 20b, an interesting phenomenon can be observed. Namely, when the laminate is subjected to only normal loads, the ( + 30) and the (–30) laminae in the laminate sustain the same amount of stress shares. If, however, the laminate is involved with an in-plane shear load, the stresses shared by the ( + 30) and the (–30) laminae are different.

559

Fig. 22a. Predicted failure envelope of xy vs xy for problem 5.

Fig. 22b. Predicted failure envelopes for problem 5, on which the ply in failure was shown.

Fig. 22c. Variation of maximum and minimum normal stresses of the fiber vs partition parameter, i, in Eq. (44) at the first-ply and the last-ply failures for problem 5, when subjected to combined tensile and in-plane shear loads.

560

Fig. 22d. Variation of maximum and minimum normal stresses of the resin vs partition parameter, i, in Eq. (44) at the first-ply and the last-ply failures for problem 5, when subjected to combined tensile and in-plane shear loads.

The principal/maximum normal stress variations in the fiber and resin materials when the first-ply and last-ply failures happened to the laminate are plotted in Figs. 22c and d. These plots correspond to the situations where the composite was subjected to the x-directional tension and in-plane shear loads. It is seen from the figures that the fiber did not fail at all, whereas the resin might fail due to either over loaded tensile or compressive load. No compressive stress occurred in the resin material of those plies which failed the first, i.e., exhibited the first-ply failure. As a matter of fact, the maximum normal stress corresponding to the first-ply failure shown in Fig. 22d was not equal to the resin first principal stress [see Eq. (21.2)]. On the other hand, the resin material of the other lamina plies was always subjected to a compressive stress component, and the composite last-ply failure was caused by this compressive stress for most of the load combinations, xy = 600 cos() (MPa) and xx = 1500 sin() (MPa). When the load combination angle  took a specific value, i.e., 67.5°, there was no tensile stress component in the resin material except for the stress component in the laminate thickness direction, which was generated due to Poisson’s ratio effect. According to the classical lamination theory, only the in-plane strains are retained. The out-off plane strains are all zero. Once the two in-plane principal stresses of the resin are obtained, the third principal stress, which is in the composite thickness direction, can be calculated using Hooke’s law and the zero strain condition in that direction. As the two in-plane principal stresses are negative, the third principal stress must be positive. This positive stress component in the thickness direction is harmful to the laminate integrity, as it may cause a laminate delamination. Detection of the laminate delamination is also possible, as described in Ref. [39]. However, no resin interlaminar strength or information related to the determination of this strength has been provided. No attempt as to detect any laminate delamination is made in the present paper. Problem 6 [(0°/ ± 45°/90°)s AS4/3501-6 laminate subjected to combined ␴yy and ␴xx loads]. It is noted that under the combined biaxial tension/compression loads, the first-ply failure envelope of this laminate also does not coincide with the last-ply failure envelope, except at two distinct points. Fig. 23a shows the two envelopes whereas Fig. 23b plots the failed ply information corresponding to the first-ply and the last-ply failure. It is noted that

561

Fig. 23a. Predicted failure envelope of xx vs yy for problem 6.

under each load combination, the present laminate generally exhibits three types of failures, i.e., the first-ply failure, the second-ply failure, and the last-ply failure. As in problem 4, the ( + 45) and the (–45) laminae involved sustained the same amount of load shares due to overall applied biaxial normal stresses on the laminate. The maximum and minimum normal stress variations in the fiber and resin when the laminate was subjected to varied biaxial tensile loads are graphed in Figs. 23c and d. A complete symmetry has been observed in these two figures. It deserves special mentioning that for this laminate, the stress level at which the constituent resin material began to yield is zero. This means that the thermal residual stresses generated

Fig. 23b. Predicted failure envelopes for problem 6, on which the ply in failure was shown.

562

Fig. 23c. Variation of maximum and minimum normal stresses of the fiber vs partition parameter, i, in Eq. (44) at the first-ply and the last-ply failures for problem 6, when subjected to combined biaxial tensile loads.

in the resin material before any external load being applied have already attained its yielding stress state. Ignorance of the thermal residual stresses would be impertinent. Problem 9 [( ± 55°)s E-glass/MY750/HY917/DY063 laminate subjected to combined ␴yy and ␴xx loads]. For this laminate, the last-ply failure envelope completely coincides with the first-ply failure envelope, as shown in Fig. 26a. Investigation on the internal stress variations indicates that when the laminate was subjected to a biaxial tensile load combination, yy = 1500 cos() (MPa) and xx = 600 sin() (MPa), with  outside 37.5°  42.5°, the fiber material did not carry much load compared to its load carrying ability especially in the range of  > 42.5°. When  varies from 40° to 42.5°, the load sharing carried by the fiber material

Fig. 23d. Variation of maximum and minimum normal stresses of the resin vs partition parameter, i, in Eq. (44) at the first-ply and the last-ply failures for problem 6, when subjected to combined biaxial tensile loads.

563 suddenly dropped, as can be seen in Fig. 26b. Thus, the resin sustained more load share and hence failed much earlier, giving rise to an earlier failure in composite. Accordingly, the failure envelope shows a sudden change, with a significant reduction in the composite load carrying ability (in the first quadrant of Fig. 26a). However, the results shown in Fig. 26b and c indicate a fact that the laminate cannot generate a delamination under the above load combination with  > 42.5°. This is because the resulting stress components of the fiber and resin materials in the thickness direction, developed due to Poisson’s ratio effect, are both negative, and hence the composite is subjected to an overall compressive stress in the thickness direction. Any delamination of the laminate must be resulted from a positive interlaminar stress [40], i.e. a positive normal stress in through the thickness direction. 5.3.2. Predictions. stress–strain curves Problem 7 [(0°/ ± 45°/90°)s AS4/3501-6 laminate subjected to uniaxial ␴xx load]. Predicted nonlinear stress–strain responses are graphed in Fig. 24. The modes and locations (ply in which failure occurs) of failure have been clearly indicated in the figure. For example, the first-ply failure of this laminate occurred at a stress level of xx = 265 MPa. The failure mode was due to the resin tensile failure, whereas the location occurred in (90°)2 laminae (i.e., the two 90° laminae failed at the stress level of xx = 265 MPa). The influence of both the resin plasticity and the laminate progressive failure on the stress–strain nonlinearity can be distinctly observed. As can be seen from the figure, the laminate second-ply (i.e., the four 45°-angle plied laminae) occurred at xx = 323.7 MPa and the last-ply (i.e., the two 0°-angle plied laminae) failed at xx = 471.3 MPa. Similarly as in problem 6, the resin plasticity took place at the very beginning of an applied load. Namely, the thermal residual stresses in the

Fig. 24. Predicted stress–strain curves, xx vs xx and xx vs yy, for problem 7. (Note MT means resin compressive failure.)

564

Fig. 25. Predicted stress–strain curves, yy vs yy and yy vs xx, for problem 8.

resin were so large that a plastic deformation had already occurred before applying any mechanical load. All the progressive failures of this laminate resulted from the resin fracture. Problem 8 [(0°/ ± 45°/90°)s AS4/3501-6 laminate subjected to combined ␴xx and ␴yy loads (␴yy /␴xx = 2)]. Both yy vs yy and yy vs xx curves are plotted in Fig. 25. While the laminate progress failure cannot be clearly visualized from the yy–yy plot, this is distinctly displayed

Fig. 26a. Predicted failure envelope of xx vs yy for problem 9.

565

Fig. 26b. Variation of maximum and minimum normal stresses of the fiber vs partition parameter, i, in Eq. (44) at the first/last-ply failure for problem 9, when subjected to combined biaxial tensile loads.

by the yy–xx curve. After the second-ply failure, the laminate stiffness in the x-direction reduced dramatically. Significant deformation occurred before the last-ply failure happened. It is noted that the stress level of the last-ply failure was only slightly higher than that of the second-ply failure. The thermal residual stresses in the present laminate were the same as in that of Problem 7.

Fig. 26c. Variation of maximum and minimum normal stresses of the resin vs partition parameter, i, in Eq. (44) at the first/last-ply failure for problem 9, when subjected to combined biaxial tensile loads.

566 Problem 10 [( ± 55°)s E-glass/MY750/HY917/DY063 laminate subjected to uniaxial ␴yy load]. Both the longitudinal and transverse stress–strain curves of the laminate subjected to the uniaxial load in the y-direction were predicted. Results are graphed in Fig. 27. There was no progressive failure pertaining to this laminate. Still, significantly nonlinear deformation occurred, as indicated in the figure, which was caused by the resin inelastic deformation. The resin plasticity occurred after yy = 47 MPa. It is noted that the failure of the present laminate with the y-directional loading was due to the resin compressive failure. On the other hand, the failure of this laminate will result from the resin tensile fracture if a uniaxial tensile load in the x-direction is applied. As the resin compressive strength is much higher than its tensile counterpart, the laminate under consideration has a much higher tensile load carrying capacity in the y direction than in the x direction. It must be pointed out that the present prediction did not incorporate any effect of a possible delamination in the laminate. If we recognize that the fiber axis of this 55° angle-plied laminate has an inclined angle of 35° with the y-direction (the loading direction), we can draw a conclusion as follows: When an angle-plied laminate is subjected to a uniaxial tensile load, a negative or positive maximum principal stress may be generated in the resin material, which has lower stiffness than the fiber, depending on whether the inclined angle between the load direction and the fiber axis is smaller or larger than 45°. As found in Ref. [39], an angle plied laminate subjected to a uniaxial tensile load with a loading direction less than 45° (as in the present case) may probably generate a delamination. Such delamination can significantly reduce the composite load carrying ability. In order to show the influence of thermal residual stresses on mechanical behavior of the composite laminate, prediction without incorporating any temperature variation has also been made, and is plotted in Fig. 27. In the prediction, all the other conditions/parameters were kept

Fig. 27. Predicted stress–strain curves, yy vs yy and yy vs xx, for problem 10. (Note MT means resin compressive failure.)

567

Fig. 28. Predicted stress–strain curves, yy vs yy and yy vs xx, for problem 11.

unchanged except that the stress-free temperature of the composite was assumed to be the same as room temperature. It is seen that the laminate elastic moduli with and without thermal residual stresses are essentially the same. However, discrepancy exists in composite inelastic and strength behavior. For this angle-plied laminate subjected to y-directional load, the composite ultimate strength decreases by 8.6% (from 321.7 to 294 MPa) if there is no temperature influence. It should be pointed out that if the laminate is subjected to x-directional load without thermal residual stresses, the resulting ultimate strength will increase by 20%. This means that the influence of thermal residual stress on the composite strength depends not only on the constituent properties and laminate geometric parameters but also on the loading direction. Problem 11 [( ± 55°)s E-glass/MY750/HY917/DY063 laminate subjected to combined ␴yy and ␴xx loads (␴yy /␴xx = 2)]. Under this fixed ratio of biaxial load combination, the present laminate also had no progressive failure. In contrast to the uniaxial tension in the previous problem, problem 10, where significant nonlinearity was observed for the laminate stress– strain response, the present laminate showed grossly linear deformation both in the x and in the y directions. This is because the resin material of the present laminate was subjected to biaxial tensile stresses combined with biaxial tensile thermal residual stresses. The thermal residual stress component of the resin in the x-direction was higher than in the y-direction. However, under present load condition, the resin mechanical stress component in the y-direction was higher than in the x-direction. Thus, the resulting von Mises equivalent stress was low even though the equivalent maximum normal stresses, defined by Eq. (21.2), was high. Not much plasticity occurred before the resin fracture failure took place. Problem 12 [(0°/90°/0°) E-glass/MY750/HY917/DY063 laminate subjected to uniaxial ␴xx load]. For this problem, the predicted stress–strain curves display a nonlinear behavior

568

Fig. 29. Predicted stress–strain curves, xx vs xx and xx vs yy, for problem 12.

slightly in the x direction but noticeably in the y direction, as indicated in Fig. 29. The nonlinearity is the effect of the first-ply failure combined with the resin plasticity which began to occur after xx = 64 MPa. In order to show the influence of thermal residual stresses on mechanical behavior of the composite laminate, predictions without incorporating any temperature variation have also been made, and are plotted in Fig. 29. The ultimate strength of this crossplied laminate increases a large amount if the composite thermal residual stresses can be reduced. Without thermal residual stress effect, this composite can elevate its ultimate load carrying capacity to 1.41 times as high as that with thermal residual stresses incorporated. Problem 13 [( ± 45°)s E-glass/MY750/HY917/DY063 laminate subjected to combined ␴yy and ␴xx loads (␴yy /␴xx = 1)]. The predicted curve is essentially a linear segment. The reason is similar to that pertaining to the problem 11. Problem 14 [( ± 45°)s E-glass/MY750/HY917/DY063 laminate subjected to combined ␴yy and ␴xx loads (␴yy /␴xx = –1)]. Significantly nonlinear deformation was observed for the predicted stress–strain curves both in the x and in the y directions, which was due to the resin plasticity. The resin began to yield at a stress level of (yy, xx) = (29.2, –29.2) (MPa). As indicated in Fig. 31, there was no progressive failure pertaining to the angle-plied laminate. An interesting phenomenon is needed to point out. It has been shown by the classical lamination theory that the behavior of the ( ± 45°)s laminate under a stress ratio of 1 : –1 condition (i.e., biaxial normal loads but different from each other in loading directions: one is tensile and another is compressive) is equivalent to the response of a unidirectional lamina subjected to a pure shear load. However, this is valid, according to the classical lamination theory combined with the bridging model, only when there is no thermal residual stress. The

569

Fig. 30. Predicted stress–strain curves, yy vs yy, for problem 13.

( ± 45°)s laminate under the stress ratio of 1 : –1 condition displays no dependency on any temperature variation, giving an ultimate strength of 126 MPa. The no dependency is valid regardless of any type of fibers and resin used. However, the unidirectional lamina having the same fiber volume fraction exhibits in-plane shear strength of 90 MPa (see Table 2). If the lamina is assumed to be free of stress at room temperature, the predicted in-plane shear

Fig. 31. Predicted stress–strain curves, yy vs yy and yy vs xx, for problem 12.

570 strength (under in-plane shear load condition) will be 126 MPa, higher than that with the thermal residual stresses.

5.4. Summary from simulations The unidirectional lamina had considerably different failure envelopes when subjected to different biaxial load combinations. If combined transverse tensile and in-plane shear loads were applied to the lamina, the resulting envelope displayed a smoothly varied, nonlinear curve. On the other hand, the failure envelope of the lamina resembled a half of a rectangle when subjected to combined longitudinal tensile/compressive and in-plane shear loads. Under biaxial tension conditions, the predicted failure envelope was influenced to a large extent by the power-index q used in Eq. (21.2). For the laminated composites, the first-ply failure was generally caused by the resin fracture, especially when the first-ply failure stress did not coincide with the last-ply failure stress. In the latter case and if the last-ply failure was induced by the fiber fracture, the last-ply failure strength was considerably higher than the first-ply failure strength. In general, the lastply failure envelope of the laminate did not coincide with its first-ply failure envelope unless the laminate consists of symmetric angle plied laminae with the same inclined angle. The influence of thermal residual stresses for composite elastic behavior is negligibly small, but is distinct or significant for composite inelastic and strength characteristics. The influence for composite strengths depends heavily on laminate lay-up configuration as well as on load form and condition. For some composites, the thermal residual stresses can increase composite strengths; whereas for others, they may significantly decrease the composite load carrying capacity. As designated by Eq. (36), a total stiffness reduction has been applied to any failed lamina for the laminate remaining analysis. This reduction may or may not cause an abrupt change in the whole stress–strain curve of the laminate, depending on failed lamina arrangement with respect to the loading direction and on the remaining laminate stiffness. If the longitudinal direction of the failed lamina has a large inclined angle with the loading direction and/or if the remaining laminate still has large stiffness in the loading direction, a smooth transition in the laminate stress–strain curve can be observed, as indicated by most stress–strain curves shown in this paper. There is only one exception that is shown in Fig. 25, where the stress–strain curve in the x-direction has a sudden change after the second-ply failure. This is because the remaining laminate, consisting of (90°)2 laminae after the second-ply failure, has considerably very small stiffness in the x-direction. On the other hand, this remaining laminate is still stiff enough in the y-direction, giving very little change in the y-directional stress–strain shape. In fact, the stiffness of the laminate [90°/ ± 45°/0°]s in the y-(i.e., 90°-)direction is mainly contributed from the (90°)2 laminae, which failed the last. Therefore, the stress–strain curve in the y-direction up to the last-ply failure is essentially linear.

6. Discussion In this section, some more discussions and remarks pertaining to the model development and to the exercise predictions are summarized.

571 (1) The key point of the present model is the bridging matrix and its explicit expression. As mentioned in Section 3, the bridging elements giving longitudinal properties can be considered as fixed. On the other hand, the transverse and in-plane shear bridging parameters, # and , in Eqs. (10.1) and (10.2) should be calibrated using experimentally measured transverse and in-plane shear moduli of the unidirectional composite. For simplicity, however, both the bridging parameters can be chosen as 0.5, giving results supported by other wellknown models predictions, see Figs. B1 and B2 in Appendix B. It is noted that the bridging parameters are material independent. Namely, once these parameters have been calibrated using measured elastic properties, if any, they can be employed in the inelastic and strength analysis of the composite. (2) Incorporation of the resin plasticity/nonlinear deformation is very important for laminate inelastic and strength analysis, as has been demonstrated in Fig. 21. It is this incorporation that makes the present bridging micromechanics model different from most other existing models. Further, the present model is consistent in that when the two constituent materials become the same, the resulting instantaneous compliance matrix of the composite, Eqs. (6), deteriorates to that of the constituent material given by any isotropic plastic flow theory used. As the resin plastic parameters are essential to the laminate strength prediction, they should be provided a priori. A uniaxial tensile and a uniaxial compressive stress–strain curves of the monolithic resin specimens fabricated in the same condition as in making the composite are sufficient for the present model application, if the resulting composite (laminate) is subjected to an in-plane load. However, neither of these stress–strain curves for any of the four resin materials has been provided in Ref. [25]. The present predictions were carried out based on the resin plastic parameters retrieved from the in-plane shear stress–strain curves of the corresponding laminae. Errors might have occurred in use of these parameters in two aspects. Firstly, any small deviation in the measured lamina stress–strain curve may bring an enlarged error to the retrieved constituent properties. Secondly, the same plastic parameters had to be used in those cases where the resin was subjected to a compressive load, even up to failure. As has been pointed out before, many a time the resin material was subjected to a significant compression even though the composite laminates were tensile loaded. In general, the stress–strain curve of an isotropic material under compression is more or less different from that under tension. This is especially true for polymer materials. Therefore, the measured compressive stress–strain curve of the resin should be also employed in prediction. (3) It has been recognized that two issues must be addressed in the laminate strength prediction. One is the stress identification, which can be achieved using the bridging model combined with the classical laminate theory, and another is the choice of a strength criterion. All the strength criteria established in the literature can be categorized into two classes. The first class criteria are applied to homogeneous, mainly isotropic, materials, and the second classes to heterogeneous composite materials. The present paper employed a generalized maximum normal stress criterion, Eqs. (21), which belongs to the first class criteria, to detect the failure of a constituent material. Evidently, the power-index q used in Eq. (21.2) is material independent. The reason for the author to choose the generalized rather than the original or classical maximum normal stress criterion is that the constituent (resin) material in a laminate is quite often subjected to the resulting biaxial tensile loads. Intuitively, the load carrying capacity of an isotropic material should be reduced a little bit when it is subjected to an equitriaxial/equibiaxial tension compared to uniaxial tension. Further, the predicted

572 curves shown in Fig. 18 also suggest that a power-index q smaller than infinity is pertinent. The use of this index is imperial, which should be correlated with experiments. With q = 3 as employed throughout in this paper, a maximum discrepancy of 26% between the resulting generalized and the classical maximum normal stress criteria can be observed only when the two principal stresses of the material are equal to each other. However, in this latter case, i.e., when the material is subjected to equibiaxial tension, the classical maximum normal stress criterion is believed to be impertinent. Therefore, the choice of q = 3 is suitable. In order to apply the generalized maximum normal stress criterion, the in situ constituent tensile and compressive strengths must be provided. As can be understood, the constituent strength values are the most critical parameters for laminate strength prediction. It is known that the in situ constituent strengths can be influenced to a large extent by the composite fabrication process. Various unforeseen parameters may affect the composite ultimate performance. Even with the same constituent materials, the same laminate geometry, and the same fabrication condition (e.g. curing temperature), different laboratory may obtain a composite with a different ultimate strength. As such, the constituent strength values should be retrieved using measured ultimate strengths of the composite under some simple load conditions if an accurate analysis/critical application is required. It must be noted that differently loaded composite strength may result in a different constituent strength, if the retrieval is performed. From Tables 1–4, we have already observed that using the resin strength retrieved from the lamina longitudinal tensile strength, a much higher than the measured transverse tensile strength of the composite could result. Therefore, some compensation for the strength values retrieved from different uniaxial strengths of the composite might have been made. Since all the stress components on each lamina are also known, the second class criteria such as Tsai–Wu and Hashin–Rotem criteria might have been incorporated in the present modeling. Comparison between use of the Tsai–Wu/Hashin–Rotem and the generalized maximum normal stress criteria will be made once the experimental data of all the exercise problems are available. (4) Application of the present theory in composite designs is expected, as a minimum number of input data only pertaining to the properties of the constituent fiber and resin materials, the laminate geometry (the fiber volume fraction, the ply angle, and the ply thickness), and the stress-free temperature are required. All these data are easily measurable using the monolithic constituent specimens, obtainable from a material data book a priori, or just used as the design parameters. The generalized maximum normal stress criterion with q = 3 can be employed, and the bridging parameters, # and , can be set to 0.5, or slightly smaller than 0.5, if no other information is provided. The resin plastic parameters, however, should be accurately specified whenever possible, as they have significant influence on the prediction. Using a PC software developed by the author (available upon request), the laminate instantaneous properties at any load level are easily obtainable. Further, the PC software for laminate properties can be simply incorporated into an FEM software such as ABAQUS using a user provided subroutine to analyze a sophisticated structure made from the composite laminate. The FEM software is employed to determine the overall stress increments applied to each composite element (Finite Element) based on its instantaneous stiffness at the previous load level. Once the new stress increments on the composite element have been known, the material software (the user subroutine) can update the instantaneous stiffness and internal stresses, check the element failure status, and calculate the elemental instantaneous

573 stiffness matrix. If the element has not failed, the updated stiffness is inputted into the FEM software again, and the calculation continues until one of the composite elements attains its last-ply failure. (5) The present theory cannot deal with damage evolution before any ply failure, and no material buckling has been taken into account. Any defect of fiber/resin interface bonding is only grossly incorporated in the bridging parameters, # and , and no further debonding is assumed before the ply failure takes place. 7. Conclusion A total number of fourteen laminated composites of different constituent materials subjected to various biaxial loads have been simulated in this paper, based on a recently developed micromechanics model, the Bridging Model, and the classical laminate theory. The most important feature of the present modeling approach is that the internal stresses generated in the constituent fiber and resin materials of the composite are fully characterized up to the composite failure. Thus, the composite properties are treated in terms of those of the constituent materials. The thermal effect/thermal residual stresses can be easily incorporated in the analysis either. All the information pertaining to the laminate progressive failure, such as the failure stress-state, the failed plies, the failure mode, the constituent stress level at this failure, etc., can be clearly identified. The modeling method is general, since it can be applied to a variety of laminated composites made from different constituent materials, as long as the constituent fibers are transversely isotropic and the resins are isotropic, and as long as the constitutive descriptions for the fiber and resin materials are available. Yet, the method is userfriendly, since only explicit formulae are employed, no iteration is involved, and a minimum number of experimental data are required. As the method is still new in its current development, further correlation between the model predictions and the experiments is published in a companion paper, Ref. [44]. References [1] Kim RY. In-plane tensile strength of multidirectional composite laminate. In: Failure analysis of composite laminates. Lancaster: Technomic Publishing Co., Inc.; 1985. [2] Rowlands RE. Strength (Failure) Theories and Their Experimental Correlation. In: Sih GC, Skudra AM, editors. Failure mechanics of composites, handbook of composites, vol. 3. New York: North-Holland; 1985. pp. 71–128. [3] Nahas MN. Survey of failure and post-failure theories of laminated fiber-reinforced composites. J Comp Tech Res 1986;8(4):138–53. [4] Echaabi J, Trochu F, Gauvin R. Review of failure criteria of fibrous composite materials. Poly Comp 1996;17(6):786–98. [5] Wolfe WE, Butalia TS. A strain-energy based failure criterion for non-linear analysis of composite laminates subjected to biaxial loading. Compos Sci Technol 1998;58:1107–24. [6] Hinton MJ, Soden PD. Predicting failure in composite laminates: the background to the exercise. Compos Sci Technol 1998;58:1001–10. [7] Soden PD, Hinton MJ, Kaddour AS. A comparison of the predictive capabilities of current failure theories for composite laminates. Compos Sci Technol 1998;58:1225–54. [8] Benzeggagh ML, Khellil K, Chotard T. Experimental determination of Tsai failure tensorial terms Fij for unidirectional composite materials. Compos Sci Technol 1995;55(2):145–56.

574 [9] Huang ZM. A unified micromechanical model for the mechanical properties of two constituent composite materials, part I: elastic behavior. J Thermoplastic Composite Materials 2000;13(4): 252–71. [10] Huang ZM. A unified micromechanical model for the mechanical properties of two constituent composite materials, part II: plastic behavior. J Thermoplastic Composite Materials 2000;13(5): 344–62. [11] Huang ZM. A unified micromechanical model for the mechanical properties of two constituent composite materials, part III: strength behavior. J Thermoplastic Composite Materials 2001; 14(1):1–16. [12] Huang ZM. A unified micromechanical model for the mechanical properties of two constituent composite materials, part IV: rubber elastic behavior. J Thermoplastic Composite Materials 2000; 13(2):119–39. [13] Huang ZM. A unified micromechanical model for the mechanical properties of two constituent composite materials, part V: laminate strength. J Thermoplastic Composite Materials 2000;13(3): 190–206. [14] Huang ZM. Micromechanical prediction of ultimate strength of transversely isotropic fibrous composites. Int J Solids Struct 2001; 38(22–23):4147 72. [15] Huang ZM. Simulation of inelastic response of multidirectional laminates based on stress failure criteria. Mater Sci Technol 2000;16(6):692–8. [16] Huang ZM. Simulation of the mechanical properties of fibrous composites by the bridging micromechanics model. Composites Part A 2001;32(2):143–72. [17] Huang ZM. Strength formulae of unidirectional composites including thermal residual stresses. Mater Lett 2000;43(1–2):36–42. [18] Huang ZM. Modeling strength of multidirectional laminates under thermo-mechanical loads. J Comp Mater 2001;35(4):281–315. [19] Huang ZM. The mechanical properties of composites reinforced with woven and braided fabrics. Compos Sci Technol 2000;60(4): 479–98. [20] Huang ZM, Ramakrishna S. Micromechanical modeling approaches for the stiffness and strength of knitted fabric composites: a review and comparative study. Composite A 2000; 31(5):479 501. [21] Huang ZM, Ramakrishna S, Tay AAO. Unified micromechanical model for estimating elastic, elasto-plastic, and strength behaviors of knitted fabric reinforced composites. J Reinforced Plastics & Composites 2000;19(8):642–56. [22] Huang ZM, Ramakrishna S, Tay AAO. Modeling of stress–strain behavior of a knitted fabric reinforced elastomer composite. Compos Sci Technol 2000;60(5):671–91. [23] Huang ZM, Ramakrishna S, Tay AAO. A micromechanical approach to the tensile strength of a knitted fabric composite. J Composite Mater 1999;33(19):1758–91. [24] Huang ZM, Ramakrishna S, Tay AAO. Micromechanical characterization for the mechanical properties of textil elastomeric composites. Mater Sci Res Int, JSMS 1999;5(3):189–94. [25] Soden PD, Hinton MJ, Kaddour AS. Lamina properties, lay-up configurations and loading conditions for a range of fiber-reinforced composite laminates. Compos Sci Technol 1998;58: 1011–22. [26] Adams DF. Elastoplastic behavior of composites. In: Sendeckyj GP, editor. Mechanics of Composite Materials. New York: Academic Press; 1974. pp. 169–208. [27] Tsai SW, Hahn HT. Introduction to composite materials. Lancaster, Basel: Technomic Publishing Co., Inc.; 1980. [28] Hyer MW. Stress analysis of fiber-reinforced composite materials. Boston: WCB, McGraw-Hill; 1997 pp. 120–124. [29] Chamis CC. Mechanics of composite materials: past, present, and future. J Comp Technol Res 1989;11:3–14. [30] Berthelot JM. Composite materials, mechanical behavior and structural analysis. Berlin, New York: Springer; 1999 pp. 158–171.

575 [31] Levin, V. M., On the coefficients of thermal expansion of heterogeneous materials. Mekhanika Tverdovo Tela, 1967. Vol. 1, p. 88. [32] Benveniste Y, Dvorak GJ. On a correspondence between mechanical and thermal effects in twophase composites. In: Weng GJ, Taya M, Abe H, editors. The Toshio Muta anniversary volume: micromechanics and inhomogeneity. New York: Springer; 1990. p. 65–81. [33] Timoshenko SP. History of strength of materials. New York: McGraw Hill; 1953. [34] Huang ZM. Micromechanical strength formulae for unidirectional composites. Materials Letter 1999;40:164–9. [35] Gibson RF. Principles of composite material mechanics. New York: McGraw-Hill, Inc.; 1994. [36] Reddy JN. Theory and analysis of laminated composite plates and shells. New York: John Wiley & Sons; 1988. [37] Huang ZM. Progressive flexural failure analysis of laminated composites with knitted fabric reinforcement. In: Proceedings of the Royal Society A: Mathematical, Physical, and Engineering Sciences [submitted]. [38] Huang ZM, Fujihara K, Ramakrishna S. Flexural failure behavior of laminated composites reinforced with braided fabrics, AIAA Journal [submitted]. [39] Huang ZM. Micromechanical life prediction for composite laminates. Mechanics of Materials 2001;33(4):185–99. [40] Pipes RB, Pagano NJ. Interlaminar stresses in composite laminates under uniform axial extension. J Comp Mater 1970;4:538–48. [41] McCullough RL. Micro-models for composite materials-continuous fiber composites, micromechanical materials modeling. In: Whitney James M, McCullough Roy L, editors. Delaware Composites Design Encyclopedia, Vol. 2. Lancaster: Technomic Publishing Co., Inc.; 1990. [42] Kriz RD, Stinchcomb WW. Exp Mech 1979;19:41. [43] Aboudi J. Effective behavior of inelastic fiber-reinforced composites. Int J Engng Sci 1984;22:439–49. [44] Huang Z. M, Correlation of the bridging model predictions of the biaxial failure strengths of fibrous laminates with experiments. Compos Sci Technol, V64, pp 529–548, 2004, see also Chapter 3.16 in this book.

576 Table 6 Predicted failure envelopes for problems 1 and 2. Problem 1

Problem 2

Quadrant 1

Quadrant 2

Quadrant 1

Quadrant 2

yy (MPa)

xy (MPa)

yy (MPa)

xy (MPa)

xx (MPa)

xy (MPa)

xx (MPa)

xy (MPa)

71.9 71.5 70.2 68.2 65.8 63 60 57.1 54.2 51.4 48.7 46.1 43.7 41.4 39.1 37 35 33 31 29.1 27.3 25.5 23.7 22 20.3 18.6 17 15.3 13.7 12 10.4 8.7 7 5.3 3.6 1.8 0

0 6.2 12.3 18 23.2 27.9 32.2 36 39.4 42.6 45.4 48 50.4 52.7 54.8 56.8 58.7 60.4 62 63.6 65.1 66.5 67.8 69.1 70.4 71.6 72.8 74 75.1 76.3 77.4 78.6 79.8 81 82.2 83.5 84.8

–114.3 –114.2 –113.9 –113.2 –112.5 –111.4 –110.1 –108.6 –107 –105.1 –103.1 –101 –98.7 –96.2 –93.5 –90.8 –87.8 –84.8 –81.5 –78.2 –74.8 –71.3 –67.7 –64 –60.3 –56.5 –52.5 –48.5 –44.3 –40.1 –35.7 –31.5 –27.8 –21.1 –12.5 –5.8 0

0 3.3 6.6 9.9 13.2 16.5 19.7 22.8 26 29 32.1 35.1 38 40.9 43.7 46.4 49.1 51.8 54.4 56.9 59.4 61.9 64.5 67 69.6 72.3 75.1 78 81.2 84.7 88.9 94.7 105.2 106.7 95.4 89.3 84.8

1500 1500.2 1500.1 1500.4 1499.4 1497 1495.6 1491 1487.1 1481.3 1474.3 1464.3 1450.6 1494.2 1492.9 1491.5 1489.3 1486.8 1484.3 1481.2 1441.5 1315.9 1199.3 1089.8 986.7 888.5 794.9 705.4 619.3 535.8 455.1 376.1 299 223 148.1 73.9 0

0 3.1 6.3 9.5 12.7 15.9 19.2 22.6 26 29.5 33 36.6 40.2 45.7 50.2 54.9 60 65.4 71.2 77.6 82.5 82.3 82.2 82.1 82 81.9 81.8 81.7 81.7 81.6 81.5 81.4 81.4 81.3 81.2 81.2 81.1

–900.5 –900.1 –900.1 –900.2 –900.1 –900.1 –899.8 –899.8 –899.8 –899.4 –899.5 –899.4 –898.9 –898.2 –897.4 –896.5 –895.5 –894.3 –893.4 –892.1 –847.5 –775.3 –707.5 –643.7 –583.5 –526.2 –471.4 –418.8 –368.2 –319 –271.2 –224.4 –178.6 –133.4 –88.7 –44.3 0

0 3.1 6.3 9.5 12.7 16 19.3 22.7 26.2 29.8 33.6 37.5 41.5 45.8 50.3 55 60.1 65.6 71.5 77.9 80.8 80.8 80.8 80.8 80.9 80.9 80.9 80.9 80.9 81 81 81 81 81.1 81.1 81.1 81.1

577 Table 7 Predicted failure envelopes for problem 3. Quadrant 1

Quadrant 2

Quadrant 3

Quadrant 4

xx (MPa)

yy (MPa)

xx (MPa)

yy (MPa)

xx (MPa)

yy (MPa)

xx (MPa)

yy (MPa)

1280 1220.5 1183.8 1152.6 1123.5 1095.1 1069 1042.7 1017.2 993.9 969.7 946.9 923.8 906.6 890.8 875.3 858.6 841.1 823.2 805.1 785.3 764 740.9 716.4 689.2 659.1 625.5 588.1 545.5 497.2 443 382.2 315.3 242.9 165.7 84.5 0

0 2.1 4.1 6.1 7.9 9.7 11.5 13.2 14.8 16.5 18.1 19.7 21.3 23.1 25 26.9 28.8 30.8 32.9 35.1 37.4 39.8 42.3 45 47.7 50.6 53.7 56.8 60 63.1 66.1 69 71.5 73.8 75.7 77.4 78.8

–800 –799.9 –799.6 –799.8 –799 –799.3 –798.5 –798.6 –798.1 –797.6 –797.6 –797.1 –796.7 –796.2 –795.7 –795.5 –794.6 –794.3 –793.4 –792.7 –791.9 –791 –789.6 –788.4 –786.7 –784.4 –781.6 –692.7 –606.7 –523.8 –443.3 –365.1 –288.7 –214 –141 –69.6 0

0 1.7 3.5 5.3 7 8.9 10.7 12.6 14.5 16.5 18.6 20.7 23 25.4 27.9 30.5 33.3 36.4 39.7 43.3 47.2 51.5 56.4 61.9 68.1 75.3 83.8 83.6 83.4 83.1 82.7 82.3 81.9 81.3 80.6 79.7 78.8

–800 –801.2 –801.6 –802.4 –802.9 –803.2 –803.7 –803.7 –804.4 –804.4 –804.8 –804.8 –805.4 –805.2 –805 –805 –804.3 –803.6 –803.3 –802.2 –802.2 –739.8 –674.9 –614 –556.3 –501.8 –449.4 –399.3 –350.9 –304.1 –258.5 –214 –170.2 –127.1 –84.5 –42.2 0

0 –5.2 –10.5 –15.8 –21.2 –26.7 –32.3 –38 –43.9 –50 –56.3 –62.8 –69.8 –76.9 –84.5 –92.7 –101.2 –110.5 –120.5 –131.3 –143.4 –144.6 –144.6 –144.6 –144.5 –144.6 –144.6 –144.6 –144.6 –144.7 –144.7 –144.8 –144.8 –144.9 –144.9 –145 –145

1280 1279.6 1279.3 1278.1 1277.8 1276.5 1275.8 1274.8 1273.3 1271.9 1270.3 1268.4 1266.6 1263.7 1261.5 1258.1 1253.8 1248.5 1220.9 1118.1 1023.1 935 852.7 775.5 702.1 633 566.7 502.9 441.8 382.4 324.8 268.6 213.4 159.4 105.8 52.8 0

0 –6.7 –13.4 –20.2 –27 –34 –41 –48.2 –55.6 –63.2 –71.1 –79.2 –87.8 –96.6 –106 –115.8 –126.2 –137.3 –146.5 –146.4 –146.3 –146.2 –146.1 –146.1 –145.9 –145.9 –145.8 –145.7 –145.7 –145.5 –145.5 –145.4 –145.3 –145.2 –145.1 –145.1 –145

578 Table 8a Predicted first-ply failure envelopes for problem 4. Quadrant 1

Quadrant 2

Quadrant 3

Quadrant 4

yy (MPa)

xx (MPa)

yy (MPa)

xx (MPa)

yy (MPa)

xx (MPa)

yy (MPa)

xx (MPa)

130.7 131.9 133.4 134.6 135.8 136.9 138.1 138.9 139.3 139.7 139.7 139.1 138.1 136.7 135 124.8 115.2 106.2 97.9 90.3 83 76.1 69.7 63.6 57.9 52.2 46.9 41.7 36.8 31.9 27.1 22.5 17.9 13.4 8.9 4.5 0

0 10.8 21.9 33.2 44.9 56.9 69.4 82.1 95.1 108.5 122.1 135.8 149.5 163.3 177.2 179.6 181.3 182.4 183.5 184.7 185.4 186 186.8 187.2 187.9 188 188.5 188.9 189.3 189.8 189.8 190.4 190.6 190.9 191.3 191.3 191.5

–183.2 –178 –172.7 –167.9 –163.1 –158.6 –154 –149.5 –145.3 –141.4 –137.5 –133.6 –129.8 –125.9 –122.1 –118.3 –114.6 –110.7 –105 –95.8 –87.8 –80 –73 –66.2 –60 –53.9 –48.2 –42.9 –37.6 –32.6 –27.6 –22.8 –18.2 –13.5 –9 –4.5 0

0 14.6 28.3 41.4 53.9 65.9 77.4 88.4 99.2 109.8 120.2 130.4 140.5 150.4 160.3 170.1 180.3 190.2 196.9 196.1 196.1 195.6 195.4 194.8 194.9 194.3 193.9 194 193.6 193.6 193.2 192.8 193 192.3 192.3 191.8 191.5

–183.2 –188.4 –193.4 –198.6 –204.3 –210.1 –216.4 –223.3 –230.5 –238.2 –246.8 –254.3 –262.3 –271.5 –282 –268.5 –242.5 –220.4 –201.2 –183.9 –168.2 –153.9 –140.6 –128.2 –116.4 –105.3 –94.9 –84.8 –75.1 –65.4 –56 –46.7 –37.4 –28.1 –18.8 –9.5 0

0 –15.4 –31.7 –49 –67.5 –87.3 –108.7 –132 –157.3 –185 –215.7 –248.2 –284 –324.3 –370.2 –386.3 –381.5 –378.7 –377.2 –376.4 –375.7 –376 –376.4 –377.4 –378 –379.2 –381.6 –383.9 –386.7 –389.1 –392.2 –394.9 –397.4 –400.5 –404 –406.6 –410

130.7 129.2 128.3 127.4 127.1 126.3 125.7 124.6 123.8 122.7 121.8 120.6 120.1 119.9 119.5 119.1 118.7 118 117.5 117.5 117.1 116.9 116.6 116.8 117.1 111.1 99.7 88.8 78.3 67.9 57.8 48 38.3 28.6 19.1 9.5 0

0 –10.6 –21 –31.5 –42 –52.5 –63.2 –73.7 –84.5 –95.3 –106.5 –117.7 –130 –143.2 –156.9 –171.4 –186.7 –202.7 –220.3 –240.4 –261.6 –285.6 –312.1 –343.7 –380.2 –400 –401 –401.9 –403.1 –403.9 –404.7 –406.1 –407.2 –408 –408.9 –410.1 –410

579 Table 8b Predicted last-ply failure envelopes for problem 4. Quadrant 1

Quadrant 2

Quadrant 3

Quadrant 4

yy (MPa)

xx (MPa)

yy (MPa)

xx (MPa)

yy (MPa)

xx (MPa)

yy (MPa)

xx (MPa)

233.3 185.4 166.3 156.8 151.5 148.7 146.6 145.2 144.1 143.1 142.1 140.7 139 137.2 135.2 129.5 124.6 120.1 115.8 112.1 108.2 104.5 101.1 97.7 94.4 91.2 94.7 167.3 145.8 124.9 98.8 73.1 52.8 36.3 22.5 10.5 0

0 15.2 27.3 38.7 50.1 61.8 73.6 85.9 98.3 111.2 124.2 137.4 150.5 163.9 177.5 186.3 196.1 206.4 217.1 229.3 241.7 255.5 270.7 287.6 306.6 328.6 380.6 757.1 751.3 742.5 691.6 618.5 561.8 517.5 481.2 450.6 425

–195.5 –191 –186.5 –182.2 –178.1 –173.9 –170 –166.1 –162.1 –158.7 –154.9 –151.1 –147.6 –143.7 –140 –136.2 –132.6 –128.8 –123.5 –115.5 –108.5 –101.6 –95.1 –88.8 –82.8 –76.8 –70.8 –64.9 –58.8 –52.8 46.4 –39.9 –33.1 –25.7 –17.9 –9.4 0

0 15.6 30.6 45 58.9 72.3 85.4 98.2 110.6 123.2 135.4 147.5 159.8 171.7 183.8 196 208.6 221.3 231.6 236.3 242.5 248.3 254.8 261.5 268.9 276.7 284.6 293.8 303.1 313.8 325 337.3 351.6 366.3 383.5 402.6 425

–195.5 –200.3 –205.1 –210.2 –215.6 –220.8 –226.7 –232.7 –239.1 –245.6 –252.3 –259.5 –267.2 –275.1 –282.7 –276 5 –256.8 –237.1 –218.9 –201.8 –185.5 –170.1 –155.6 –141.7 –128.5 –115.9 –103.7 –92 –80.7 –69.8 –59.3 –49 –38.9 –29 –19.2 –9.6 0

0 –16.4 –33.6 –51.9 –71.3 –91.8 –113.9 –137.6 –163.1 –190.8 –220.6 –253.3 –289.3 –328.6 –371.1 –397.8 –404 –407.4 –410.5 –412.9 –414.4 –415.7 –416.5 –417.1 –417.4 –417.3 –416.9 –416.7 –415.8 –415.3 –414.9 –414.4 –413.6 –412.9 –411.9 –411.6 –411

233.3 256.6 251.8 206.5 183.8 169.5 160.2 152.6 147.1 142.2 138.2 134.8 132.1 130.2 128.2 126.5 125 123.3 121.8 121.1 120 119.2 118.2 117.9 117.7 111.6 100.1 89.1 78.4 68.1 58 48.1 38.3 28.7 19.1 9.6 0

0 –21 –41.3 –51 –60.8 –70.5 –80.5 –90.2 –100.4 –110.4 –120.9 –131.6 –143 –155.5 –168.3 –182 –196.7 –211.8 –228.4 –247.7 –268.1 –291.2 –316.6 –347.1 –382.4 –401.8 –402.4 –403.3 –404.1 –404.9 –405.7 –407.1 –407.7 –408.5 –409.4 –410.6 –411

580 Table 9 Predicted failure envelopes for problem 5. First-play failure envelopes

Last-ply failure envelopes

Quadrant 1

Quadrant 3

Quadrant 2

Quadrant 4

xx (MPa)

xy (MPa)

xx (MPa)

xy (MPa)

xx (MPa)

xy (MPa)

xx (MPa)

xy (MPa)

191.5 191.3 191.3 191.3 190.6 190.4 189.3 188.8 187.9 187.1 185.8 184.5 183.2 181.3 179.8 177.7 175.4 173.3 170.4 167.9 164.9 161.6 158.9 156.4 153.2 150.1 146.4 142 134.9 120.6 106 90.8 74.8 58 40.1 20.9 0

0 3.3 6.7 10.1 13.4 16.9 20.3 23.8 27.4 31 34.7 38.4 42.3 46.2 50.4 54.5 58.9 63.5 68.2 73.3 78.6 84.3 90.8 98.2 106.2 115.3 125.6 137.1 148.3 153 158.2 163.8 169.6 176.1 183.3 191 199.4

–410 –400.6 –391 –381.2 –372.3 –363.7 –355.5 –346.7 –338.8 –330.7 –322.6 –314.9 –307 –299.4 –291.2 –283.2 –275.4 –266.9 –258.4 –250 –240.7 –231.6 –222 –211.7 –201.2 –190 –178.3 –165.9 –152.7 –138.5 –123.2 –106.8 –89.2 –70 –47.7 –22.8 0

0 7 13.7 20.1 26.3 32.2 38.1 43.7 49.3 54.8 60.2 65.6 70.9 76.3 81.6 86.9 92.4 97.8 103.4 109.1 114.8 120.7 126.8 132.9 139.4 146 153 160.2 167.8 175.7 183.9 192.7 202.3 212.6 218.2 208.6 199.4

425 424.6 424.4 424.8 424.5 424.2 424 423.5 423.3 423.1 422.8 422.2 421.8 421.3 420.6 420.1 419.4 419.1 420 424.3 416.2 407.9 399.5 380.9 361 341 320.1 217.9 223 187 154.4 124.3 97.4 72.2 48.1 24.2 0

0 7.4 14.9 22.4 29.9 37.6 45.4 53.4 61.6 70.1 78.9 87.9 97.4 107.4 117.8 128.9 140.8 153.6 168 185.2 198.4 212.6 228.2 239.2 250.1 262 274.6 210.5 245.1 237.3 230.5 224.4 221 219.5 219.8 221.4 224.6

–411 –416.1 –421.9 –420.4 –406.7 –394.9 –384.4 –373.9 –364.1 –354.8 –345.8 –336.6 –327.8 –318.8 –309.6 –300.7 –291.9 –282.4 –273.3 –263.8 –253.9 –244.1 –233.7 –222.7 –211.7 –199.9 –187.6 –174.5 –160.7 –146 –130.2 –113.2 –94.7 –74.5 –51.2 –25 0

0 7.3 14.8 22.1 28.7 35 41.2 47.2 53 58.8 64.5 70.1 75.7 81.2 86.7 92.3 98 103.5 109.3 115.2 121 127.3 133.5 139.8 146.7 153.6 161 168.5 176.7 185.2 194.3 204.2 214.9 226.4 234.3 229 224.6

581 Table 10a Predicted first-ply failure envelopes for problem 6. Quadrant 1

Quadrant 2

Quadrant 3

Quadrant 4

yy (MPa)

xx (MPa)

yy (MPa)

xx (MPa)

yy (MPa)

xx (MPa)

yy (MPa)

xz (MPa)

265.2 263.7 262.6 261.3 260 258.5 256.9 255.2 253.3 251.3 249.1 246.9 244.6 241.5 238.9 235.8 232.3 228.6 224.3 209.4 194.9 180.9 167.3 153.9 141.2 128.6 116.1 104.1 92.2 80.5 68.8 57.3 45.8 34.4 23 11.5 0

0 11.5 23 34.4 45.8 57.3 68.8 80.5 92.2 104.1 116.1 128.6 141.2 153.9 167.3 180.9 194.9 209.4 224.3 228.6 232.3 235.8 238.9 241.5 244.6 246.9 249.1 251.3 253.3 255.2 256.9 258.5 260 261.3 262.6 263.7 265.2

–554.8 –543.1 –524.4 –517.1 –509.7 –501.8 –493.8 –486 –478.1 –470.1 –462.2 –454.5 –446.2 –434.5 –396.5 –362.4 –331.9 –304.1 –278.3 –255.1 –233.2 –212.8 –193.9 –176 –159 –143.1 –127.8 –113.3 –99.3 –85.8 –72.7 –59.9 –47.5 –35.3 –23.4 –11.6 0

0 23.7 45.9 68.1 89.9 111.2 132.3 153.2 174 194.7 215.5 236.6 257.6 276.8 277.6 278.1 278.5 278.6 278.3 278.4 277.9 277.4 276.9 276.3 275.4 275 274.1 273.5 272.9 272 271.2 270.2 269.4 268.5 267.4 266.1 265.2

–554.8 –565.9 576.6 –586.1 –594 –601.8 –610.1 –618.8 –627.7 –637.5 –647.8 –658.9 –671.7 –685.8 –701.5 –718.8 –738.2 –760.6 –785.2 –696.9 –619.4 –551.5 –491.2 –436.9 –387.8 –343 –302.1 –264.1 –228.5 –195.1 163.5 –133.4 –104.7 –77.2 –50.4 –24.7 0

0 –24.7 –50.4 –77.2 –104.7 –133.4 –163.5 –195.1 –228.5 –264.1 –302.1 –343 –387.8 –436.9 –491.2 –551.5 –619.4 –696.9 –785.2 –760.6 –738.2 –718.8 –701.5 –685.8 –671.7 –658.9 –647.8 –637.5 –627.7 –618.8 –610.1 –601.8 –594 –586.1 –576.6 –565.9 –554.8

265.2 266.1 267.4 268.5 269.4 270.2 271.2 272 272.9 273.5 274.1 275 275.4 276.3 276.9 277.4 277.9 278.4 278.3 278.6 278.5 278.1 277.6 276.8 257.6 236.6 215.5 194.7 174 153.2 132.3 111.2 89.9 68.1 45.9 23.7 0

0 –11.6 –23.4 –35.3 –47.5 –59.9 –72.7 –85.8 –99.3 –113.3 –127.8 –143.1 –159 –176 –193.9 –212.8 –233.2 –255.1 –278.3 –304.1 –331.9 –362.4 –396.5 –434.5 –446.2 –454.5 –462.2 –470.1 –478.1 –486 –493.8 –501.8 –509.7 –517.1 –524.4 –543.1 –554.8

582 Table 10b Predicted last-ply failure envelopes for problem 6. Quadrant 1

Quadrant 2

Quadrant 3

Quadrant 4

yy (MPa)

xx (MPa)

yy (MPa)

xx (MPa)

yy (MPa)

xx (MPa)

yy (MPa)

xz (MPa)

471.2 389.2 357.8 340.3 328.1 318.7 311 304 297.3 290.1 283.1 276 269.2 261.8 254.6 247.2 239.9 232.1 224.3 212.7 201.3 189.7 178.3 166.8 155.4 143.7 132 120.2 108.2 95.9 83.3 70.6 57.9 44.8 31.3 17 0

0 17 31.3 44.8 57.9 70.6 83.3 95.9 108.2 120.2 132 143.7 155.4 166.8 178.3 189.7 201.3 212.7 224.3 232.1 239.9 247.2 254.6 261.8 269.2 276 283.1 290.1 297.3 304 311 318.7 328.1 340.3 357.8 389.2 471.2

–854.8 –857.2 –859.1 –861 –863.5 –863.4 –842.7 –806.5 –772.8 –741 –709.8 –680.5 –651.6 –622.8 –582.3 –544.6 –509.9 –477.5 –446.9 –417.8 –390.3 –363.8 –339.6 –318.9 –280.6 –249 –222.6 –199.5 –179.2 –162.1 –145.9 –127.2 –102.2 –77.2 –51.8 –26.1 0

0 37.4 75.2 113.3 152.3 191.4 225.8 254.3 281.3 306.9 331 354.3 376.2 396.7 407.7 417.9 427.8 437.5 446.9 455.9 465.1 474.1 484.9 500.6 486 478.3 477.4 481.5 492.4 514.2 544.4 573.7 579.9 586.1 592.1 597 471.2

–854.8 –852.8 –850.8 –849.5 –847.3 –845.9 –843.8 –842.7 –841.2 –840.4 –839.2 –838 –836.9 –835 –832.9 –829.8 –825.5 –820.7 –785.2 –752.1 –692.7 –636.8 –583.2 –531.9 –483.2 –436.3 –391.3 –348.1 –306.2 –265.7 –226.1 –187.5 –149.4 –111.8 –74.4 –37.2 0

0 –37.2 –74.4 –111.8 –149.4 –187.5 –226.1 –265.7 –306.2 –348.1 –391.3 –436.3 –483.2 –531.9 –583.2 –636.8 –692.7 –752.1 –785.2 –820.7 –825.5 –829.8 –832.9 –835 –836.9 –838 –839.2 –840.4 –841.2 –842.7 –843.8 –845.9 –847.3 –849.5 –850.8 –852.8 –854.8

471.2 597 592.1 586.1 579.9 573.7 544.4 514.2 492.4 481.5 477.4 478.3 486 500.6 484.9 474.1 465.1 455.9 446.9 437.5 427.8 417.9 407.7 396.7 376.2 354.3 331 306.9 281.3 254.3 225.8 191.4 152.3 113.3 75.2 37.4 0

0 –26.1 –51.8 –77.2 –102.2 –127.2 –145.9 –162.1 –179.2 –199.5 –222.6 –249 –280.6 –318.9 –339.6 –363.8 –390.3 –417.8 –446.9 –477.5 –509.9 –544.6 –582.3 –622.8 –651.6 –680.5 –709.8 –741 –772.8 –806.5 –842.7 –863.4 –863.5 –861 –859.1 –857.2 –854.8

583 Table 11 Predicted first-ply and last-ply failure envelopes for problem 9. Quadrant 1

Quadrant 2

Quadrant 3

Quadrant 4

yy (MPa)

xx (MPa)

yy (MPa)

xx (MPa)

yy (MPa)

xx (MPa)

yy (MPa)

xz (MPa)

321.5 334.2 348.2 362.9 379.6 397.8 418.7 442 469.4 500.7 538.3 584.1 640.9 713.5 810.6 885 899.7 911.3 357.1 309.4 288 266.9 245.8 224.1 202.2 180.3 158.3 136.8 116.3 97.1 79.8 63.8 49.1 35.2 22.5 10.8 0

0 5.8 12.2 19.1 26.8 35.3 44.9 55.8 68.3 83 100.4 121.6 148 181.8 227 271.6 302 334 142.8 135.1 137.3 139.2 140.4 140.7 140.1 138.6 135.8 132.1 127.8 123.2 119.2 115.2 11.1 106.9 102.8 99.3 95.8

–224.5 –216.3 –208.2 –201.3 –194 –187.4 –181.6 –175.5 –169.6 –164 –159.1 –153.5 –148.5 –143.4 –138.4 –133.3 –128.7 –123.5 –118.8 –113.8 –108.6 –103.8 –98.7 –93.2 –886 –82.4 –76.7 –70.6 –64.5 –58 –51.1 –43.9 –36.3 –28.1 –19.4 –10.1 0

0 3.8 7.3 10.6 13.7 16.6 19.5 22.1 24.7 27.2 29.7 32 34.3 36.5 38.8 40.9 43.2 45.3 47.5 49.7 51.8 54.1 56.4 58.5 1 63.3 65.8 68.2 70.9 73.6 76.3 79.3 82.3 85.5 88.7 92.1 95.8

–224.5 –233.3 –243.1 –253.8 –265.4 –278.7 –293.2 –310 –329.4 –352 –378.8 –411.6 –452.5 –505.2 –575.9 –587.5 –597.5 –603.5 –607.1 –608.4 –606.5 –600.5 –589.6 –571.4 –542.7 –498.9 –384.6 –300.6 –236.7 –186.1 –145.3 –111.5 –82.7 –57.9 –36.3 –17.1 0

0 –4.1 –8.5 –13.4 –18.7 –24.7 –31.4 –39.1 –48 –58.3 –70.7 –85.7 104.5 –128.7 –161.3 –180.3 –200.5 –221.2 –242.8 –265.6 –289.1 –313.1 –336.8 –358.8 –376 –383.4 –329.9 –290.3 –260.1 –236.1 –216.9 –201.1 –187.5 –175.9 –165.8 –156.9 –149

321.5 310.2 298.9 289 279.2 269.9 261.3 252.7 244.8 237 229.3 221.8 214.3 207.5 200.3 193.2 186.5 179.5 172.5 165.5 158.4 151.3 144 136.5 128.7 120.7 112.6 104.1 95.3 86 76.1 65.6 54.4 42.5 29.5 15.5 0

0 –5.4 –10.5 –15.2 –19.7 –23.9 –28 –31.9 –35.6 –39.3 –42.8 –46.2 –49.5 –52.9 –56.1 –59.3 –62.6 –65.8 –69 –72.3 –75.5 –78.9 –82.2 –85.7 –89.2 –92.8 –96.6 –100.5 –104.7 –109.1 –113.6 –118.3 –123.5 –129.1 –135.1 –141.7 –149

584 Table 12 Predicted stress–strain curves for problems 7, 8, and 10. Problem 7

Problem 8

Problem 10

xx (%)

yy (%)

xx (MPa)

yy (%)

xx (%)

yy (MPa)

xx (%)

yy (%)

yy (MPa)

0 0.0006 0.0204 0.0403 0.0602 0.0802 0.1002 0.1203 0.1404 0.1606 0.1808 0.2011 0.2214 0.2417 0.262 0.2824 0.3028 0.3233 0.3437 0.3643 0.3848 0.4054 0.4258 0.4462 0.468 0.4931 0.5184 0.544 0.5697 0.5974 0.6295 0.6616 0.6937 0.7258 0.7577 0.7897 0.8217 0.8538 0.8859 0.9181 0.9503 0.9823 0.9942

0 –0.0002 –0.0059 –0.0116 –0.0173 –0.0231 –0.0288 –0.0345 –0.0403 –0.0461 –0.0518 –0.0576 –0.0634 –0.0691 –0.0748 –0.0806 –0.0864 –0.0923 –0.0981 –0.1039 –0.1097 –0.1156 –0.1214 –0.1273 –0.1369 –0.1551 –0.1734 –0.1922 –0.2113 –0.2267 –0.2326 –0.2386 –0.2446 –0.2503 –0.256 –0.2617 –0.2674 –0.2732 –0.279 –0.2848 –0.2907 –0.2964 –0.2985

0 0.3 12 23.7 35.3 47 58.7 70.3 82 93.7 105.3 117 128.7 140.3 152 163.7 175.3 187 198.7 210.3 222 233.7 245.3 257 268.7 280.3 292 303.7 315.3 327 338.7 350.3 362 373.7 385.3 397 408.7 420.3 432 443.7 455.3 467 471.3

0 0.0005 0.0102 0.0199 0.0296 0.0393 0.049 0.0587 0.0684 0.0781 0.0879 0.0976 0.1074 0.1171 0.1269 0.1366 0.1464 0.1561 0.1659 0.1757 0.1855 0.1952 0.2049 0.2147 0.2244 0.2339 0.2435 0.253 0.2626 0.2721 0.2816 0.2912 0.3008 0.3104 0.3199 0.3294 0.339 0.3485 0.358 0.3667 0.3753 0.3839 0.3923 0.4021

0 0.0001 0.0025 0.0049 0.0073 0.0096 0.012 0.0144 0.0167 0.0191 0.0214 0.0238 0.0261 0.0285 0.0308 0.0331 0.0354 0.0377 0.04 0.0423 0.0445 0.0467 0.0488 0.051 0.0531 0.0553 0.0575 0.0597 0.0618 0.064 0.0662 0.0683 0.0704 0.0725 0.0746 0.0766 0.0786 0.0806 0.0826 0.0881 0.0939 0.0998 0.1058 0.1848

0 0.3 7 13.7 20.3 27 33.7 40.3 47 53.7 60.3 67 73.7 80.3 87 93.7 100.3 107 113.7 120.3 127 133.7 140.3 147 153.7 160.3 167 173.7 180.3 187 193.7 200.3 207 213.7 220.3 227 233.7 240.3 247 253.7 260.3 267 273.7 279

0 –0.0009 –0.0327 –0.0646 –0.0964 –0.1282 –0.16 –0.1978 –0.2395 –0.2831 –0.3289 –0.3802 –0.4338 –0.4897 –0.5552 –0.628 –0.705 –0.7995 –0.9081 –1.0337 –1.1971 –1.396 –1.6481 –1.9936 –2.4346 –2.9757 –3.5965 –4.2635 –4.9515 –5.3289

0 0.0015 0.0538 0.1062 0.1585 0.2108 0.2632 0.3201 0.3796 0.44 0.5016 0.5662 0.632 0.6987 0.7708 0.8467 0.9246 1.0121 1.107 1.2109 1.3353 1.4784 1.6494 1.8693 2.1362 2.4497 2.7981 3.1647 3.538 3.7417

0 0.3 12 23.7 35.3 47 58.7 70.3 82 93.7 105.3 117 128.7 140.3 152 163.7 175.3 187 198.7 210.3 222 233.7 245.3 257 268.7 280.3 292 303.7 315.3 321.7

585 Table 13 Predicted stress–strain curves for problems 11, 12, 13, and 14. Problem 11

Problem 12

Problem 13

Problem 14

yy (%) xx (%) yy (MPa) xx (%) yy (%) xx (MPa) xx (%) xx (%) yy (MPa) xx (%) yy (MPa) 0 0.0021 0.0229 0.0437 0.0645 0.0853 0.1061 0.127 0.1478 0.1686 0.1894 0.2102 0.2306 0.2509 0.2711 0.2913 0.3116 0.3318 0.352 0.3722 0.3923 0.4125 0.4326 0.4525 0.4724 0.4923 0.5122 0.5321 0.5519 0.5718 0.5916 0.6114 0.6312 0.651 0.6708 0.6901 0.7093 0.7286 0.7479 0.7671 0.7863 0.8055 0.8247 0.8324

0 0.0006 0.007 0.0134 0.0198 0.0262 0.0327 0.0391 0.0455 0.0519 0.0583 0.0647 0.0714 0.0782 0.0849 0.0917 0.0984 0.1051 0.1118 0.1185 0.1252 0.1318 0.1385 0.1451 0.1517 0.1583 0.1649 0.1715 0.1781 0.1847 0.1913 0.1978 0.2044 0.2109 0.2175 0.2239 0.2304 0.2369 0.2433 0.2498 0.2562 0.2627 0.2691 0.2717

0 0.7 7.3 14 20.7 27.3 34 40.7 47.3 54 60.7 67.3 74 80.7 87.3 94 100.7 107.3 114 120.7 127.3 134 140.7 147.3 154 160.7 167.3 174 180.7 187.3 194 200.7 207.3 214 220.7 227.3 234 240.7 247.3 254 260.7 267.3 274 276.7

0 0.0011 0.0403 0.0795 0.1187 0.1579 0.1971 0.2379 0.2803 0.3228 0.3654 0.4076 0.4489 0.4903 0.5318 0.5733 0.6153 0.6584 0.7016 0.7449 0.7919 0.8399 0.8875 0.9351 0.9829 1.0308 1.0788 1.1269 1.1751 1.2234 1.2718 1.3203 1.3689 1.4173 1.4655 1.5137 1.562 1.6104 1.6589 1.7074 1.7559 1.8045 1.8532 1.8908

0 –0.0001 –0.0048 –0.0094 –0.014 –0.0186 –0.0233 –0.0271 –0.0304 –0.0336 –0.0368 –0.0398 –0.0419 –0.0441 –0.0462 –0.0484 –0.0505 –0.0524 –0.0542 –0.0561 –0.0637 –0.0725 –0.0806 –0.0888 –0.0971 –0.1054 –0.1139 –0.1224 –0.131 –0.1397 –0.1485 –0.1574 –0.1664 –0.175 –0.1831 –0.1913 –0.1995 –0.2078 –0.2162 –0.2246 –0.2331 –0.2416 –0.2502 –0.2569

0 0.3 12 23.7 35.3 47 58.7 70.3 82 93.7 105.3 117 128.7 140.3 152 163.7 175.3 187 198.7 210.3 222 233.7 245.3 257 268.7 280.3 292 303.7 315.3 327 338.7 350.3 362 373.7 385.3 397 408.7 420.3 432 443.7 455.3 467 478.7 487.7

0 0.0005 0.0138 0.0272 0.0405 0.0538 0.0671 0.0805 0.0938 0.1071 0.1205 0.1338 0.1472 0.1606 0.1739 0.1873 0.2007 0.2141 0.2274 0.2408 0.2541 0.2675 0.2808 0.2941 0.3074 0.3206 0.3339 0.3472 0.3604 0.3736 0.3869 0.4001 0.4133 0.4265 0.4398 0.4529 0.4659 0.4789 0.492 0.505 0.5165

0 0 0.2 0.0016 4.7 0.033 9.2 0.0644 13.7 0.0958 18.2 0.1272 22.7 0.1586 27.2 0.19 31.7 0.2214 36.2 0.2528 40.7 0.289 45.2 0.338 49.7 0.3886 54.2 0.4406 58.7 0.5003 63.2 0.5639 67.7 0.629 72.2 0.7134 76.7 0.8084 81.2 0.9298 85.7 1.0865 90.2 1.3347 94.7 1.6702 99.2 2.1367 103.7 2.6316 108.2 3.1371 112.7 3.647 117.2 4.1587 121.7 4.671 126.2 5.1834 130.7 5.6958 135.2 6.2081 139.7 6.7203 144.2 7.2324 148.7 7.7445 153.2 8.2565 157.7 8.7685 162.2 9.2804 166.7 9.7923 171.2 10.1762 175.2

0 –0.0016 –0.033 –0.0644 –0.0958 –0.1272 –0.1586 –0.19 –0.2214 –0.2528 –0.2874 –0.331 –0.3764 –0.4233 –0.477 –0.5343 –0.5935 –0.67 –0.7566 –0.8678 –1.0126 –1.244 –1.5612 –2.0086 –2.4892 –2.9846 –3.4873 –3.994 –4.5026 –5.0125 –5.523 –6.0339 –6.5451 –7.0565 –7.568 –8.0796 –8.5912 –9.1029 –9.6146 –9.9984

0 0.2 3.5 6.8 10.2 13.5 16.8 20.2 23.5 26.8 30.2 33.5 36.8 40.2 43.5 46.8 50.2 53.5 56.8 60.2 63.5 66.8 70.2 73.5 76.8 80.2 83.5 86.8 90.2 93.5 96.8 100.2 103.5 106.8 110.2 113.5 116.8 120.2 123.5 126

586 Appendix A Let V , denote the volume of the representative volume element (RVE) of a UD composite, as shown in Fig. A1. The volumes of the fiber and the resin in the RVE are V f and V m, respectively. Suppose that the i th point-wise stress in the RVE is ¯ i which may be different at a different point. The volume-averaged stress i of the composite is defined as i =

冕 冋冕 冕 册 冉 冊冉 冕 冊 冉 冊冉 冕 冊

1 V

+

¯ i dV =

V

V m V

1 V m

1 V

¯ idV +

V f

¯ i dV =

V m

¯ i dV = Vf  fi + Vmmi,

V f V

1 V f

¯ i dV

V f

i = 1, 2, . . . , 6,

V m

where Vf ( = V f /V  ) and Vm( = V m /V  ) are the volume fractions of the fiber and the resin respectively, and  fi and mi are the volume-averaged constituent stresses. Similarly, we can derive an identity for volume averaged strains among i ,  fi , and mi. Thus, Eqs (B1) and (B2) (see Appendix B) hold in any case. Further, the constitutive relationship between volume averaged stresses and strains has the same form as that between point-wise stresses and strains, because a compliance matrix is independent of volume averaging. This shows that Eqs. (B3), (B4), and (B5) are correct. It is noted that only two material phases, i.e., the fiber and the resin, have been assumed for the composites considered in this paper. This means that no interface region between the fiber and the resin is taken into account.

Fig. A1. A representative volume element (RVE) of a UD composite.

Appendix B

Let us consider a general three-dimensional stress state. The volume averaged stresses and strains in a representative volume element (RVE) of the lamina satisfy (see Appendix A) {d} = Vf {d f } + Vm{dm},

(B1)

587 and {di } = Vf {d f } + Vm{dm }.

(B2)

The constitutive equations correlating the averaged stresses and strains in different phases of the RVE are expressed as {d f } = [S f ]{d f },

(B3)

{dm} = [S m]{dm},

(B4)

{d} = [S ]{d}.

(B5)

and

Suppose that the averaged stresses in the fiber and the resin can be correlated by {dm} = [Aij ]{d f }.

(B6)

Substituting Eq. (B6) into Eq. (B1) and inverting the resulting equation yields {d f } = (Vf [I ] + Vm[Aij ])–1{d},

(B7.1)

where [I ] is a unit matrix, whereas substituting (B7.1) into (B6) gives {dm} = [Aij ](Vf [I ] + Vm[Aij ])–1{d}.

(B7.2)

Further, substituting (B3) and (B4) into (B2) and making use of (B6), one obtains (B5), or Eq. (6) (note that in the present case, all the matrices involved in (6) should be in 6
6). Let us consider an elastic deformation first. In such case, the overall compliance matrix of the lamina, Eq. (6), reads

[S ] =

冋

[Sij ] 0

册

0 , [Sij ]

(B8.1)

where 1 E11



v12 E11

1 E22

(Sij ) =

symmetry

v12 E11 v23 ,  E22 1 E22 

(B8.2)

and 1 G23

(Sij ) =

0

0

1 G12

0

symmetry

1 G12

,

(B8.3)

588 Note that the material parameters E22, G23, and v23 are not all independent but are related by G23 =

E22 . 2(1 + v23)

(B9)

Therefore, there are only five independent elements in the bridging matrix, [Aij ], in (B6). The other non-zero elements are determined by substituting (B6) into (6) and by making the resulting compliance matrix to be symmetric, i.e., Sji = Sij ,

(B10)

i, j = 1, 2, . . . , 6

Note that the constituent compliance matrices, [S f ] and [S m], have the same form as that of [S ], given in Eqs. (B8.1)–(B8.3). Hence, the most general form of the bridging matrix (in an elastic region) should read [note that some of the bridging elements, Aij’s, in the following should be identical with some of those in Eq. (5)]

[Aij ] =

A11 A12 A13 A21 A22 A23 A31 A32 A33

.

A44

(B11)

A55 remaining zero

A66

As G12 is an independent modulus, the bridging element A55 = A66 must be independent. In light of Eq. (B9), i.e., G23 is not independent, the other four independent bridging elements should be among Aijs where i 3 and j 3. We may thus set A21 = A31 = 0, and take A11, A22, A33, and A32 to be independent. Substituting the chosen [Aij ] into Eq. (6) and imposing that Sji = Sij for all i, j = 1, 2, 3, three algebraic equations are obtained as follows: 11 A12 + 12 A13 + 13 A23 = p1,

(B12.1)

21 A12 A23 + 22 A13 + 23 A23 = p2,

(B12.2)

31 A12 A23 + 32 A12 + 33 A13 + 34 A23 = p3

(B12.3)

In Eqs. (B12.1)–(B12.3), the parameters ijs and pis are given by: 11 = (Vf + Vm A33)(S f11  S m11), 12 =  Vm(S f11  S m11)A32, 13 =  Vm(S f12  S m12)A32, 21 = Vm(S f11  S m11), 22 =  (Vf + Vm A22)(S f11  S m11), 23 = [(Vf + Vm A11)(S f12  S m12)  Vm(S f13  S m13)A32], 31 = Vm(S f12  S m12), 32 = (Vf + Vm A33)(S f31  S m31)

589 33 =  [(Vf + Vm A22)(S f12  S m12) + Vm(S f31  S m31)A32], 34 =  (Vf + Vm A11)(S f22  S m22), p1 = (Vf + Vm A33)(S f12  S m12)(A11  A22)  (Vf + Vm A11)(S f13  S m13)A32, p2 = (Vf + Vm A22)(S f13  S m13)(A33  A11), p3 = (Vf + Vm A11)[(S f23  S m23)(A33  A22)  (S f33  S m33)A32] As the Eqs. (B12.1) (B12.3) are nonlinear, there might exist two sets of solutions to A12, A13, and A23. Choosing A12 as the primary variable, two solutions of it are given, respectively, by AI12 =

 b + 兹b2  4ac , 2a

(B13.1)

 b  兹b2  4ac AII12 = , 2a

where a = 21 2,

b = 21#2 + 22 1 + 23 2,

c = 22#1 + 23#2  p2,

#1 =

[13( p3/31  p2/21)  p1(34/31  23/21)] , 13(33/31  22/21)  12(34/31  23/21)

(B13.2)

#2 =

[ p1(33/31  22/21)  12( p3/31  p2/21)] , 13(33/31  22/21)  12(34/31  23/21)

(B13.3)

1 =

[  13(32/31) + 11(34/31  23/21)] , 13(33/31  22/21)  12(34/31  23/21)

(B13.4)

2 =

[12(32/31)  11(33/31  22/21)] . 13(33/31  22/21)  12(34/31  23/21)

(B13.5)

With the formulae (B13.1)–(B13.5), the other two dependent variables are obtained as A13 = #1 + 1 A12, A23 = #2 + 2 A12

(B13.6)

The independent elements, A11, A22, A33, and A32, are expected to depend on the elastic properties of the resin and the fibers, and on the fiber packing geometry (the relative position of the fibers embedded in the resin, the fiber volume fraction, the fiber cross-sectional shapes, etc.). When the properties of the two materials become the same, the bridging matrix, [Aij ], must be identical (of unit matrix). Hence, the general forms of the independent elements are always expressible as the power series of the material properties, i.e. A11 = 1 + #11(1  E m/E f11) + · · · ,

(B14.1)

A22 = 1 + #21(1  E m/E f22) + · · · ,

(B14.2)

A32 = #31(1  E /E ) + #32(1  v /v ) + · · · ,

(B14.3)

A33 = 1 + #41(1  E /E ) + · · · ,

(B14.4)

A55 = A66 = 1 + #51(1  G m/G f12 ) + · · ·

(B14.5)

m

f 11

m

m

f 12

f 22

590 where #ijs only depend on the fiber packing geometry but are independent of material properties. For clarity, we call #ijs as bridging parameters. The most rigorous method to determine the bridging parameters is through experiments. Supposing that the five elastic constants of the composite have been measured, the bridging parameters can be determined using some best approximation, such as the least square, techniques. However, explicit expressions for them are much more important in application. The most significant feature is that when the bridging parameters are determined using an elastic deformation condition, they remain unchanged during an inelastic deformation. This is because the fiber packing geometry does not change or only varies very little when the composite deforms from the elastic region to the inelastic one. As the composite elasticity theory has already been fairly well established, we may use it to explicitly determine a set of bridging parameters. Thus, let us imagine that a representative volume element is composed of a concentric cylinder (Fig. A1). Based on this, some rigorous analyses have been done and it was found that [28]: dm12 = A66 d f12,

(B15.1)

where

A66 =

冉 冊

1 Gm 1+ f 2 G 12

(B15.2)

Comparing (B15.2) with (B14.5), we see that #51 = –0.5

(B16)

Further, experiments have shown that the overall longitudinal stress of a unidirectional fiber reinforced composite is comparable with the stress of the fiber in the same direction, whereas the overall transverse stress of the composite is comparable with the resin stress in that direction. It is thus reasonable to assume that the averaged normal stresses between the fiber and the resin are correlated by dm11 = A11 d f11 + A12 d f22 + A13 d f33,

(B17.1)

dm22 = A22 d f22,

(B17.2)

dm33 = A33 d f33

(B17.3)

Because of the axi-symmetry of the concentric cylinder geometry, we may further assume that A33 = A22. Substituting so defined Aijs, i.e., A21 = A31 = A23 = A32 = 0

and

A33 = A22

(B18)

into Eqs. (B13), it is found that A13 = A12 = (S f12  S m12)(A11  A22)/(S f11  S m11)

(B19)

591 Hence, there are only two independent elements, A11 and A22, to be defined. Again, substituting (B18) into Eq. (6) and making some manipulation, the longitudinal Young’s modulus is obtained as E11 =

(Vf + Vm A11) (Vf S f11 + Vm A11S m11)

(B20)

It is well known that the rule of mixture approach gives quite accurate approximation to the composite longitudinal modulus. Supposing that the modulus defined by (B20) is equal to that given by the rule of mixture formula, we get A11 = E m/E f11,

(B21)

#11 = –1

(B22)

or

It is noted that with this set of Aijs, the resulting longitudinal Poisson’s ratio, v12, is exactly the same as that given by the rule of mixture formula, i.e., v12 = Vf v f12 + Vmv m. There remains A22, which defines the transverse modulus E22, to be determined. Many different micromechanical formulae have been proposed for the transverse modulus E22. In fact, one of the main motivations of different micromechanics approaches is to give a distinguished expression for the transverse modulus [41]. Tsai and Hahn [27] chose A22 = 0.5 in a modified rule-of-mixture formula and much better estimations were found for some composites. In light of the fact that A22 = 1 must be valid when the fiber and the resin become the same, a formula similar to (B15.2) is chosen for A22, i.e.,

A22 =

冉 冊

1 Em 1+ f 2 E 22

(B23)

which corresponds to #21 = –0.5 in (B14.2). Finally, using condition (B9), the bridging element A44 is found to have the same expression as A22, i.e., Eq. (B23). By means of the bridging matrix defined above, a set of new formulae for the five engineering moduli of the unidirectional fibrous composite are derived as

E22 =

E11 = Vf E f11 + VmE m,

(B24)

v12 = Vf v df12 + Vmvm,

(B25)

(Vf + Vm A11)(Vf + Vm A22) , {(Vf + Vm A11)(Vf S f22 + A22VmS m22) + Vf Vm(S m21  S f21)A12} (G f12 + G m) + Vf (G f12  G m) , (G f12 + G m)  Vf (G f12  G m)

(B27)

0.5(Vf + Vm A22) . Vf (S  S f23) + Vm A22(S m22  S m23)

(B28)

G12 = G m

G23 =

(B26)

f 22

592 Eqs. (B24) and (B25) are the rule of mixture formulae for composite longitudinal Young’s modulus and Poisson’s ratio, which are sufficiently accurate. Eq. (B27) is the result of an exact elastic solution for the overall in-plane shear modulus (G12) of the composite, see Ref. [28]. The accuracy of formula (B26), which is obtained based on Eq. (B23), can be seen from the comparison between predicted and experimental results for the transverse modulus of a glass/epoxy composite shown in Fig. B1 (with “beta = 0.5”). Some further remarks deserve mentioning. It has been recognized that the composite longitudinal property (E11 together with v12) is the least “case sensitive.” However, the composite transverse and in-plane shear properties are much more dependent on the in situ conditions involved, such as fabrication defects, processing condition, fiber-resin interface bonding, fiber arrangement, fiber cross-sectional shape, etc. In order to account for these variations, the corresponding independent bridging elements, i.e., A22 and A66, may be chosen as variable. This can be achieved by using, e.g., the following formulae, A22 = A33 = A44 = # + (1  #) A55 = A66 =  + (1  )

Em , E f11

Gm , G f12

0 ≤ # ≤ 1, 0 ≤  ≤ 1.

(B29) (B30)

The parameters # and  can be calibrated using measured transverse Young’s modulus and inplane (longitudinal) shear modulus. The transverse Young’s modulus and transverse shear modulus formulae, Eqs. (B26) and (B28), are valid for any A22 used. The longitudinal shear modulus, Eq. (B27), however, should be changed accordingly to G12 =

Vf + Vm A66 . Vf /G f12 + Vm A66/G m

(B31)

Fig. B1 shows the influence of different bridging parameters # (designated as “beta” in the figure) used on the predictions for the transverse Young’s modulus, E22, of a glass/epoxy UD composite. Except for experimental data which were taken from Ref. [27], predictions from three other micromechanics models are also shown in Fig. B1 for comparison. One is the rule of mixture model [27], another is the Chamis model [29], and the third is the Hill–Hashin– Christensen–Lo model as presented in Ref. [30]. It is seen that for the considered composite, the bridging model predictions based on 0.4 # 0.5 are in closest agreement with the experiments. The smaller the bridging parameter # used the stiffer the predicted transverse modulus will be. Fig. B2 plots the influence of different bridging parameters  (designated as “Alf ” in the figure) used on the predictions for the longitudinal shear modulus, G12, of another glass/epoxy UD composite. The experimental data from Ref. [27] as well as predictions from the three other micromechanics models, i.e., the rule of mixture, the Chamis, and the Hill–Hashin– Christensen–Lo models, are also indicated in Fig. B2. The prediction by the bridging model with  = 0.5 for the lamina longitudinal shear modulus is exactly the same as that by the Hill– Hashin–Christensen–Lo model. For the composite under consideration, all the bridging model predictions with 0.3 <  < 0.6 are acceptable. The same conclusion is that the smaller the bridging parameter  used the stiffer the predicted longitudinal shear modulus will be. Finally, let us investigate the influence of the bridging parameter # on the prediction for the transverse shear modulus, G23. The material used is a carbon/epoxy system. Kriz and

593

Fig. B1. Predicted and measured [27] transverse moduli of a glass/epoxy UD composite vs fiber volume fraction. The used material parameters are: E f = 73.1 GPa, v f = 0.22, E m = 3.45 GPa, and v m = 0.35.

Stinchcomb [42] measured various modulus data of the resulting UD composite with different fiber contents. The elastic properties of the carbon fibers and the epoxy matrix used in the present prediction were taken from Aboudi [43], and are listed in Table B1. The predicted

Fig. B2. Predicted and measured [27] in-plane shear modulus of a glass/epoxy UD composite vs fiber volume fraction. The material parameters used are: G f = 30.2 GPa and G m = 1.8 GPa.

594 Table B1 Constituent properties of a carbon/epoxy composite [43].

Fiber Resin

E11 (GPa)

E22 (GPa)

G12 (GPa)

v12

v13

232 5.35

15 5.35

24 1.8

0.279 0.49

0.49 0.49

results for the transverse shear moduli, G23, versus fiber volume fractions by using the rule of mixture and the Chamis model formulae and by using the bridging model formula with different bridging parameters rR (designated as “beta”) are graphed in Fig. B3. Kriz and Stinchcomb’s experimental data are also shown in the figure. It is noted that for the transversely isotropic carbon fiber reinforced composite, the Hill–Hashin–Christensen–Lo’s model is not applicable to the transverse shear modulus calculation [30]. The figure indicates that the Chamis model over- whereas the rule of mixture model underestimated the transverse shear modulus of this composite system. On the other hand, the bridging model with 0.3 # 0.6 showed good correlation. Having validated the correctness and accuracy of the independent bridging matrix elements, Eqs. (B 15.2), (B21), and (B23), in an elastic region, we can now easily extend them to a plastic region based on a logical consideration. As the bridging matrix correlates the stress states generated in the fiber and the resin materials, it can only depend on the physical and geometrical properties of the constituent materials in the composite. As long as the bridging matrix has been determined using an elastic deformation condition, only the physical properties involved need to be changed when any constituent material undergoes a plastic

Fig. B3. Predicted and measured [42] transverse shear modulus of a carbon/epoxy UD composite vs fiber volume fraction. The used material parameters are given in Table B1.

595 deformation. The geometrical properties, i.e., the fiber volume fraction, the fiber arrangement in the resin, the fiber cross-sectional shape, etc., do not change or only vary by negligibly small amount. Therefore, the independent elements of the bridging matrix should be given by A11 = Em /Ef 1, A22 = A33 = A44 = # + (1  #) A55 = A66 =  + (1  )

Em , Ef 2

Gm , Gf

(B32.1) 0 < # < 1, 0 <  < 1.

(B32.2)

(B32.3)

In the above, Em, Gm, Ef 1, Ef 2, and Gf are called effective moduli and are defined as

再

E m, when me ≤ mY , E mT , when me > mY

(B33.1)

0.5E m/(1 + v m), when me ≤ mY , E mT /3, when me >  mY

(B33.2)

Em =

Gm =

再

Ef 1 =

Ef 2 =

Gf =

E f11, when  fe ≤  fY , E fT , when  fe >  fY

再

(B33.3)

再

E f22, when  fe <  fY , E fT , when  fe >  fY

(B33.4)

G f12, when  fe ≤  fY E fT /3, when  fe >  fY

(B33.5)

再

where, for simplicity, an isotropic hardening has been assumed for the fiber material, and E fT represents the hardening modulus of the fiber (i.e., fibers can be either transversely isotropic with linearly elastic until rupture or isotropically elastic-plastic). e is the von Mises effective stress of the material defined using three principal stresses of the material, (1), (2), and (3) ((11) ≥ (2) ≥ (3)), via

e =

冑

1 [((1)  (2))2 + ((2)  (3))2 + ((3)  (1))2]. 2

(B34)

596 The dependent elements of the bridging matrix have to be determined by solving equations (B10). It must be pointed out that the bridging matrix has the following

[Aij ] =

A11 A12 A13 A14 A22 A23 A24 A33 A34 A44 remaining zero

A15 A25 A35 A45 A55

A16 A26 A36 A46 A56 A66

(B35)

rather than Eq. (B11). Only when both the constituents are in elastic deformation and the independent bridging elements are defined using (B32.1)–(B32.3), the Eqs. (B35) and (B11) are equivalent. If the composite deteriorates to a planar problem, the bridging matrix [A], as given in Eq. (5), together with the independent bridging elements, Eqs. (7.1)–(7.4) or (7.1), (10.1), (10.2), and (7.4), is apparently correct.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

597

CHAPTER 3.17

Expanding the capabilities of the Ten-Percent Rule for predicting the strength of fibre–polymer composites L. J. Hart-Smith* Phantom Works, The Boeing Company, Huntington Beach, CA 90808-4345, USA Abstract The Ten-Percent Rule, for predicting the strength and stiffness of fibre polymer composites on the basis of simple rule-of mixtures formulae, is here extended beyond its prior restriction to only fibre patterns in the 0°, ± 45°,90° family of balanced laminates. This has been accomplished by characterizing the implicit, but previously unused, lamina failure envelope associated with the model, and increasing the transverse strains-to-failure in the fibres to prevent prematurely predicted matrix failures undercutting fibre-dominated strengths. The use and validity of such an extension is demonstrated for the composite materials and fibre patterns involved in an international exercise (organized by Hinton, Soden and Kaddour). The theory is applied also to a ± 55° glass-epoxy laminate, and a 30°, ± 90° glassepoxy laminate. The simplifications remain valid only for highly orthotropic materials, like fibre polymer composites. Keywords: A. Glass fibres; A. Carbon fibres; C. Laminates; Ten-Percent Rule

1. Introduction This paper is the result of questions posed to the author about whether or not more could be done with his Ten-Percent Rule (see Ref. [1]) for predicting the strength of fibre-polymer composites than had originally been intended. This simple rule-of-mixtures approach had originated circa 1970, when there were neither pocket calculators nor desk-top computers, and when the computer codes of the contemporary interactive composite failure theories left much to be desired. (This was before the introduction of the now widely accepted stratagem of replacing all of the measured matrix-dominated properties by other strengths and stiffnesses

* Tel.: + 1-714-896-5852; fax: + 1-714-896-6505. E-mail address: [email protected] (L. J. Hart-Smith).

598 which cannot actually be measured but which produce more acceptable estimated strengths.) When first proposed, the Ten-Percent Rule was limited to fibre-dominated strengths of welldesigned laminates with fibre patterns confined to the widely used 0°/ ± 45°/90° family of balanced laminates. Within these two restrictions, the method was so simple that the factors to be applied to the measured reference strengths and stiffnesses of unidirectional laminae could be evaluated mentally. (A companion theory, with no simplifying assumptions to complicate its encoding for computers, was developed in parallel by the author who, therefore, had seen no reason1 to consider a computer-coded version of the Ten-Percent Rule.) Over the years, the Ten-Percent Rule has been refined, to reduce the number of measured properties needed to only the lamina modulus parallel to the fibres and the tensile and compressive strengths in the same direction. This paper represents an effort to extend the applicability of the theory to other fibre angles and to permit it to be encoded for use on computers. The lamina failure envelope implied by the Ten-Percent Rule was identified and, on the basis of hand solutions for the 0°/ ± 45°/90° quasi-isotropic laminate, a simple modification of the prescribed transverse strain-to-failure was deduced to eliminate compatibility-of-deformations problems. The solution of the remaining problems in the exercise conducted by Mike Hinton, Peter Soden, and Sam Kaddour [3], for the ± 55° and 90°/ ± 30° glass epoxy laminates, has confirmed that no further modifications were necessary.

1

At the end of the 1970s, the first Lear Fan (see Ref [2]) was sized by this method alone, by Brian Spencer (an experienced analyst of composite structures), because the computer program used to size the subsequent airframes was not yet operational. Years later, an exDouglas colleague at Boeing, Adrian Barraclough, was so impressed by the simplicity and reliability of the method that he suggested it should always be used as a sanity check on the output of the many computer-predicted laminate strengths. More recently, the capabilities of this model were illustrated for the one problem it should be capable of solving as part of a comparison between various composite failure theories [3]. This was for a (0°/ ± 45°/90°)s quasi-isotropic carbonepoxy laminate. The agreement with the author’s other analyses was excellent. Indeed, the agreement was so good as to cause one of the contest organizers, Sam Kaddour, to remark that it was a shame the model could not be extended to the other two sets of test data also. Surely there had to be an embedded failure criterion within the model that could be applied more widely? This was an intnguing suggestion; only two weeks earlier, at a lecture on this subject at the University of California at Santa Barbara, Prof. Keith Kedward had questioned why a different theory was needed merely to permit its encoding. Without this encouragement, particular by Sam, the author would still be convinced that a theory as simple and approximate as his Ten-Percent Rule could not be transformed into a quasi-scientific method for predicting the strength of fibre polymer composite laminates. Yet, if the Ten-Percent Rule could be expanded in the manner suggested, it would open up the possibility of reliable simple, one-shot analyses needing only half the experimental data called for by conventional composite failure theories. Better yet, since these predicted strengths did not involve any measured matrix-dependent properties, it might put an end, for once and for all, to one of the worst legacies of traditional interactive composite failure theories the myth that a change in resin, for common fibre reinforcement, created a new composite material requiring millions of dollars in qualification and characterization testing before it could be used in production or for repairs.

599 2. The lamina failure envelope implied by the Ten-Percent Rule The reference properties needed for the Ten-Percent Rule [1] in its original (limited) context are as follows. Only three measurements are needed. These are: the longitudinal modulus EL,

(1) t L

the longitudinal tensile strength F ,

(2)

and the longitudinal compression strength F cL,

(3)

all for a 0° lamina. The corresponding major Poisson’s ratio vLT may be measured. If no direct measurements are available, the typical value 0.3 is a suitable substitute. The predicted strengths are not sensitive to what are usually only small variations from this number. The following four properties are assumed, as the basis of the Ten-Percent Rule. Here, FL is the numerically greater of F tL and F cL: the transverse modulus ET( = EL/10),

(4)

the extensional modulus of an entirely ± 45° laminate, E ± 45 = ET ( = EL/10),

(5)

the transverse lamina strengths, F tT = F cT = F tL/10 or F cL/10, whichever is greater,

(6)

and the uniaxial strength of ± 45° piles F ± 45 = FL/10, for both tension and compression.

(7)

Two more properties are derived, for this further extension of the Ten-Percent Rule; the matrix-dominated in-plane shear modulus GLT = G0/90 =

E ± 45 EL ≈ = 0.0278 EL, 2(1 + v ± 45) 36

(8)

and the shear strength of a 0°, 90°, or 0°/90° laminate FLT = F s0/90° =

F t,c ± 45 = 0.05F tL or 0.05F cL, 2

(9)

whichever is greater.2 The Poisson’s ratio vLT for a ± 45° laminate can be deduced from a simple formula given in Eq. (12) of Ref. [4]. It is approximately 0.8, and would be 1.0 in the limiting case of a zerostiffness matrix. This characterization implies a specific relation between the strains-to-failure under longitudinal and in-plane-shear loads. Given that the lamina shear strength FLT is prescribed to be 0.05EL, and the shear modulus as EL/36, the corresponding peak shear strain is

LT = 1.8tL, or 1.8cL, whichever is greater.

2

(10)

The selection of the greater strength is based on the logic that, if the compressive strength of a ± 45° laminate were less than its tensile strength, the higher strength could not have been achieved if the lower value represented a shear failure. There is no significant longitudinal load in the fibres in either such test.

600

Fig. 1. Stress-based failure envelope for unidirectional lamina when F tL > F cLL, according to the original Ten-Percent Rule.

(Any difference between tL and cL must necessarily be associated with brittle fracture or compressive instability of the fibres, neither of which would affect the matrix. This is why the higher of these two strains is adopted rather than the lower one.) The corresponding lamina failure envelope, on the stress plane, is shown in Fig. 1. The failure envelope is presented with cut-offs in the 2nd and 4th quadrants for carbon-fibre reinforced laminates and without them for glass-fibre reinforcements. This relatively new distinction, since the preparation of Ref. [1], is explained in Ref. [5]. It is needed to account for differences between the transverse strains in the lamina and in the fibres. The slope of the shear cut-offs in the 2nd and 4th quadrants of the lamina-level strain plane equivalent to the stress plane in Fig. 1 is only about 30°, rather than the 45° in the generalized maximum-shearstress failure model. This slope becomes very much closer to the latter value after the theory is modified in the manner described later. The effect of the cut-offs is most pronounced for fibre-dominated in-plane-shear strengths. With the cut-off, per the original mental-arithmetic formulation of the Ten-Percent Rule, the fibre-dominated in-plane-shear strength of an all ± 45° laminate would be

F s± 45 =

冉 冊

1 1 + 0.1 FL = 0.275FL (with shear cut-offs) 2 2

(11)

while, without the cut-off, it would be twice as high, at F s± 45 = 0.55FL (without shear cut-offs).

(12)

These two shear strengths are derived from the longitudinal strengths this theory would predict for a 0°/90° laminate under equal-and-opposite tension and compression; they are not derived directly from Fig. 1. The preceding characterization is precisely how the Ten-Percent Rule should be applied in its original guise. However, as the worked examples show, minor but significant modifications are needed to the transverse strengths to avoid compatibility-of-deformations problems when the method is expanded in the manner developed here. Otherwise, predictions of transverse

601 failures in other plies would precede development of the strengths of the reference plies shown in Fig. 1. This is explained and justified later in the paper. Unlike the many abstract mathematical interactive failure theories for homogeneous anisotropic solids, which the author has condemned as not having the slightest relevance to heterogeneous fibre polymer composites, the Ten-Percent Rule is mechanistic and the modes of failure are easily identified. (By definition, the constituents of the composite CANNOT be identified within a truly homogeneous material.) The horizontal lines in Fig. 1 refer exclusively to failures in the resin matrix, while the vertical and sloping lines refer exclusively to fibre failures. It is even possible to distinguish between three common modes of fibre failures – brittle fracture under tension, from flaws; compressive instability under compression; and shear, for any combination of stresses not pre-empted by one of the other two failure modes. From this point on, for this paper at least, the approximate failure envelope characterized in Fig. 1 is treated mathematically just like any other empirical or mechanistically based envelope. While seeming to be very similar, it is not exactly the same as either the author’s generalization of the maximum-shear-stress criterion [6], or the truncated maximum-strain failure envelope [7], in which matrix-dominated properties can be input separately. In these other models, the matrix strengths and stiffnesses should be based on secant properties associated with the actual strains-to-failure whether they be established by matrix characteristics of the ply under consideration or by the strains-to-failure of fibres in adjacent plies. To actually apply this new failure envelope, it is first necessary3 to convert the stress-based failure envelope in Fig. 1 to one on the lamina strain plane, with every straight line defined by the co-ordinates of two points each. The results of such calculations are shown in Fig. 2, on the basis of standard inverse stress–strain relations, as given by Jones in Ref. [8]:

再冎

1 = 2

1 E1 v12  E1

v12 E1 10 E1



再冎

1 . 2

(13)

The co-ordinates of all of these points are as follows, in terms of the tensile reference strain tL = F tL/EL. Point A: 1 = tL,

2 = –vLTtL

冋 册

Point B: 1 = 1 

3

vLT t L, 10

(14)

2 = (1  vLT)tL

(15)

The need for this step derives from the two possible failure mechanisms permitted by the present model – the fibre dominated failure in Figs. 1 and 2, and the possible failures by in-plane shear of the matrix whenever there are fibres in too few directions to prevent it. Compatibility of deformations can be assured only in terms of strains whenever nonlinear behaviour of the matnx is included in the analysis. Only when a single failure mechanism governs for all states of biaxial stress and strain is it even possible to operate with only a stress-based failure criterion.

602

Fig. 2. Strain-based failure envelope for unidirectional lamina when F tL > F cL, according to the TenPercent Rule.

Point C: 1 = 

Point I: 1 = 

vLT t L, 10

Point L: 1 = 

Point F: 1 =

(16)

冉冏 冏 冊 冉 冏 冏冊 冋冏 冏 冉 冏 冏冊册 冋 冏 冏 冉 冏 冏冊册 冏冏 冏冏 冉冏 冏 冊 冉 冏 冏冊 cL vLT t + L, tL 10

2 = 1 + vLT

cL vLT cL + 1 t t L 10 L

Point K: 1 = 

Point J: 1 = 

2 = tL

cL t L, tL

cL vLT t  L, t L 10

vLT t L, 10

冋 册

Point G: 1 = 1 +

2 = vLT

tL,

tL

2 = vLT

cL cL + 1 t t L L

cL t L tL

2 =  1  vLT

2 =  tL

vLT t L, 10

cL tL

2 =  (1 + vLT)tL.

cL tL

tL

tL

(17)

(18)

(19)

(20)

(21)

(22)

603 While not actually corner points on the failure envelope, the equal-biaxial-strain points B and E for the full axial stresses tL and cL are useful for subsequent calculations. So are the equaland-opposite biaxial strain points G and H. The former points can be identified through use of Eq. (13) as occurring when

冉 冊

2 1  vLT = , 1 10 + vLT

(23)

whence for

冋 冉 冊册

Point B: 1 = 2 = 1  vLT

1 + vLT 10 + vLT

tL,

(24)

and for Point E: 1 = 2 = 

冏 冏 冋 冉 冊册 cL tL

1  vLT




1 + vLT 10 + vLT

tL.

(25)

Similarly, points G and H are located where

冉 冊

2 1  vLT = , 1 10  vLT

(26)

whence, for

冋 冉 冊册

Point G: 1 =  2 = 1 + vLT

1  vLT 10  vLT

tL,

(27)

and for Point H: 1 =  2 = 

冏 冏 冋 冉 冊册 cL tL




1 + vLT

1  vLT 10  vLT

tL.

(28)

With the further standardizing approximation that vLT = v12 is approximately 0.3,4 the expressions for the co-ordinates of points B can be simplified as 1 = 2 = 0.962136tL, almost directly above the uniaxial stress Point A. Point G is likewise located at 1 =  2 = 1.021649tL, almost directly below Point A. The lines GAB and JLH in Fig. 2 are clearly very close to vertical on the strain plane, according to the Ten-Percent Rule, even though these are actually constant-stress limits. The distinction between constant-stress and

4

The precise values of Poisson’s ratios are used here for the specific problems to facilitate comparisons between the various failure models, even though the discrepancy is very small. In the normal course of events, a standard value of 0.3 would be used, particularly when no test data were available.

604 constant-strain lines for longitudinal loads in fibre–polymer composites is quite insignifiicant. In contrast with the corresponding failure envelope for the truncated maximum-strain failure model [5], the various lines in Fig. 2 are not quite horizontal, not quite vertical, and not sloping at 45°. Nevertheless, strong similarities to both of the composite failure theories assessed in Refs. [5] and [9] are quite clear. It should be noted that the corner points B and J in the 1st and 3rd quadrants of Fig. 2 lie off the equal-biaxial-strain axis.

3. Test problem No. 9A: biaxial (␴x ⴚ ␴y) failure envelopes for (0°/ ± 45°/90°)s carbonepoxy laminate, according to the original formulation of the Ten-Percent Rule This problem is particularly simple to solve because the laminate strengths are entirely fibre dominated. This is precisely the kind of laminate for which the analysis method was always intended to be applied. The analysis is included here to enable an assessment to be made of how much the new approach changes the predicted answers. According to Eqs. (4)–(7), the uniaxial strengths of this laminate are (1
1.0 + 3
0.1)/4 = 0.325 times as strong as the unidirectional plies under uniaxial tension and compression stresses, and are (1
1.0 + 2
0.55 + 1
0.1)/4 = 0.55 as strong as the uniaxial plies under equal biaxial tension and compression stresses of the same sign. The factor 0.55 for the contribution of the ± 45° plies derives from the fact that it must be the same as for an equal mixture of 0° and 90° plies, i.e. (1
1.0 + 1
0.1)/2 = 0.55. Differences are permitted between the tensile and compressive reference properties of the unidirectional lamina which, of course, can be expected to vary with the operating environment as well. The Ten-Percent Rule merely establishes factors to be applied to these lamina reference strengths. Given the tensile and compressive unidirectional strengths of 1950 and 1480 MPa supplied in Ref. [3] for the reference lamina, the extremities of the failure envelope for a quasi-isotropic laminate made from this carbon-epoxy material are associated with the following stresses. Tensile uniaxial strength, 0.325
1950 = 633.8 MPa Compressive uniaxial strength, 0.325
1480 = 481.0 MPa Tensile biaxial strength, 0.55
1950 = 1072.5 MPa Compressive biaxial strength, 0.55
1480 = 814.0 MPa In-plane shear strength, 633.8/2 = 316.9 MPa Believe it or not, this is the complete set of calculations needed to construct the entire failure envelope when using the original formulation. (It is only slightly more complicated, as described in Ref. [1], when the numbers of 0° and 90° plies differ.) The method really is incredibly simple. The failure envelope is completely fibre dominated, being described in three-dimensional form in Fig. 3. The height of the shear-stress plateau is half of the uniaxial tension strength, or 0.1625 times as strong as a unidirectional lamina under a 0° tensile load. [This same factor could alternatively have been calculated as the average shear strength of ± 45° and 0°/90° laminates, i.e. (0.275 + 0.5)/2/2 = 0.1625. The original version of the Ten-

605

Fig. 3. Three-dimensional drawing of stress-based failure envelope for carbon-epoxy laminate, according to the original forrnulation of the Ten-Percent Rule.

Percent Rule is very consistent, even if it does not pay proper attention to compatibility of deformations.] The failure envelope is entirely flat faceted, rectangular in cross section, and pointed at its ends. The lines in the 2nd and 4th quadrants of the plan view (xy = 0) are shown as nonparallel joining the uniaxial tensile and compressive strengths together, as the author did in Ref. [10] before he learned how to formulate his generalization of the maximum-shear-stress failure criterion on the strain, rather than stress, plane. This is consistent with the original formulation of the model, but it is now known that these two lines should be drawn at 45°, parallel to each other, which would result in the kinks in the envelope lying off the stress axes for compression-dominated loads. This improvement is introduced later in this section.5 Had this been a failure envelope for a glass-fibre-reinforced laminate rather than one made from carbon fibres, the (roughly) 45° cut-offs for shear (tension/compression) loads would have been omitted and the failure envelope completed by projecting from the equal-biaxial strengths through the uniaxial strengths until the lines crossed. The shear-strength plateau would have been omitted for the same reason, being replaced by a ridge running orthogonal to the equal-biaxial-stress line. The ridge would be offset from the vertical axis through the origin whenever the tensile and compressive lamina (and laminate) strengths differed. An ancillary formula derived for use with the Ten-Percent Rule (to overcome a slight conservatism in regard to uniaxial strengths predicted by the rule-of-mixtures formula) states that, for doubly symmetric fibre patterns, the uniaxial strength is (1  v) times the biaxial strength, there being only one Poisson’s ratio for such laminates [1]. In this case, since v is 0.33 for quasi-isotropic laminates according to the Ten-Percent Rule (see Eq. (12) in Ref. [4]), the uniaxial strengths in tension and compression (based on the biaxial strengths above of 1072.5 MPa in tension and 814 MPa in compression) would be re-estimated as 715.0 and 542.7 MPa, respectively. The in-plane shear strength would be raised to 357.5 MPa, likewise.

5

A better failure envelope, in which the vertical sides in the 2nd and 4th quadrants are parallel and inclined at 45° to the reference axes, is shown here in the Appendix, based on an intermediate formulation of the Ten-Percent Rule.

606 These values, based on these calculated biaxial strengths, are even more accurate than those based on the laminate uniaxial strengths. It is noteworthy that application of the Ten-Percent Rule, in its original form, requires knowledge of only the unidirectional lamina strengths in tension and compression. All other strengths are deduced from these two measurements. Not even the elastic constants need be known.

4. Additional equations needed for formal mathematical analyses The first relations needed are those transforming the strains between the lamina (fibre) (1–2) and laminate (x–y) co-ordinate systems (see Fig. 4). Quoting from Jones’s book [8], x cos2  y = sin2 

xy 2 sin2  cos 

sin2  cos2  –2 sin  cos 

–sin  cos  sin  cos  cos2   sin2 

1 2

12

(29)

1 cos2  2 = sin2 

12 –2 sin  cos 

sin2  cos2  2 sin  cos 

sin  cos  –sin  cos  cos2   sin2 

x y .

xy

(30)

and

On the base plane of the laminate failure envelopes, there is zero shear strain xy and, on parallel planes, xy = constant = . ¯ It follows from the last of these equations, then, that the inplane shear strain acting on the lamina varies according to the relation

12 =

1 [ ¯  2 sin  cos  (1  2)]. (cos   sin2 ) 2

Fig. 4. Sign convention and identification of lamina and laminate coordinate systems.

(31)

607 Substitution of Eq. (31) into Eq. (30) yields the following solutions.

再冎

x 1 = 2 y (cos   sin2 )

冋

cos2  –sin2 

–sin2  cos2 

–sin  cos  sin  cos 

册

1 2

¯

(32)

for all angles except  = ± 45°, for which the denominator would be zero in both equations. Additional formulae are needed to locate the intercepts on the xy = 0 laminate plane equivalent to planes for which 12 = constant in the lamina reference system. According to the last of Eq. (13), then,

12 = LT = 2 sin  cos  (y  x),

(33)

whence

y = x +

LT . 2 sin  cos 

(34)

Computing the overall laminate stresses associated with these strains requires further equations, which are also available from Jones’s book. For balanced symmetric laminates, the lamina stresses are related to the lamina strains by 1 Q11 2 = Q12 12 0

Q12 Q22 0

1 2 ,

12

0 0 Q66

(35)

where

Q11 =

E1 , 

Q12 =

v12E2 v21E1 = ,  

Q22 =

E2 , 

Q66 = G12,

 = 1  v12v21.

(36)

These equations need to be re-expressed with respect to axes inclined by an angle  from the fibre direction, with the sign convention adopted in Fig. 4, so that the properties of laminates containing fibres in more than one direction can be established. Jones gives the following results, simplified here by the introduction of the notation s = sin , c = cos . ¯ 11 x Q ¯ 12 y = Q ¯ 16 xy Q

¯ 12 Q ¯ 22 Q ¯ 26 Q

¯ 16 Q ¯ 26 Q ¯ 66 Q

(37) x y ,

xy

(38)

608 where ¯ 11 = Q11c4 + 2(S12 + 2Q66)s2c2 + Q22s4 Q ¯ 12 = (Q11 + Q22  4Q66)s2c2 + Q12(s4 + c4) Q ¯ 22 = Q11s4 + 2(Q12 + 2Q66)s2c2 + Q22c4 Q ¯ 16 = (Q11  Q12  2Q66)sc3 + (Q12  Q22 + 2Q66)s3c Q

.

(39)

¯ 26 = (Q11  Q12  2Q66)s3c + (Q12  Q22 + 2Q66)sc3 Q ¯ 66 = (Q11 + Q22  2Q12  2Q66)s2c2 + Q66(s4 + c4) Q For balanced symmetric laminates, containing an equal number of plies in each of the ±  directions, certain simple relationships follow from these and the inverse relations given by Jones. The key to the derivation of these expressions is that in-plane-shear stresses are developed within each lamina in order that there be no in-plane shear strain when stresses are applied only in the x and y directions. (In contrast with this, the corresponding formulae for off-axis laminates with fibres in only one of the two directions is associated with the absence of shear stress and the presence of shear strain whenever the lamina is loaded in only the x or y direction.) When only a x stress is acting, and there is no shear strain xy, Eq. (38) yields that ¯ 12 /Q ˜ 22. vxy = Q

(40)

Similarly, when only a y stress is applied, it follows that ¯ 12 /Q ¯ 11. vyx = Q

(41)

Two sets of equations can be derived for the Young’s moduli; one from Eq. (39) and the other from the inverse relations cited in [8]. The set to be used here is the former, with ¯ 11  Ex = Q

¯ 212 Q , ¯ 22 Q

and

¯ 22  Ey = Q

¯ 212 Q . ¯ 11 Q

(42)

The in-plane shear stiffness is unchanged by the combination of the two sets of fibres, because there are no odd  terms in the expression relating the in-plane shear stress and strain. ¯ 66. Gxy = xy / xy = Q

(43)

The remaining coefficients A¯ 16 and A¯ 26 are zero for balanced laminates, into which category all of the fibre patterns considered here fall. It now remains only to integrate the individual lamina stiffnesses through the thickness of the laminates, using the well-known formula for the coefficients Aij of the membrane-stress terms in the stiffness matrix.

冘 N

Aij =

1

¯ ij )k tply, (Q

(44)

609 The traditional engineering constants then follow as vxy =

A12 , A22

vyx =

A12 , A11

Ex =

A11 A212  , t A22t

Ey =

A22 A212  , t A11t

and

Gxy =

A66 1

(45)

in which t is the total laminate thickness. These equations permit a simple but powerful check on the consistency of the simplifying assumptions throughout these analyses, by comparing the ratio E ± 45 /E0 with the value of 0.1 assumed for the rule-of-mixtures procedure. From Eqs. (45), (39), (4), and (8)

冉 冊 冋 冉 冊 册 冋 冉 冊 册

E ± 45  EL vLT 2 1 EL =
1+2 + +  E0 EL 4 10 36 10 4

1+

1 4 2vLT  + 10 36 10

2

.

(46)

vLT 2 1 + 1+2 + 10 36 10

With the further standard assumption that vLT 0.3, so that  = 0.991, E ± 45 1 = E0 4

冋

1.270111 

1.0498892  1.270111

册

(47)

= 0.100565.

A similar calculation verifying that E90 /E0 equalled 0.1 would confirm only the absence of errors in the algebraic manipulations, but the ratio E ± 45 /E0 is not predetermined. Consequently, agreement almost within half a percent provides very real validation for the initial simplifying assumptions – and assurance that the new formulation of the Ten-Percent Rule analysis will give the same answers as the original rule-of-mixtures version. A further important sanity check concerns the 0.55 factor for equal biaxial loads on ± 45° laminates. This can be assessed by consideration of the 0°/90° laminate under the same equal biaxial loads. According to Eqs. (38) and (14), the fibres in an all-0° lamina under a purely 0° load would fail at a stress given by ¯ 11L + Q ¯ 12T = (Q ¯ 11  vLTQ ¯ 12)0. uniaxial = Q

(48)

Now, if the fibre is to fail at the same axial stress under equal biaxial strain, the associated biaxial strain  must be given by the similar relation ¯ 11 + Q ¯ 12) = uniaxial, biaxial = (Q

with

˜ 12 + Q ¯ 22). 90,biaxial = (Q

(49)

It then follows from these two equations that, for a 0°/90° laminate, 0/90,biaxial =

冉 冊冉

冊

¯ 11 + 2Q ¯ 12 + Q ¯ 22 ¯ + 2Q ¯ 12 + Q ¯ 22) 0 + 90 Q (Q =  = 11 2 2 2

冉

冊

˜ 11  vLTQ ¯ 12 Q 0,uniaxial (50) ¯ 11 + Q ¯ 12 Q

whence, from Eqs. (37) and (4),  ± 45,biaxial  ,biaxial (1 + 2v21 + 0.1)(1  v12v21) 1.16
0.991 = = = = 0.5580, 0,uniaxial 0,uniaxial 2(1 + v21) 2
1.03 0 90

(51)

for vLT = 0.3, which is quite consistent with the approximate value of 0.55 used with the original (rule-of-mixtures) formulation of the theory.

610 Eq. (45) also permit the derivation of a useful approximation for the Poisson’s ratio of laminates in the 0°/ ± 45°/90° family.

再 冋 册 冋冉 冊 册 冋 册冎 再 冋 册 冋 冉 冊 册冎 冉 冊 冉 冊 冉 冊 %0°

vxy = v0,90 =

vLT % ± 45 + 10 4

1+

1 % ± 45° %0° + 10 4

vLT 1 4 vLT  +2 + %90° 10 36 10 10

(52)

vLT 2 + 1+2 + %90°[1] 10 36

and, since %0° = 100  %90°  ± 45°, vLT +

v0,90 =

% ± 45° 40

1+

1 2vLT 4   10 10 36

.

% ± 45° 1+ 40

(53)

9 1 2vLT 4 + 1+  + %90° 10 10 36 100

With the same further assumptions as above, v0,90 =

0.3 + 0.023247(% ± 45°) , 1 + 0.028753(% ± 45°) + 0.09(%90°)

(54)

which satisfies the necessary sanity checks for all-0° and all-90° fibre patterns, predicting that v0,90 = 0.3120 for a quasi-isotropic laminate, and v0,90 = 0.6773 for a completely % ± 45° laminate (increasing towards a maximum possible value of 0.8056 as the in-plane shear stiffness GLT decreases steadily with progressively higher strains). The prediction for a 0°/90° laminate is 0.033 if vLT is 0.3. This formula does not reduce to the earlier simple formula derived by the author in Ref. [4], for nearly quasi-isotropic laminates made from strong, stiff fibres in a soft resin matrix, i.e., v0,90 ≈

1

冉 冊

.

(55)

%90° 1+4 % ± 45°

However, the value predicted, that v0,90 = 0.3333 for the quasi-isotropic fibre pattern, is quite similar, and Eq. (55) can be used with confidence for laminates with a reasonable minimum fibre content in all four directions. Eq. (54) provides more accurate predictions for fibre patterns that are far from quasi-isotropic than can be obtained with Eq. (55).

5. Test problem No. 9B: biaxial (␴x ⴚ ␴y) failure envelopes for (0°/ ± 45°/90°)s carbonepoxy laminate, according to new formulation of the Ten-Percent Rule This problem solved earlier using the Ten-Percent Rule in its original guise is solved again, below, using the implied lamina failure envelope in exactly the same manner as for any other formal lamina failure envelope. The shear cut-offs in the 2nd and 4th quadrants are included in the solution, because this is a carbon-epoxy laminate. (They are omitted for the two remaining fibreglass-epoxy laminates, in accordance with the assessment of this issue provided in Ref. [5].)

611 Construction of the failure envelope using the new process follows standard practices, except for complications due to the singularity in Eqs. (19) and (20) for  = ± 45°. The following relations are used in their place. For  = ± 45°,

12 = y  x regardless of the value of xy,

xy = 1  2 regardless of the value of 12, and x + y = 1 + 2.

(56)

Prior solutions of this problem have revealed that the 0° and 90° plies define most of the failure envelope, with the only likely involvement of the ± 45° plies being in the form of a matrix-shear cut-off. (Any matrix shear cut-offs for the 0° and 90° plies are perpendicular to the xy axis, off the base plane of the failure envelope.) Symmetry of the failure envelope with respect to the equal-biaxial-strain diagonal reduces the number of points for which actual calculations are needed to only A, B, C, K, J and F in Fig. 1, which are the same as A, B, C, K, J and F in Fig. 2. The numerical values of the co-ordinates of these points are unchanged between lamina and laminate strain planes for 0° plies, so the process is particularly simple. Thus, since tL = 0.0138 and cL = –0.01175, and vLT = 0.28 per the data supplied in Ref. [3],  = 0.992 so that, for the 0° plies, Point A: Point B: Point C: Point K: Point J: Point F:

x = + 0.01380, x = + 0.01341, x = –0.0003964, x = –0.01181, x = –0.01136, x = + 0.0003864,

y = –0.003864 y = + 0.009936 y = + 0.01380 y = + 0.005340 y = –0.01051 y = –0.01380.

The corresponding values for the 90° plies are established by interchanging the x and y values. The ± 45° plies are critical only for equal-biaxial strains, unless in-plane shear loads are applied. In addition, there are two 45°-sloping lines associated with possible matrix shear failures in the ± 45° plies, intercepting the axes in accordance with the first of Eq. (43) when

12 = 1.8tL = ± 0.02484. The corresponding strain-based laminate failure envelope is shown in Fig. 5. The matrixshear limits clearly lie well outside those set by the fibres, and can henceforth be neglected. However, it is also apparent that this failure envelope predicts that absolutely no fibres will fail under tensile lamina loads aligned with the fibres except for minute areas near the positive strain axes. In the jargon of the interactive failure theories of which the author disapproves, the analysis would apparently predict almost universal “first-ply” failures in the matrix under transverse loads throughout the tension–tension (lst) quadrant. This is quite at odds with the solution shown in Fig. 3. One is forced to conclude that the failure model in Figs. 1 and 2 does NOT represent the Ten-Percent Rule, after all. (This finally explains the need for the empirically increased transverse strengths in the author’s stress-based failure model in Ref. [11]. Without them, the BLACKART computer code would have been just as unreliable as the theories it was intended to replace. However, the current work has revealed that the entire failure envelope should have been expanded – and by a precise amount – not merely by any minimum amount to render fibre and matrix failures non-interactive in the 1st and 3rd quadrants.)

612

Fig. 5. Failure envelope for quasi-isotropic (0°/ ± 45°/90°)s carbon-epoxy laminate, on the laminate strain plane.

The kind of abnormality evident in Fig. 5 is one reason why the author had not previously tried to develop a computer-code version of the Ten-Percent Rule and had, instead, tried to develop a scientifically more precise model for that purpose, as in Refs. [6] and [7]. As originally envisaged, the Ten-Percent Rule derived its simplicity without unnecessary loss of accuracy by avoiding any equations requiring the satisfaction of compatibility of deformations. Fortunately, a very simple modification of the present graphical model can make it consistent with the original rule-of-mixtures formulation. If the strains due to transverse loads are all increased by the factor (1 + vLT), leaving the transverse modulus unchanged, Fig. 5 would become totally fibre dominated. Figs. 1 and 2 are therefore replaced by Figs. 6 and 7. Because none of the stiffnesses are being changed, this modification does not constitute a replacement of the original formulation. It may be looked upon as addressing the issue of compatibility of deformations, which could not be considered in the original formulation. Loads in transverse plies are still assigned to be ten percent of those in the longitudinal plies, for common longitudinal and transverse strains, even though the transverse fibre strengths (at a higher transverse strain beyond the capacity of transverse fibres) are increased in the ratio (1 + vLT). (The transverse and in-plane matrix strengths are unchanged.) The transverse-failure points on the fibre-failure envelope are identified by primes. The corresponding matrix-failure points in Figs. 6 and 7 are identified by the same letters without primes, as in Figs. 1 and 2. All that this modification ensures is that failure can continue to be predicted by the strain in the longitudinal plies, when the fibres fail, and need not be undercut by predictions of earlier matrix failures in transverse plies, because it is now possible to distinguish between the two.

613

Fig. 6. Modified stress-based failure envelope for unidirectional lamina when F tL > F cL, corrected to override premature predictions of transverse failures.

Admittedly, it is still necessary to differentiate between possibly real matrix failures, along the line BCH, and real fiber failures along the lines BCH or BCD, for example.6 Subject to

Fig. 7. Modified strain-based failure envelope for unidirectional lamina when F tL > F cL, corrected to override premature predictions of transverse. 6

An attempt was made to remove any ambiguity by consistently defining the primed failure points in Figs. 6 and 7 as being applicable to both fibre and matrix failures. However, a comparison with the solutions made using the other models (Refs. [5] and [9]) indicated that the matrix failures so predicted would then be excessive. In addition, such a change could be perceived as a fundamental change in the TenPercent Rule, rather than as merely a minor modification to overcome some numerical problems in applying it.

614 the obvious limitations of the Ten-Percent Rule in regard to predicting all real matrix failures, the choice between one or other failure mechanism is normally quite clear in the laminatelevel failure envelope. Whenever there is a fibre-failure segment inside or barely outside the corresponding matrix-failure prediction, one should assume that the fibre-failure prediction governs and ignore the predicted matrix failures. Conversely, when there are fibres in the laminate in so few directions that predictions of matrix failures by this method lie well inside the predictions of fiber failures, for at least some portion(s) of the failure envelope, one should accept these predicted matrix failures as being more reliable than simply ignoring them completely. Point B in Fig. 7 now lies beyond Point B, which governs for equal biaxial strains, solving the problem of ensuring the prediction of fibre failures for biaxial loads. The line CD, when transposed for a 90° ply, now passes imperceptibly outside Point A for a 0° ply, ensuring that the estimated transverse ply strengths for this model do not undercut the fibre-dominated uniaxial strength predictions, either. This is the reason for selecting this particular amplification factor (1 + vLT), to permit equal longitudinal and transverse strains, with failure in the fibres, at the equal-biaxial-strain points B and E in Fig. 2, instead of predicting that only matrix failures were possible there, as Fig. 5 would suggest. Any lesser amplification factor would restrict the transverse strain at those points below the longitudinal strain. This, in turn, would impose a limit on the longitudinal strain in transverse fibres, for those stress states, which was contrary to the basis of the original Ten-Percent Rule. The failure envelopes in Figs. 1 and 2 must be discarded, despite their apparent plausibility. All of the remaining analyses will be based on the failure model depicted in Figs. 6 and 7 that permits two distinct possible failure mechanisms under transverse loads. This arbitrary increase in transverse strains-to-failure may seem to some as justifying the corresponding techniques used to enhance strength predictions with existing composite failure models by use of progressive-failure or ply-discounting techniques. On the contrary, although the effects are similar, the contexts are very different. The advocates of progressive-failure analyses with interactive failure models have justified their approach by maintaining that such matrix failures actually do occur – at the stress levels they predict. (Otherwise their theories would be inevitably invalidated. It is the author’s view that none of those predicted first-ply failures, as they are customarily called, has ever been validated experimentally. The fact that there can be subsequent real matrix failures in no way validates these premature predictions.) Here, the reason for this modification of the assumed transverse strain at failure is to create a failure model making predictions as close as possible to those predicted on the basis of Eqs. (4)–(7), which do not actually imply that the matrix must fail immediately after the fibres fail. All they stipulate is that the amount of transverse load which is carried without matrix failure is one tenth of the longitudinal load in each ply – all the way to failure of the fibres. (This is why only the strains-to-failure, and the associated strengths, were increased here, leaving the transverse moduli unchanged.) The modification of Figs. 1 and 2 into Figs. 6 and 7 is no more or less scientific than the original Ten-Percent Rule, which has never been portrayed as anything but a valuable approximate analysis method. Whether or not these changes to the transverse, and in-plane-shear, fibre strengths are scientifically valid is immaterial. They represent the minimum changes needed to ensure that this new representation of the TenPercent Rule predicts essentially the same strengths as the original formulation for fibre patterns which are known (or believed to be) totally fibre dominated.

615 Eqs. (14)–(22) are therefore complemented by the following set, for fibre-dominated failures. The earlier equations continue to be applied for predicting matrix failures. Point A: 1 = tL,

2 = –vLTtL

(57)

冋 冉 冊册 冋 册

Point B: 1 = 2 = 1  vLT

1 + vLT 10 + vLT

Point B: 1 = 1  (1 + vLT)

vLT t L, 10

Point C: 1 =  (1 + vLT)

冉冏 冏 冋冏 冏 冏冏 冏冏冋 冉冏 冏

Point I: 1 = 

Point K: 1 = 

Point L: 1 = 

(58)

2 = tL

cL tL

冊 冉 冉 冏 冏冊册 冏冏 冉 冊册 冊 冉

1 = vLT




1  vLT

冉

vLT t  L, 10

Point G: 1 = 1 + (1 + vLT)

Point H: 1 =  2 =

冋

冏 冏冊 冉 冏 冏冊

tL,

cL tL

tL

2 = (1 + vLT) 

cL t L tL

1 + vLT 10 + vLT

cL vLT t  (1 + vLT) L, t L 10

Point F: 1 = (1 + vLT)

(60)

2 = (1 + vLT) + vLT

cL vLT cL + (1 + v ) 1  LT tL 10 tL

cL t  L, tL

(59)

2 = (1 + vLT)tL

cL vLT t + (1 + vLT) L, t L 10

Point E: 1 = 2 = 

Point J: 1 = 

vLT t L, 10

tL

冊

vLT t L, 10

1 + vLT  (1 + vLT)

vLT 2  (1 + vLT) 10

册

tL 

(62)

(64)

冏 冏冊

2 =  (1 + vLT)  vLT

cL tL

tL

(65)

(66)

2 =  (1 + 2vLT)tL v2LT 10

tL

(63)

tL

2 =  (1 + vLT)tL

cL tL

(61)

(1 + vLT) t L. 2

(67)

(68)

616 The slope of the cut-offs in the 2nd and 4th quadrants is not exactly 45°, as it is in the author’s generalization of the maximum-shear-stress failure criterion in Ref. [5]. It would be, if the transverse Poisson’s ratio vTL were absolutely zero. However, the slope is very much closer to 45° on the lamina strain plane than for the model shown in Fig. 2. For conventional carbonfibre-polymer composites, therefore, the slope is now only minutely steeper than 45°. It would be almost precisely 90° for glass-fibre laminates, for which there should be no cut-off. It is also necessary to confirm that there are no implied changes to the matrix-dominated in-plane-shear strength that might result from the increase in transverse fibre strength introduced above. It should be noted that none of the lamina or laminate stiffnesses, whether fibre- or matrix-dominated, has been changed by this modification. Not even the fibredominated in-plane-shear strength of a ± 45° laminate is affected, because the strain at failure is still restricted by the longitudinal strain in the fibres. The greater transverse strain capability simply cannot be exercised in a fibre-dominated laminate. On the other hand, if the matrixdominated strength of 90° and ± 45° laminates were increased by the factor (1 + vLT), one would need to implement the same increase in the in-plane shear strength of 0° or 90° laminae with respect to Eq. (9). The selectively modified corner-point strains for the 0° ply in the 0°, ± 45°, 90° quasiisotropic laminate are then as follows. Point A: Point B: Point B: Point C: Point K: Point E: Point J: Point F: Point G: Point I:

x = + 0.01380, x = + 0.01332, x = + 0.01331, x = –0.0004903, x = –0.01182, x = –0.01134, x = –0.01126, x = + 0.0004946, x = + 0.01429, x = –0.01224,

y = –0.003836 y = + 0.01332 y = + 0.01380 y = + 0.01764 y = –0.005886 y = –0.01134 y = –0.01437 y = –0.01766 y = –0.02147 y = + 0.02090.

Henceforth, the specific values of the Poisson’s ratios given by the organizers for unidirectional laminae have been used (in this case 0.28), to facilitate comparisons between the theories, even though the author normally uses the value 0.3 as a standard. The strainbased laminate failure envelope for these 0° plies, and those in the 90° direction, is plotted in Fig. 8. It is clear that, unlike Fig. 5, all predictions for these two fibre directions are for fibre failures. Figure 8 also includes the predictions for failure of the ± 45° plies, a rectangular box defined by fibre-failure lines through the same Points B and E as for the 0° and 90° plies in conjunction with sides defined by the same matrix shear failures as calculated earlier, crossing the axes at strains of + 0.02484. The ± 45° ply failure envelope clearly plays no part in this laminate failure envelope, at least not on the base plane. Naturally, these fibres would eventually become critical under in-plane-shear loads. The fiber-dominated in-plane-shear strength of the laminate is most easily assessed in terms of the –45° sloping line, for equal and opposite strains, through the origin in Fig. 8, at the point where it crosses the 0°/90° failure envelope at ± Point H in Fig. 7.

617 The strain-based failure envelope in Fig. 8, incorporating the modifications indicated above, can now be converted to the laminate stress plane instead by use of the following standard equations. x A 1 11 y = A12 t xy 0

A12 A22 0

0 0 A66

x y ,

xy

(69)

where the Aij coefficients can be evaluated by using Eqs. (44) and (39). In this case, for a quasi-isotropic 0°, ± 45°,90° laminate,

冘

¯ 11 = Q

¯ 12 = Q

1 [(EL + ET  4GLT) + 6vLTET] 8

A11 A22 = = t t

A12 = t

A66 = t

冘 冘

1 [3EL + 2(vLTE2 + 2GLT) + 3ET] 8

.

(70)

¯ 66 = 1 [(1  vTL)EL + (1  vLT)ET + 4GLT]. Q 8

Fig. 8. Improved failure envelope for quasi-isotropic (0°/ ± 45°/90°)s carbon-epoxy laminate, on the laminate strain plane.

618 Incorporating the simplifications and assumptions of the Ten-Percent Rule, this set of equations becomes

冋 冉 冊 册 冋冉 冊 冉 冊册 冋冉 冊 冉 冊 册

A11 A22 EL = = t t 8

A12 EL = t 8

A66 EL = t 8

3.0 + 2

1. + 0.1 

1

vLT 2 + + 0.3 = 0.436704EL 10 36

4 vLT +6 36 10

= 0.145864EL

.

(71)

vLT 1  vLT 4 + + = 0.145420EL 10 10 36

Given that the unidirectional lamina modulus for this carbon-epoxy material is given as 126 GPa in Ref. [3], the stress–strain transformations for this case, equivalent to Eq. (71), are then

再冎冋

x 55.025 = y 18.379

册再 冎

18.379 55.025

x . y

(72)

The stresses corresponding with the strains in the table above can now be computed. Thus, for the corner points located by the 0° ply in the quasi-isotropic laminate, Point A: Point B Point B Point C Point K Point E Point J Point F

x = + 688.3 MPa, x = + 977.7 MPa, x = + 985.8 MPa, x = + 297.4 MPa, x = –541.9 MPa, x = –832.4 MPa, x = –883.5 MPa, x = –297.4 MPa,

y = + 41.01 MPa y = + 977.7 MPa y = + 1003.9 MPa y = + 962.9 MPa y = + 108.1 MPa y = –832.4 MPa y = –997.8 MPa y = –962.9 MPa.

The corresponding stresses for the 90° ply are established by interchanging the x and y stresses for each point in turn. The combination of these strength limits is plotted in Fig. 9, along with the two predictions from the original analysis method. The lines projected from the equal-biaxial-compression point E through the uniaxial compression points intersect the close-to-45° sloping shear cut-offs at x and y stresses of ( + 108.1, –541.9) and (–541.9, + 108.1) MPa, respectively. An indication of how close the sloping shear cut-off is to 45° is given by comparing the precise predicted shear strength of ± 327.0 MPa with the value ± 323.6 MPa computed to correspond with an assumed 45° slope. The inoperative failure envelope for the ± 45° plies is included to confirm how strongly fibre-dominated this laminate is. The fibre-failure lines for the ± 45° plies pass through the equal-biaxial strain points B and E, at a slope of –45°. The matrix shear-failure lines for the ± 45° plies cut the axes at stresses of ± 910.3 MPa, for a shear strain of 1.8
tL, per Eqs. (35) and (70). The corner points for this rectangular failure envelope for the ± 45° plies are at (1432.8, 522.5) MPa and (–377.3,

619 –1287.6) MPa, along with their points of symmetry across the + 45° diagonal. The predicted matrix shear failures lie well outside the fibre-failure envelope for the truncated lamina failure envelope. However, they might need to be considered for the untruncated version needed for fibreglass-epoxy laminates. A refined solution is included for the equal-biaxial-tension strain Point B because the material property data supplied indicate that the unidirectional lamina is strain hardening, with the modulus increasing from 126 GPa for small strains to 141 GPa at failure. The prediction for Point B should therefore be increased in this ratio, to 1094.0 MPa, agreeing more closely with the 1072.5 MPa prediction by the original method, which would automatically include the strain-hardening effect. The uniaxial tension strength could be modified likewise, with only the modulus of the 0° plies increased, since the other plies are not strained to failure. An estimate of the magnitude of such an increase can be obtained from the original formulation of the Ten-Percent Rule, i.e. this strength would likewise be increased by the factor 141 + 3
0.1 126 = 1.09. 1 + 3
0.1

Fig. 9. Failure envelopes for quasi-isotropic (0°/ ± 45°/90°)s carbon-epoxy laminate, on the laminate stress plane.

620 However, even without such a refinement, the currently predicted uniaxial strength of 648.1 MPa is clearly compatible with the upper and lower bounds of 715 and 633.8 MPa predicted by the original formulation. The predicted equal biaxial compression strength for the laminate, of 832.4 MPa, is equally consistent with the earlier prediction of 814.0 MPa. The predicted uniaxial compression strength of 575.3 MPa actually lies slightly outside the earlier estimates of 481.0 and 542.7 MPa. [The reason for this discrepancy appears to lie in the greater ratio of the lamina stresses (1950 versus 1480 MPa) used with the original version of the method than for the strains (0.0138 versus 0.01175) used with the current formulation, because of the strain hardening of the unidirectional lamina under tensile loads.] The new failure envelope in the 2nd and 4th quadrants is within l/7th of a degree of being straight, at a slope of 45°, and is, therefore, consistent with the original relationship between in-plane-shear and uniaxial-tension strengths given in Eq. (9). The new failure envelope bulges outward imperceptibly, confirming that fibres rather than the matrix govern even in these two quadrants. This is significant, because it shows that increasing the transverse strains by the factor (1 + vLT) is the absolute minimum increase capable of suppressing the prediction of matrix failures after the manner of Fig. 5. Any greater increase would have no effect on the predictions for the 1st and 3rd quadrants, but would have a weak effect in the form of a kink in the fibre shear cut-offs in the 2nd and 4th quadrants. The new formulation of the TenPercent Rule proposed here, for carbon-epoxy laminates, is thus remarkably similar to the truncated maximum strain failure model evaluated in Ref. [9]. The consistency between predicted in-plane shear strengths, using the original mathematical approach on the one hand, and the new graphical method with the expanded transverse strains on the other, confirms that the lines CD and FA in Figs. 6 and 7 are so located as to make the techniques equivalent. Any different degree of expansion would undermine the consistency. The author’s understanding of this subject must have improved since the preparation of Fig. 15 in Ref. [11], in which the expansions were confined to the 1st and 3rd quadrants. The earlier comment that the transverse strain amplification invoked here is quite dissimilar in nature from that used for traditional progressive-failure analyses can now be elaborated upon. In the first place, the amount of strain increase required here is barely 25%, while other published works cite a need for reductions in transverse stiffness by as much as a factor of 100, corresponding with a hundred-fold increase in transverse strain needed to fail the matrix.7 Factors as large as these are needed to transform interactive failure theories into something akin to the noninteractive netting theory. The present model is totally noninteractive. With

7

In the computer code listed in Appendix B of Ref 12, subroutine ANALYS on p. B-39 contains a section near the bottom of p. B-40 entitled “Call Stiffness Reduction Model.” This, in turn, calls for subroutines PSMA and/or PSMB, listed on p. B-51. These subroutines contain the self-evident instructions e2(k) = 0.01*e2(k)

and

g12(k) = 0.01*g12(k)

to reduce the transverse stiffness and in-plane shear stiffness of each ply by arbitrary factors of 100. The author has found no restriction to prevent these subroutines being called more than once. Nowhere in the report is this factor tied to any measurements of reduction in stiffness of the matrix.

621 elliptical failure envelopes, degradation factors of only 2 or 3 still leave the failure envelope strongly interactive, with much of the failure envelope as dominated by transverse failures as in Fig. 5 – and sensitive to whatever matrix-dominated properties were used as input. Each of the solutions in Fig. 9 can now be compared with the corresponding predictions from other composite failure theories, particularly with those which, by the time the “last-ply” strengths have been estimated, are in reality no less approximate and empirical than the values predicted by the Ten-Percent Rule. The analysis of this quasi-isotropic carbon-epoxy laminate was used to establish how to represent the Ten-Percent Rule as a complete failure envelope in such a way that it would essentially replicate the predictions of the earlier formulation. The two remaining fibreglassepoxy laminates in this exercise will now be used to verify that no further modifications are necessary.

6. Test problem No. 6: biaxial (␴y ⴚ ␴x) failure envelope for ( ± 55°)s E-glass-epoxy laminate

Prior solutions of this problem in Refs. [5] and [9] have revealed that the failure envelope for this particular laminate, on both the stress and strain planes, is a simple skewed parallelogram, with the width established uniquely by the matrix shear strain at failure and the length (ends) defined by the biaxial tension and compression strengths, which are fibre dominated. This problem is so simple that the application of the “new” failure model in Figs. 6 and 7 can be demonstrated by hand calculations. The strains can be transformed from lamina to laminate axes using Eq. (32). Here,  = ± 55°, so, on the base plane of the laminate failure envelope ( ¯ = 0),

再冎

冋

x 0.3290 = –2.9238 y –0.6710

册再 冎

–0.6710 0.3290

1 . 2

(73)

These equations are sufficient to locate the fibre-failure cut-offs on the laminate strain plane. The matrix failure cut-offs are 45°-sloping lines parallel to the equal-biaxial-strain axis, given by Eq. (34) as

y = x ±

LT . 0.93963

(74)

In this case, given the unidirectional strains-to-failure for the individual laminae of tL = 0.02807 in tension and cL = –0.01754 in compression, and the lamina Poisson’s ratio of 0.278, the lamina strains in Eqs. (57) to (67) are located on the laminate strain plane as follows.

622 Point B: Point I: Point J: Point G:

x = + 0.02903, x = + 0.09778, x = –0.04490, x = –0.1137,

y = + 0.02611 y = –0.07557 y = –0.002639 y = + 0.09904.

The parallel lines BG and IJ in Figs. 6 and 7 define the fibre-limited strengths in tension and compression, respectively. Two other parallel lines, BI and JG establish the transverse strain limits perpendicular to the fiber axes. (The shear cut-offs AF and DC are not applied because this is a fibre-glass reinforced laminate, not one full of carbon fibers.) The remaining possible failure mode is by matrix shear, defined by lines sloping at + 45°. These are derived as follows. Eqs. (34) and (10), predict intercepts on the laminate strain axes of ± 1.8L/ 0.93963 = 1.9155L. In this case, L = tL = 0.02807, so the intercepts on the x and y laminate strain axes are at + 0.05377. The net result of these predicted strength limits is shown in Fig. 10, in the form of a basic skewed parallelogram, with two opposite corners truncated. The various modes of failure are identified in the figure. (Because the Ten-Percent Rule is noninteractive, it is possible to differentiate between fibre failures and between the multiple kinds of matrix failures.) It remains now to convert this into a stress-based failure envelope. Since all plies have the same thickness for the ± 55° laminate under consideration, the following specific equations apply to the present problem, when the simplifications and assumptions of the Ten-Percent Rule are introduced into Eqs. (36), (39) and (44).

Fig. 10. Failure envelope for ± 55° E-glass-epoxy laminate, on the laminate strain plane.

623

A11 =

E tL laminate 

A12 =

E tL laminate 

A22 =

E tL laminate 

A66 =

E tL laminate 

冋 冉 冊 冉 冊册 冋冉 冊 冉冊 册 冋 冉 冊 冉 冊册 冋冉 冊 冉冊 册 (1.0)c4 + 2

1.0 +

1 4 2 2   sc + (s4 + c4) 10 36 36

(1.0)s4 + 2

1.0 +

vLT 2 2 2 1 4 + sc + s 10 36 10

,

(75)

vLT 2 2 2 1 4 + sc + c 10 36 10

1 vLT 2 2 2  2  sc + (s4 + c4) 10 10 36 36

where, for  = 55°, c4 = 0.1082,

s4 = 0.4503,

s2c2 = 0.2208,

 = 1  0.1,

(s4 + c4)-0.5585,

v2LT = 0.9907. (76)

Hence, since vLT = 0.278, A11 = 0.1915E tL,

A12 = 0.2357E tL,

A22 = 0.4955E tL,

and

A66 = 0.2355E tL.

(77)

In this case, the data supplied in Ref. [3] include a longitudinal modulus EL of 45.6 GPa, so that A11/t = 8.726 GPa,

A12/t = 10.748 GPa,

A22/t = 22.589 GPa,

and

A66/t = 10.743 GPa. (78)

values remarkably close to those estimated precisely in Refs. [5] and [9]. The stress–strain transformation equations for this laminate then follow from Eq. (68) as

再冎冋

x 8.726 = y 10.748

册再 冎

10.748 22.589

x . y

(79)

The stresses at the corner-points identified in Fig. 6 corresponding with the strains recorded in Fig. 10 are thus Point B: Point I: Point J: Point G: Point C:

x = + 534.0 MPa, x = + 40.97 MPa, x = –420.2 MPa, x = + 72.84 MPa, x = + 230.6 MPa,

y = + 901.9 MPa y = –656.0 MPa y = –542.2 MPa y = + 1015.7 MPa y = –56.91 MPa.

The additional points locating the matrix shear-strength cut-offs in terms of x and y laminate stresses are established as follows. The strain points (x = + 0.05377, y = 0) correspond with stresses of (x = ± 469.2 MPa, y = + 577.9 MPa). The strain points (x = 0, y = + 0.05377)

624 correspond with stresses of (x = + 577.9 MPa, y = ± 1214.6 MPa). The lines so defined have a slope of 30° from the y axis and intercept the axes at x = ± 131.6 MPa and y = + 225.3 MPa. The x and y co-ordinates of the corner points of the parallelogram defined by the various intersecting lines are as follows, in MPa: ( + 642.7, + 875.1), ( + 412.7, + 931.8), (–444.8, –536.1), and (–214.8, –593.0). The upper right corner is truncated by predicted transverse-tension failures in the matrix along the line from ( + 483.8, + 914.3) MPa to ( + 268.9, + 235.1) MPa. The lower left corner is truncated similarly, but to a far lesser extent because of the specified weakness of the laminate in compression, along the line from (–370.0, –554.6) MPa to (–268.9, –235.1) MPa. Fig. 11 depicts the resulting stress-based failure envelope for the ± 55° glass-epoxy laminate, according to the Ten-Percent Rule, as modified above. (Note that the axes have been interchanged from the normal convention at the request of the contest organizers.) This solution is very similar, in both form and size, to the equivalent results given in Refs. [5] and [9]. The failure envelope is basically a skewed parallelogram, with the short ends defined by fibre-dominated biaxial tension and compression strengths and the long sides by matrix shear failures. There is a significant zone of predicted matrix failures by cracking under transverse tension in the top-right of this figure, and a much smaller zone in the bottom-left corner for matrix failures under transverse compression. (This last zone would be longer if a better uniaxial compression strength had been used in the analysis.) The original Ten-Percent Rule has never implied that the plies necessarily fail when loaded transversely beyond one tenth of the load needed to fail them under longitudinal loads, or in shear beyond one twentieth of that same reference strength, so whether or not these are zones of real matrix failures is open to question. (The transverse-strain limits were originally set to estimate

Fig. 11. Failure envelope for ± 55° E-glass-epoxy laminate, on the laminate stress plane.

625 contributions of loads within transverse plies to overall fibre-dominated laminate strengths.) Unlike the generalized maximum-shear-stress failure model, however, the Ten-Percent Rule model has no provision for adjusting any predicted matrix failures. One must either accept them or reject them, based on the likelihood or otherwise of fibres in other plies to arrest such damage and maintain such matrix-dominated strengths. The results plotted in Fig. 11 permit the identification of the governing modes of failure, as a function of the biaxial strain combinations. By definition, interactive failure theories based on assumed homogenized “composite materials” cannot make such distinctions.8 In contrast with the fully fibre-dominated failure envelope for the quasi-isotropic laminate discussed in the preceding section, this failure envelope is bounded on two sides by matrix failures – because of the absence of fibres in a sufficient number of directions to limit all in-plane strains. The ability to even make predictions about the strength of such a laminate – without any measured matrix-dominated material properties as inputs – represents a major enhancement of the capabilities of the Ten-Percent Rule, with respect to what was originally expected of it. The predicted fibre failures compare very well with those in Refs. [5] and [9], so the ability of the new formulation to accept fibre angles outside the original 0°, ± 45°,90° family has also been confirmed.

7. Test problem No. 4: biaxial (␴y ⴚ ␴x) failure envelope for (90°/ ± 30°)s glass-epoxy laminate If the characterization, and single modification, of the Ten-Percent Rule described above is sufficient to form the basis of a reliable analysis method for the strength of fibrous composite laminates, this could be verified by not needing any more modifications to successfully solve the final task. As before, the derivation of the solution to this problem must start on the strain plane, using the results of the previous sections, to ensure compatibility of deformations while identifying which constituent fails, and how, for each state of biaxial stress. The ± 55° laminate characterized above was clearly subject to matrix failures around parts of the failure envelope, because there were fibers in only two directions. It was therefore appropriate to analyze the laminate in terms of the corner points defined in Figs. 1 and 2. The 90°/ ± 30° laminate, with fibres in three directions, on the other hand, is clearly fibredominated throughout. Therefore, the analysis must be conducted in terms of the transversely expanded failure envelopes in Figs. 6 and 7 instead. Nevertheless, any checks for possible real matrix failures must obviously be carried out using the original failure envelopes in Figs. 1 and 2. Both transverse limits on lamina strain are therefore included in this next analysis. Figs. 12 and 13 show, to the same lamina-strain scale, the failure possibilities for 90° and ± 30° layers, in both the fibres and the matrix.

8

Nevertheless, the undeniable fact is that computer codes based on such theories do differentiate between fibre and matrix failures. Otherwise, progressive failure analyses would not be possible. The changeover from fibre to matrix failures in Fig. 11 is clear and unambiguous. But on what basis does one select the changeover points on a smooth continuous ellipse? It is for this reason that the TenPercent Rule should not be reformulated as something more akin to the smooth von Mises criterion than to the flat-faceted Tresca model akin to that used here.

626

Fig. 12. Failure envelope for 90° unidirectional lamina, on laminate strain plane.

Fig. 13. Failure envelope for ± 30° unidirectional laminae, on laminate strain plane.

627 Fig. 12 is merely a mirror image version of Fig. 7, rotated around the equal-biaxial-strain diagonal. The measured lamina strains-to-failure are given in Ref. [3] as tL = 0.02132 and cL = –0.01065, and the lamina Poisson’s ratio as 0.278. The significant strain corner points for the 90° plies, equivalent to those in Fig. 7 for fiber and matrix failures in the 0° plies, are then:

x = + 0.01539, x = + 0.02428, x = –0.01836, x = –0.02725, x = + 0.02132, x = + 0.03021, x = –0.02429, x = –0.03317,

Point B: Point I: Point J: Point G: Point B: Point I: Point J: Point G:

y = + 0.02073 y = –0.01124 y = –0.01006 y = + 0.02191 y = + 0.02056 y = –0.01141 y = –0.009893 y = + 0.02208.

No matrix shear failures in the 90° plies can occur on the base plane of the failure envelope ( xy = 0). Fig. 13 derives from the application to Fig. 7 of the strain-transformation relations in Eq. (32) which, on the base plane of the laminate failure envelope ( ¯ = 0), for  = ± 30°, are

再冎 冋

x 0.75 = 2.0 y –0.25

册再 冎

–0.25 0.75

1 . 2

(80)

These equations are sufficient to locate the fibre- (and possible matrix-) failure cut-offs on the laminate strain plane, as tabulated below, using the specific value of Poisson’s ratio, 0.278, rather than the generic value of 0.3. Point B: Point I: Point J: Point G: Point B: Point I: Point J: Point G:

x = + 0.02340, x = –0.02900, x = –0.00591, x = + 0.04649, x = + 0.02018, x = –0.03222, x = –0.002696, x = + 0.04970,

y = + 0.01273 y = + 0.04204 y = –0.02251 y = –0.05183 y = + 0.02170 y = + 0.05102 y = –0.03148 y = –0.06080.

The matrix-failure cut-offs for the ± 30° plies are 45°sloping lines parallel to the equalbiaxial-strain axis, given by Eqs. (34) and (70) as

y = x ±

LT = x ± 2.07846
1.278
tL, 0.86603

(81)

so the intercepts on the axes are at strains of ± 0.044313. Fig. 14 is an overlay of Figs. 12 and 13, identifying which failures occur first for the various biaxial stresses, distinguishing between possible fibre and matrix failures and between the two fibre directions.

628

Fig. 14. Failure envelope for 90°/ ± 30° E-glass-epoxy laminate, on laminate strain plane.

The two failure envelopes for the 90° and ± 30° layers necessarily cross at the equalbiaxial-strain points Point B: x = y = 0.02058 and Point E: x = y = –0.01028. The form of this combined failure envelope is basically a skewed parallelogram, with a nearly horizontal top and bottom defined by fibre failures in the 90° plies, and sloping sides. The leftside is clearly defined by compressive failures of the fibres in the ± 30° plies, while the right-hand side appears to be limited almost simultaneously by the transverse tension strength of the 90° plies and by the longitudinal tension strength of the ± 30° plies. Possible matrix failures are confined to two regions. One is the equal-biaxial-strain zone, affecting all plies equally, and the other is the transverse-tension load on the 90° plies, regardless of the orthogonal stress component. It seems likely that real non-catastrophic matrix failures would precede the actual fibre failures since the stiffness of ± 30° plies perpendicular to the 90° direction is far less than of 0° plies (in other laminates), which would restrain matrix cracks between 90° fibres far more effectively. This level of explanation of possible modes of failure, at the laminate level, is simply unobtainable from traditional interactive composite failure criteria, because the discrete fibre and matrix constituents have been replaced by a single homogenized “material.”

629 These laminate strains can now be converted to laminate stresses as follows. The laminate stiffness matrix coefficients for use in Eq. (69) are

A11 1 = {[16ET] f90 + [9EL + 6(vLTET + 2GLT) + ET] f ± 30}, t 16

A12 1 = {[16vLTET] f90 + [3(EL + ET  4GLT) + 10vLTET] f ± 30}, t 16 A22 1 = t 16

再

冋 冉

[16EL] f90 + EL + 6

冊 册 冎

,

(82)

vLT EL + 2GLT + 9ET f ± 30 , 10

A66 1 = {[16GLT] f90 + [3(EL + ET  2vLTET  2GLT) + 10(GLT)] f ± 30}, t 16

where  = 1  vLTvTL and f.. represents the fraction of fibres in the appropriate direction. Here, per Ref. [3], f90 = 0.1727 and f ± 30 = 0.8273. On introducing the approximations of the Ten-Percent Rule, these expressions become

A11 EL = t 16

再冋 册 冋 冉 冊 册 冎

A12 EL = t 16

再冋 册 冋 冉

16 vLT 2 1 f90 + 9 + 6 + + f ± 30 = 0.51875EL, 10 10 36 10

冊 册 冎

16 1 4 vLT f90 + 3 1 +  + vLT f ± 30 = 0.17405EL, 10 10 36

A22 EL = t 16

再

冋 冉 冊 册 冎

A66 EL = t 16

再冋 册 冋 冉

[16] f90 + 1 + 6

vLT 2 9 + + f ± 30 = 0.29898EL, 10 36 10

冊 册 冎

16 1 2 4 f90 + 3 1 +  vLT + f ± 30 = 0.17381EL. 36 10 10 36

,

(83)

630 Since the unidirectional lamina stiffness is quoted as 53.48 GPa in Ref. [3], where the Poisson’s ratio vLT is cited as 0.278, so that  = 0.988916, the stiffness matrix for this 90°/ ± 30° laminate is x 27.677 y = 9.308 xy 0

9.308 15.989 0

0 0 9.295

x y .

xy

(84)

The stresses corresponding with the strains in the table above can now be computed. There are more entries needed than before, because of the fibres in two distinct directions. Thus, on the base plane of the basic failure envelope ( xy = 0), Point B (90°): Point B ( ± 30°): Point B (90°): Point B ( ± 30°): Point B (90° & ± 30°): Point I (90°): Point I ( ± 30°): Point I (90°): Point I ( ± 30°): Point J (90°): Point J ( ± 30°): Point E (90° & ± 30°): Point G (90°): Point G ( ± 30°):

x = + 619.0 MPa, x = + 765.9 MPa, x = + 781.5 MPa, x = + 760.6 MPa, x = + 761.3 MPa, x = + 567.4 MPa, x = – 411.4 MPa, x = + 729.9 MPa, x = – 416.8 MPa, x = – 764.3 MPa, x = – 367.7 MPa, x = – 380.3 MPa, x = – 712.7 MPa, x = + 809.7 MPa,

y = + 474.4 MPa y = + 421.2 MPa y = + 527.3 MPa y = + 534.8 MPa y = + 520.7 MPa y = + 46.24 MPa y = + 402.3 MPa y = + 98.78 MPa y = + 515.8 MPa y = – 384.3 MPa y = – 528.5 MPa y = – 260.1 MPa y = + 44.22 MPa y = – 509.5 MPa.

These stresses, omitting the inoperative cut-offs, are plotted in Fig. 15, with the constituent and mode of failure identified on each line. This failure envelope is basically a simple skewed parallelogram. The x and y co-ordinates of the corner points are as follows, in MPa: ( + 761.3, + 520.7),(–399.3, + 145.5), (–380.3, –260.1), and (780.3, 115.1). Cut-offs would have been predicted on the steeper lines in the upper left and lower right corners if this had been a carbon-epoxy laminate instead of one made from glass fibres. The failure envelope is predicted to be almost entirely fibre dominated, as one would expect for a laminate with triangulated fibre patterns. There is, however, a small truncation shown below the top line, representing transverse tension limit set for the 90° fibres, extending between the stresses of x and y = (766.6, 404.1) and (729.9, 98.8) MPa, as shown. Possible, or even likely, matrix failures under transverse tension are predicted between the 90° fibres along the almost horizontal stress line between x and y = (619.0, 474.7) and (567.4, 46.24) MPa, and between the ± 30° fibres along the almost vertical line between x and y = (765.9, 421.2) and (437.2, 415.9) MPa. The inoperable transverse-tension failures in the ± 30° fibres are predicted to occur along the nearly vertical line between x and y = (760.6, 534.8) and (–416.8, 515.8) MPa. Fig. 15 also contains the predicted locations for matrix shear failures in the ± 30° plies, passing through the points x = 1,226.5 MPa, y = 412.5 MPa and x = – 412.5 MPa, y = – 708.5 MPa, and their complementary co-ordinates, crossing the y axis at ± 623.4 MPa and the x axis at  426.4 MPa.

631

Fig. 15. Failure envelope for 90°/ ± 30° E-glass-epoxy laminate, on laminate stress plane.

Because the Ten-Percent Rule is mechanistic, rather than interactive, further information can be deduced about this failure envelope, even though it cannot be quantified. Near the upper-left corner of the basic failure envelope, the axially compressed 90° fibres cannot be expected to develop their measured reference compression strength when there are such strong indications of matrix damage, in transverse tension between all fibres and in shear between the ± 30° fibres. One should therefore expect a significant truncation of the failure envelope in this region. In the lower-left corner, however, the matrix is predicted to not be anywhere near critical in any possible failure mode, so these predictions in Fig. 15 are likely to be confirmed by test. The 90° fibres will be stable in the bottom-right corner of the failure envelope, but the ability of the ± 30° fibres to withstand axial compress loads would be diminished by in-plane-shear damage to the matrix in those plies. One might well expect some small truncation of the basic failure envelope there. In the upper-right corner of the failure envelope, all fibres are loaded primarily by longitudinal tension. Consequently, the predicted ultimate strengths will be attained, although it is likely that these will be preceded by detectable matrix damage. A comparison of this solution with the corresponding predictions in Refs. [5] and [9] shows very great similarities. One can conclude therefore that, apart from the need to increase the tolerable transverse stresses and strains in the fibres to preclude predicting physically unrealistic premature failures, the Ten-Percent Rule can be easily extended to solve problems with fibre-polymer composites having lay-ups far beyond what was the original planned coverage of the method. Nevertheless, it should not be regarded as competition to computer coded models of properly formulated mechanistic failure models. On the other hand, and very

632 significantly, it can be used when these cannot because of the absence of, or uncertainty about, some of the input data. 8. Test problem No. 5: biaxial (␴x ⴚ ␶xy) failure envelope for (90°/ ± 30°)s glass-epoxy laminate The remaining failure envelope requested in Ref. [3] is a different section cut through the complete three-dimensional surface for same laminate as in the previous problem. Fig. 16 presents the solution to this problem, based on the preceding calculations. The additional strengths needing to be calculated are for the intercepts on the xy axis of the failure surfaces for the ± 30° plies under biaxial tension and compression loads. The intercepts are given by the following expression, derived from the first two of Eqs. (30).

xy = ±

 , sin  cos 

(85)

where  is the magnitude of the 1 and 2 strains on the diagonal 2 = –1 in Fig. 7. (No points off that diagonal can be transformed to lie on the xy strain axis.) In this case, the critical points are at G and H, the co-ordinates of which are given in Eqs. (27) and (28). Using the specified value vLT = 0.278, Eqs. (27), (28) and (85) establish the magnitudes of the xy shear strains as 0.050302 for tension loads and 0.025128 for compression. Hence, per the last of Eqs. (86), the intercepts on the xy stress axis are at 467.5 and 232.9 MPa, respectively. The corresponding x stresses for xy = 0 can be interpolated from the data used to construct Fig. 15 as 424.3 MPa in x tension and –392.5 MPa in compression. Were a shear failure to occur within the 90° plies, it would do so at a shear strain, with respect to x and y axes given by Eq. (70) as 1.8tL, i.e. 0.038376. The corresponding laminate shear stress is 356.7 MPa, which is almost the same as the corresponding strength limit of 375.5 MPa established in Refs. [3] and [7] according to the generalized maximum-shear-stress and maximum-strain failure models,

Fig. 16. (x–xy) Section cut through failure envelope for (90°/ ± 30°)s glass-epoxy laminate.

633 yet the Ten-Percent Rule involves no measured matrix properties. These potential matrix shear failures between the fibers in the various plies are included in Fig. 16, lying far outside the predicted fibre failures except for the upper-right corner of the envelope. 9. The remaining test problems9 The original request for further solutions to problems in Ref. [3] involved mainly stress–strain curves and 3 lamina biaxial failure envelopes (Test Problems 1, 2 and 3). For all these remaining cases, the in-plane-shear strength FLT was taken here as F tL/20 and the transverse lamina strengths F tT and F cT were taken here as F tL/10. According to the present theory, the behavior of Test Problem No. 1 would be a totally noninteractive relation between the inplane-shear strength FLT and the transverse lamina strengths F tT and F cT. This is plotted in Fig. 17. Problem No. 2 involved what is, to the Ten-Percent Rule, an equally non-interactive relation between this same in-plane-shear strength and the longitudinal lamina strengths F tL and F cL. This is plotted in Fig. 18. Test Problem No. 3 is the (2  1) failure envelope depicted here in Fig. 1 for a unidirectional lamina. (Fig. 7 represents a necessary change from properties which would be measured to properties needed to make the analytical model predict reliable strengths for multidirectional structural laminates.) Fig. 19 is a quantified version of Fig. 6, and may be compared with other solutions for this problem drawn to the same scale, even though it will not match any measured transverse ply properties.

Fig. 17. (2–12) Section cut through lamina failure envelope for unidirectional glass-epoxy laminate.

9

After the submittal of this manuscript at the same time as all the other Part A papers were prepared, before the experimental results had been distributed, the organizers requested that the author add solutions for the stress–strain curves that he had not originally included because they were simple straight-line results and seemingly not very informative. Presumably, the organizers want to compare as many as possible of these additional predictions in their summary papers, Ref [13]. These predictions by the author have been included, while those for the other theories he reported on were not, because there are no prescribed matrix strength limits with his other models.

634

Fig. 18. (1–2) Section cut through lamina failure envelope for unidirectional T300/914C carbon/ epoxy lamina.

In their Test Problems Nos. 7 and 8, the organizers requested uniaxial and biaxial tensile stress–strain curves for the ( ± 55°)s glass-epoxy laminates made from E-glass/MY750/ HY917/DY063 composite material. The first request is easily satisfied, since it is for the intercept on the y axis of the failure envelope in Fig. 11. This had earlier been calculated to be at (y /x = 225.3/0) MPa. The associated laminate strains then follow from the inverse of Eq. (79) as (y, x = 0.02410, –0.02968). The corresponding linear stress–strain curves are plotted in Fig. 20. The corresponding stress–strain curves for biaxial loading in the ratio (y /x = 2/1) starts with the establishment of the failure stresses by linear interpolation along the fiber-failure line in Fig. 11 (line AB in Fig. 7). The stresses are (y, x = 920.1, 460.1) MPa. The associated laminate strains then follow from the inverse of Eq. (79) as (y, x = 0.03762, 0.006163). The corresponding linear biaxial stress–strain curves are plotted in Fig. 21. Test Problems Nos. 10 and 11 pertain to stress strain curves for the AS4/3501-6 quasiisotropic (0°/ ± 45°/90°)s carbon-epoxy laminates. The first request, for a uniaxial load (y /x = 1/0), is trivial. For this model, it is a straight line, with no predicted hiccups, to the uniaxial-stress failure point at stresses of 648.1 MPa, as calculated earlier for test problem 9B. The associated longitudinal and transverse strains, in the laminate, are (y, x = 0.0l327,

Fig. 19. (1–2) Section cut through lamina failure envelope for unidirectional glass/epoxy laminate.

635

Fig. 20. Stress-strain curve for (y /x = 1/0) loading of ( ± 55°)s glass-epoxy laminates.

–0.00443) since the Poisson’s ratio for the laminate is given as A12 /A22 = 18.37881/ 55.02467 = 0.33340. The fiber strain-to-failure is less than the maximum value of 0.0138 because the 0° fibers are biaxially stressed in the quasi-isotropic laminate and Point A in Fig. 6 now lies off the laminate uniaxial stress line. This prediction is plotted in Fig. 22.

Fig. 21. Stress–strain curve for (y /x = 2/1) loading of ( ± 55°)s glass-epoxy laminates.

636

Fig. 22. Stress-strain curve for (y /x = 1/0) loading of AS4/3501-6 quasi-isotropic (0°/ ± 45°/90°)s carbon-epoxy laminates.

The related request for the stress–strain curves for (y /x = 2/1) loading of the same laminate is a little more difficult to establish. The biaxial stresses are established by linear interpolation (along the line AB in Fig. 7), on which point B is located at biaxial stresses of (977.7, 977.7) MPa in Fig. 9. The predicted failure stresses are (y, x = 793.49, 396.74) MPa. The corresponding laminate-level strains at failure then follow from the inversion of Eq. (72) as (y, x = 0.013521, 0.002694). These linear biaxial stress–strain relations are plotted in Fig. 23. The final three problems (Nos. 12–14) posed by the organizers of this contest, all for glassfibre-reinforced epoxy laminates, are also amenable to solution by the Ten-Percent Rule. The first of these is for the strength of a ± 45° laminate under equal-biaxial stresses. The strength is the same as for a 0°/90° laminate under the same loads. The strength follows directly from Eq. (52), given the reference lamina strengths of 1280 and 800 MPa in tension and compression, respectively. The biaxial strengths would thus be + 714.2 and –446.4 MPa, respectively. The strains at failure would be given by Points B and E in Fig. 7. The strain is given in Eq. (58), as 0.0.96540 = 0.02710 in tension and –0.01693 in compression. The linear stress–strain relation depicting this solution is given in Fig. 24. The next request is for the same glass-epoxy laminate subjected to equal and opposite tension and compression loads. This is equivalent to pure shear applied to a 0°/90° laminate, with direct strains equal to half the shear strain. According to Eq. (10), then, the direct strains at failure will be + 0.90 = 0.02526. The tensile and compressive stresses will be equal in magnitude to the shear stress of G , where the shear modulus G is given by Eq. (8) as EL /36, or 1.267 GPa and the shear strain is twice the direct strain, or ± 0.05052. The laminate strength under this particular biaxial load is therefore 64.0 MPa, as shown in Fig. 25. The

637

Fig. 23. Stress–strain curve for (y /x = 2/1) loading of AS4/3501-6 quasi-isotropic (0°/ ± 45°/90°)s carbon-epoxy laminates.

Fig. 24. Stress-strain relation for equal biaxial loads applied to ± 45° glass-epoxy laminate.

638

Fig. 25. Stress–strain relation for equal-and-opposite biaxial loads applied to ± 45° glass-epoxy laminate.

in-plane shear strength cited by the organizers for a unidirectional lamina made from this material is 73 MPa. The final exercise requested by the organizers is the prediction of the longitudinal and transverse stress–strain curves for a uniaxial load applied to a 0°/90° laminate of the same fibreglass-epoxy composite. The failure strain has been provided as 0.02807, and the laminate modulus is given above as 0.558 EL, where the lamina modulus is given as 45.6 GPa. The Ten-Percent Rule would predict a linear stress–strain curve to failure, with only minute transverse strains in comparison with the longitudinal strains. The peak transverse strain caused by the Poisson contractions would be defined by Eq. (53); in this case the primary Poisson’s ratio would be 0.05045 (using the value 0.278 provided for vLT), and the peak transverse strain would be –0.001419. The failing stress would then be 714.2 MPa. These predictions are plotted in Fig. 26.

10. Conclusions Despite its humble origins, the Ten-Percent Rule for predicting the strength of fibre polymer composites has been shown here, and in earlier applications to real composite structures, to have at least as much capability as the best of the macromechanical analysis methods. The real virtues of the Ten-Percent Rule are that it is easily comprehended, relatively unambiguous to apply, and has been shown to be realistic (at the ultimate failure level) in a great many cases. Perhaps the greatest virtue of the Ten-Percent Rule, other than its simplicity, is that there is no opportunity to fiddle the predictions as so many other authors have done under the mask of progressive-damage analysis. Is it really credible that changes in the soft matrix properties

639

Fig. 26. Stress-strain relation for uniaxial load applied to 0°/90° glass-epoxy laminate.

can really have a great influence on laminate strengths that are dominated by the strong, stiff fibers? The original formulation of this theory was as a mental-arithmetic technique for fibredominated patterns in the 0°, ± 45°,90° family of balanced laminates. With only a minor modification, it has been extended here to cover fibres at all orientations and even some of the matrix failures. The matrix failures predicted by the Ten-Percent Rule are a consequence of the formulation of the fibre failures in the model, and not the result of a scientific assessment of matrix failures. As such, their accuracy is open to question, as are most predicted matrix failures. It would obviously be possible to devise a better characterization of matrix failures, one that could be superimposed on the present predictions of fibre failures. This would need to cover both non-catastrophic initial failures, which might precede fibre failures, and eventual complete failure. However, the virtue of this model is not absolute precision; it is reasonable accuracy from an extremely simple model. Until such time as the knowledge of matrix failures has improved to the level that a comparably simple model could be prepared, it is recommended that no efforts be made to “improve” the fit of this model to matrix-failure data. With the exception that predicted matrix failures under transverse may need to be over-ridden if there are fibres orthogonal to the cracks that would inhibit the spread of such cracks, all of its predictions, for fibre failure and matrix failures in shear, tension, and compression, should be accepted as design ultimate strength in every analysis. While the analysis of fibre failures with the Ten-Percent Rule is simple and straightforward, for both the vertical and sloping lines in Fig. 6, there is some ambiguity about the original and expanded horizontal lines representing transverse failures. The author would recommend that analyses be completed with both possible transverse limits included before a decision is made as to which to accept and which, possibly, to reject. It is clear that the

640 original cut-offs should be employed whenever there are no fibres in the laminate that could suppress or limit such cracking of the matrix. It is equally clear the original transverse limits cannot be used universally, as a “conservative” solution, since doing so would often conceal a more plausible solution. If matrix damage is predicted by the original unexpanded transverse limits, it is reasonable to over-ride it as suggested here when it occurs while the critical fibers are loaded by tension, as was done for the 0°, ± 45°/90° carbon-epoxy laminates under biaxial tension. On the other hand, if matrix damage is predicted in combination with critical fibres loaded in compression, it might be more plausible to believe such lower predicted strengths, as hinted at for the 90°, ± 30° glass-epoxy laminates. Some ambiguities will remain, as for matrix cracks between 90° fibres that are restrained not by stiff 0° fibres, but by the softer ± 30° plies that could easily permit structurally significant matrix failures before the restraining fibres failed. While it may seem to some that the present modifications of the transverse properties are no less unscientific than similar techniques of which the author remains very critical, the contexts are entirely different. The Ten-Percent Rule has never been portrayed as anything more than a useful approximation. The purpose of this paper is not to develop yet another “scientific” composite failure criterion, but to see how much the existing Ten-Percent Rule can be extended without losing its inherent advantages of simplicity and a lack of ambiguity. With these simple modifications, it is now clear that the capabilities of the original Ten-Percent Rule can be expanded appreciably – far beyond what was envisaged back in 1970. Without them, no such expansion is practical – which in no way detracts from the very significant capabilities it already had long before the preparation of this paper. Slightly different formulations are needed to cope with biaxial stresses of opposite signs for carbon- and glass-fibre-reinforced polymer composites, because of differences between the transverse strains in the lamina and the fibres. (Although not stated clearly, the Ten-Percent Rule is still limited to composites of strong, stiff fibres in soft matrices. It never will be applicable to the likes of metal-matrix composites.) When first formulated, the Ten-Percent Rule needed an additional reference strength, that of the in-plane shear-strength of a ± 45° laminate. An earlier improvement to the model, made in the early 1980s, was a formula to predict this property on the basis of longitudinal measured properties alone. This derived from the author’s recognition that fibres were failing by shear. Since then, one of the great appeals of the classical formulation of the Ten-Percent Rule was its need for barely half of the material properties required by other more formal theories. This simplicity has been retained throughout the new extensions of its capabilities as a pseudobonafide composite failure criterion. This means that reliable solutions can now be predicted for problems that could not hitherto be solved with other theories because of a lack of test data. This may well turn out to be the greatest attribute of the new formulation. Significantly, the properties not needing to be established experimentally for the TenPercent Rule match almost perfectly the inputs that are used and then discarded in the many interactive failure theories that the author has condemned over the years. Surely this cannot be a mere coincidence. The completion of the fourteen Test Cases requested by the organizers of this comparison between different fibre polymer composite failure criteria [13] provided a further valuable insight into the process. Although this was by far the last of the three papers commenced, the other two being Refs. [3] and [7], it was the first to be completed. The reason for this is likely

641 to persist in future work. The Ten-Percent Rule is unambiguous. There is neither need nor opportunity to adjust the matrix-dominated properties to seek a “better” solution; all such inputs are prescribed by this model. With almost all other composite failure theories, whether physically realistic or not, such a modification is not only desirable; it is necessary, just to produce an acceptable answer. It may well transpire that this feature will enable more reliable composite structure designs to be created with the Ten-Percent Rule than are produced by even more physically realistic failure models. Nevertheless, it must be remembered that the primary function of the Ten-Percent Rule is to predict the strength of well-designed fibredominated laminates. It should not be expected to compete with physically realistic models than really can identify the initiation and propagation of matrix failures. References [1] Hart-Smith LJ. The Ten-Percent Rule for preliminary sizing of fibrous composite structures. McDonnell Douglas Paper MDC 92K0024, SAWE Paper No. 2054, presented to 51st SAWE International Conference, Hartford, Connecticut, 18–20 May, 1992; published in SAWE Journal, Winter 1992, pp. 29–45; also published by Aerospace Materials, Vol. 5, No. 2. August–October 1993. pp. 10–16. [2] Hart-Smith LJ. Design and Development of the First Lear Fan All-Composite Aircraft. Douglas Paper 8184, Presented as Keynote address to Institution of Mechanical Engineers Conference on Advanced Composites, London; 7–8 March 1989. [3] Soden PD, Hinton MJ, Kaddour AS. Lamina properties and lay-up configurations and loading conditions of a range fibre reinforced composite laminates. Compos Sci Technol 1998;V58: 1011–22. [4] Hart-Smith LJ. Simplified estimation of stiffness and biaxial strengths for design of carbon-epoxy composite structures. Douglas Paper 7548, presented to Seventh Conference on Fibrous Composites in Structural Design, Denver, Colorado, 17–20 June 1985; in Proceedings, AFWALTR-85-3094, pp. V(a)-17 to -52. (Subsequently cleared for unlimited distribution.) [5] Hart-Smith LJ. Predictions of a generalized maximum-shear-stress failure criterion for certain fibrous composite laminates, McDonnell Douglas Paper MDC 97K0011, companion paper to the present. Composites Science and Technology 1998;58(7):1179–208. [6] Hart-Smith LJ. Fibrous composite laminate strength predictions demystified. Douglas Paper 8430, presented to 11th European SAMPE meeting, Basel, Switzerland, 29–31 May 1990. In: Hornfeld HL, editor. Proceedings. pp. 365–380. [7] Hart-Smith LJ. The truncated maximum-strain composite failure model. Composites 1993;24(7):587–91. [8] Jones RM. Mechanics of composite materials. Washington (DC): Scripta; 1975. [9] Hart-Smith LJ. Predictions of the original and truncated maximum-strain failure models for certain fibrous composite laminates. McDonnell Douglas Paper MDC 97K0010, companion paper to the present. Composites Science and Technology 1998;58(7):1151–78. [10] Hart-Smith LJ. Simplified estimation of stiffness and biaxial strengths of woven carbon-epoxy composites. Douglas Paper 7632, presented to 31st International SAMPE Symposium and Exhibition, Las Vegas, Nevada, 7–10 April 1986. In: Closed Session Proceedings, pp. 83–102 (paper subsequently cleared for unlimited release). [11] Peterson DA, Hart-Smith LJ. A rational development of lamina-to-laminate analysis methods for fibrous composites. Douglas Paper 7928, presented to 9th ASTM Symposium on Composite Materials: Testing and Design, Sparks, Nevada, 27–29 April 1988. In: Garbo SP, editor. Proceedings, ASTM STP 1059, February 1990. pp. 121–64. [12] Sun CT, et al. Comparative evaluation of failure analysis methods for composite laminates. FAA Technical Report DOT/FAA/AR-95/109, 1996.

642 [13] Hinton MJ, Kaddour AS, Soden PD. A comparison of the predictive capabilities of current failure theories, judged against experimental evidence. Compos Sci Technol, V62, pp 1752–1797, 2002, see also Chapter 6 in this book.

Appendix. Corrigenda There is an error in Fig. 10 of Ref. [1], in regard to the truncations associated with a compressive laminate strength less than that in tension. Both ends of the failure envelope should have been pointed, as at the equal-biaxial-tensile-strain point, since all fibres are necessarily equally critical at the equal-biaxial-compression point also. The corrected figure is shown on here.

Fig. A1. Corrected perspective view of failure envelope for (0°/ ± 45°/90°)s composite laminate, according to the Ten-Percent Rule.

Section 4 A Comparative Study of Failure Theories and Predictions for Fibre Polymer Composite Laminates: Part (A)

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Failure Criteria in Fibre Reinforced Polymer Composites © 2004 QinetiQ Ltd. Published by Elsevier Ltd. All rights reserved

CHAPTER 4.1

A comparative study of failure theories and predictions for fibre polymer composite laminates: Part (A) A. S. Kaddoura, M. J. Hintonb and P. D. Sodenc a

QinetiQ, Farnborough, Hampshire, UK b QinetiQ, Fort Halstead, Kent, UK c UMIST, Manchester, UK

Abstract This article presents a unique and comprehensive comparison of the predictive capabilities of internationally recognised failure theories for fibre reinforced polymer composite laminates. The theories were applied by their originators (or their collaborators) to carefully selected test cases that covered a wide range of lay-ups, materials and in-plane loading conditions. Key features in each theory are identified including: types of failure models employed, whether linear or nonlinear analysis was carried out, reliance on software and numerical methods, allowance for thermal stresses and identification of modes of failure. The results (initial and final failure envelopes and representative stress–strain curves) have been superimposed to show similarities and differences between the predictions of the various theories. In addition, bar charts have been constructed to demonstrate the levels of agreement between the predicted initial and final failure stresses and strains. Sources of discrepancies between the predictions of the various failure theories are discussed.

Introduction There have been a number of previous articles surveying work on the biaxial behaviour of composites and comparing failure theories for fibre reinforced plastic laminates. These include reviews presented by Chamis [1], Sandhu [2], Owen and Griffiths [3], Soni [4], Tsai [5,6], Rowlands [7], Nahas [8], Chen and Mathews [9] and chapters in textbooks on composites, e.g. Tsai [6] and Daniel and Ishai [10]. Despite the extensive work reported over a period covering the last four or five decades on the subject of failure in composites, a meeting of experts as recently as 1991, Ref. [11], had indicated that there is no faith in the current failure criteria. In an attempt to establish the predictive capability of current design and failure methodologies, an activity, known as the World-Wide Failure Exercise, Ref. [11], was launched and organised by the present authors. In the exercise, Ref. [11], the originators of a variety of failure theories have used their own theory to predict the performance of specified carbon and glass fibre reinforced epoxy

645 laminates subjected to a range of biaxial loads, using the same given material properties, laminate arrangements and loading conditions, Ref. [12]. In all, fourteen Test Cases were analysed, all of which were defined by the organisers of the exercise. Because laminates are often able to carry higher loads after the first damage occurs, the participants were asked to predict initial and final failure loads and, in some cases, stress–strain curves up to final failure. Each contributor has described their theory and analysed fourteen different cases of biaxial failure envelopes and stress–strain curves in some detail in a separate paper, Refs [13–24, 26–30]. Their predictions were made ‘blindly’, i.e. without prior knowledge of the experimental data, see also Refs [25,31]. This article is aimed at providing a systematic comparison of the ‘blind’ predictions in order to identify some of the major similarities and differences between the theories supplied. The predictions from each contributor are superimposed onto combined graphs to facilitate direct comparison. Reasons for the key overall differences between the predictions are then discussed. 2. Description of the failure theories employed in the exercise 2.1. Identification of the theories Numerous failure theories are available for fibre-reinforced composites, Refs [1–10]. The contributors to this exercise were selected to represent a variety of the most important approaches, Ref. [11]. Table 1 lists the participants and indicates the nineteen approaches they employed. For the sake of simple identification and quick referencing, each of the theories is referred to by a single name in this book, see the last column in Table 1. For those papers that are single-authored, the name of the theory is that of the author. However, for those papers that are co-authored, the name of one key author is given, although this key author may not be the sole originator of the theory. For example, the strain energy based theory employed in the paper by Wolfe and Butalia, Ref. [23], is referred to as Wolfe’s theory although it originated from the previous collaboration with Sandhu and colleagues at Wright Patterson Air Force Base, see for instance Ref. [2]. Some of the participants (Hart-Smith, Sun and Chamis) presented more than one method of solving the test problems. Hart-Smith provided three contributions to the current exercise. One is a Generalized Tresca model, Ref. [18], which considers shearing of the fibres as one of the dominant failure modes and is referred to here as Hart-Smith(1) theory. The second paper, Ref. [17], contains an interpretation of the widely used Maximum Strain and Truncated Maximum Strain failure theory. This Maximum Strain theory is referred to here as HartSmith(2) theory. The Truncated Maximum Strain theory is a subset of this. The third submission, Hart-Smith(3), describes a simple and easy-to-use theory, based on an extension of Hart-Smith’s original ‘Ten-Per-Cent rule’, Ref. [26]. Sun and Tao, Ref. [21], presented results for linear and nonlinear analyses which are referred to as Sun(L) and Sun(NL), respectively. Their detailed nonlinear analysis, which is based on finite element analysis and allows for elastic/plastic material properties and progressively increasing matrix crack density, was used only to predict stress–strain curves. Their linear analysis was applied to generate the failure envelopes and stress–strain curves. Chamis, Ref. [13], and his group (Gotsis and Minnetyan) at NASA employed two computer programs (a) ‘Integrated Composite Analyser’ (ICAN) and (b) ‘Composite Durability

646

Table 1 Summary of the participants and approaches represented in the exercise. Contributor(s)

Organisation

Approach represented

Theory designation

Chamis C C, Gotsis P K and Minnetyan L, Ref. [13] Hart-Smith L J, Ref. [18] Hart-Smith L J, Ref. [17] Eckold G C, Ref. [14]

NASA Lewis, Cleveland, USA

– ICAN (micro-mechanics based) – CODSTRAN Generalised Tresca theory Maximum Strain Theory British Standard pressure vessel design codes British Aerospace, In-house design method Physically based ‘Damage Mechanics’ Physically based 3-D phenomenological models Maximum strain energy method, due to Sandhu Linear and nonlinear analysis (nonlinear is FE based) Development of Maximum stress theory

– Chamis(1) – Chamis(2) Hart-Smith(1) Hart-Smith(2) Eckold

Edge E C, Ref. [15] McCartney L N, Ref. [16] Puck A and Schürmann H, Ref. [19] Wolfe W E and Butalia T S, Ref. [23] Sun C T and Tao J X, Ref. [21] Zinoviev P, Grigoriev S V, Labedeva O V and Tairova L, Ref. [24]. Tsai S W and Liu K-S, Ref. [22]. Rotem A, Ref. [20]. Hart-Smith L J, Ref. [26]. Cuntze R and A Freund, Ref. [28] Bogetti T, C Hoppel, V Harik, J Newill and B Burns, Ref. [27] Mayes S J and A C Hansen, Ref. [29] Z-M Huang, Ref. [30]

Boeing, USA Boeing, USA AEA Technology, UK British Aerospace, Military Aircraft Division, Warton, UK National Physical Laboratory, London, UK Technische Hochchule, Darmstadt, Germany Department of Civil Engineering, Ohio State University, Ohio, USA Purdue University School of Aeronautics & Astronautics, West Lafayette, Indiana, USA Institute of Composite Technologies, Orevo, Moskovkaya, Russia Aeronautics and Astronautics Department, Stanford University, California, USA Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa, Israel Boeing, USA MAN Technologies, Germany

Edge McCartney Puck Wolfe Sun(L) Sun(NL) Zinoviev

Interactive progressive quadratic failure criterion Interactive matrix and fibre failure theory

Tsai Rotem

Ten-Per-Cent rule Failure mode concept (FMC)

Hart-Smith(3) Cuntze

U.S. Army Research Laboratory, AMSRL-WM-MB, Aberdeen Proving Ground, MD 21005

3-D Maximum strain

Bogetti

U.S. Naval Surface Warfare Center, West Bethesda, MD, and Alfred University Department of Engineering Mechanics, Tongji University, Shanghai, China

Multi-continuum micro-mechanics theory

Mayes

Anisotropic plasticity and generalised max stress

Huang

647 Structural Analyser’ (CODSTRAN), which is integrated with a finite element package that allows analysis of complex structures. Results are presented for each program. ICAN results are referred to as Chamis(1) and CODSTRAN results as Chamis(2).

2.2. Characteristics of the theories Each of the theories can be characterised by a number of key features which are categorised in Table 2 as follows: 2.2.1. Method of analysis Column 2 of Table 2 indicates whether classical laminate theory or a finite element (FE) program was used. All of the contributors utilised classical laminate theory as the principal ‘calculation engine’ on which to base their theory. Sun(NL), Mayes and Chamis also used finite element codes. 2.2.2. Type of analysis This refers to whether linear or nonlinear material properties were considered. Chamis, Eckold, Hart-Smith, McCartney, Tsai, Sun(L) and Zinoviev used linear analysis of material properties whereas the rest (Edge, Rotem, Puck, Wolfe, Sun(NL), Bogetti, Cuntze, Huang and Mayes) used nonlinear analysis. 2.2.3. Thermal stresses A significant portion of the participants (Eckold, Hart-Smith, Rotem, Wolfe, Zinoviev and Bogetti) did not include residual thermal stresses while the others did, but not all in the same way or to the same extent. For example, Huang was the only contributor to provide an analysis that considers residual stresses at the micro-mechanics (i.e. resin and fibre) level. 2.2.4. Modes of failure The level of ability to identify specific modes of failure encountered during loading is indicated in Column 5. Almost all of the theories were able to discriminate between two or more modes of failure. Various modes of failure and failure criteria were postulated in the theories and this detail is further amplified in Table 3. The modes of failure basically range from fibre failure (tension, compression or shear) to matrix failure (due to transverse tension, transverse compression, shear or a combination of these three). In most cases the theories used the lamina properties (provided to all participants) from which to predict the modes of failure rather than constituent (fibre or matrix) properties. However, Mayes, Huang and Chamis used the constituent fibre and matrix properties (also provided to all participants) for determining failure. Although the primary failure criterion employed by Tsai (see Table 3) does not distinguish between modes of failure, a secondary criterion is used to distinguish between fibre and matrix failure to direct the subsequent path of the analysis procedure.

648

Table 2 A summary of key features of the theories used by contributors. Contributor

Method

Type of analysis

Thermal stresses

Failure Modes

Micromechanics

Degradation model

Failure criterion

Computer program used

Chamis(1) Chamis(2) Eckold Edge Hart-Smith(1) Hart-Smith(2) Hart-Smith(3) McCartney

CLT CLT + FE CLT CLT CLT CLT CLT CLT

Linear Linear Linear Nonlinear Linear (c) Linear (c) Linear Linear

(a) (a) No Yes No No No Yes

Yes(b) Yes(b) No Yes Yes Yes Yes Yes

Yes Yes No No Yes Yes No No

No Yes No Yes No No No Yes

Micromechanics based Micromechanics based BS4994 Grant-Sanders Generalised Tresca criteria. Maximum strain theories Ten-per-cent rule. Fracture mechanics

Puck

CLT

Nonlinear

Yes (d)

Yes

Yes

Yes

Puck’s theory

Rotem Sun(L) Sun(NL)

CLT CLT(e) CLT + FE (f)

Nonlinear Linear Nonlinear

Yes Yes Yes

Yes Yes No

Yes No No

Yes Yes Yes

Tsai Wolfe Zinoviev Bogetti Mayes Cuntze Huang

CLT CLT CLT CLT CLT + FE CLT CLT

Linear Nonlinear Linear Nonlinear Nonlinear Nonlinear Nonlinear

Yes(g) No No No (a) Yes(d) Yes(a)(h)

Yes Yes Yes Yes Yes Yes Yes

Yes No No No Yes No Yes

Yes Yes Yes Yes Yes Yes Yes

Rotem theory Rotem-Hashin theory Plasticity model based on Hill’s yield Tsai-Wu quadratic theory Sandhu’s strain energy model Maximum stress theory 3D Maximum strain theory Multi-continuum theory Failure mode concept (FMC) Generalised max stress + plasticity

ICAN CODSTRAN None Modified ESDU package None None None Program developed at NPL, UK FRACUAN developed in Kessel, Germany In-house program In-house program ABAQUS program

(a) (b) (c) (d) (e) (f) (g) (h)

Not in all cases. The theory identifies failure modes but Chamis chose not to present them for some of the cases he analysed. Secant properties rather than initial properties are occasionally used in the analysis. Only part of the thermal residual stresses was considered. Used to generate the failure envelopes and stress–strain curves. The finite element (FE) analysis was used only to generate the stress–strain curves. Tsai introduced a certain amount of moisture to compensate for the thermal stresses. Huang attempted to consider the micro thermal stresses generated in the constituents.

Mic-Mac In-house program STRAN software In-house program FE-based In-house program In-house program

Table 3 Modes of failure and failure criteria used by the participants. (See the participants’ papers in Chapters 3 and 5 for definition of the parameters used in this table). Mode of failure

Failure criterion

Fibre failure

±

Theory

K1f I 21f + K4f I4f = 1 where ± K1f = ±

1 1 , K4f = 2 and I1f and I4f are fibre stress S 211f S 12f

Mayes

invariants. Fibre failure

eq =

再

Huang

(1), when (2) ≤ 0, [((1))q + ((2))q] , when (6) > 0, 1 < q ≤ & 1 q

where (1) and (2) are fibre stresses along and transverse to the fibre direction

冉

冊

Fibre failure in tension

1 v 1 + f 12 mf 2 = 1 1T Ef 1

Puck

Longitudinal tension failure

1 = XT (and Cuntze E储, 1 = XT )

Longitudinal tensile failure

1 = 1T (and Eckold’s 1 = 0.004)

Zinoviev, Rotem, Sun(L), Edge, Hart-Smith(3), Cuntze Hart-Smith(2), Eckold and Bogetti Tsai

Fibre tension/compression (No modes of failure identified in primary criterion) Fibre failure (in tension and compression) Fibre failure in compression

1 XT XC

2

+

兰ˆ 1 1 d1 兰u1 1 d1

1 1C

1 +

2 YTYC

2

+

m1

i = 1,2,6

vf 12 m f 2 Ef 1

冊冉冊

1 1 1 1 2F1212 12  1 +  2 + + XT XC YT YC S 12 兹XT XCYTTC

兰ˆ i i di 兰ui i di

mi

=1

Wolfe

≥ 0.1

= 1  (10 21)2

1 = XC (and Eckold 1 = XT)

2

Puck

Zinoviev, Edge, Rotem, Sun (L), Hart-Smith(1)(3), Cuntze and Eckold

649

Longitudinal compressive failure

冉 冊冉 冊冉 冊 冉 冊 冉 冋 册 冒冘冋 册 冏冉 冊冏

650

Table 3 Continued. Mode of failure

Failure criterion

Theory

Longitudinal compressive failure Shear of fibres Transverse tensile failure

1 = 1C (and Eckold 1 = 0.004)

Hart-Smith(2), Eckold and Bogetti Hart-Smith(1) Zinoviev, Edge, Eckold and Sun(L) McCartney

Non-conventional Tresca type criterion, see Ref. [18] 2 = YT

Transverse tensile cracking  + kt > Transversal tensile failure (IFF1) Transverse tensile failure

冑

2 Eff ⊥ · R t⊥

4p

+ ¯ 0 where 2 is fracture energy, see also Ref. [16] 1 1  EA(2p) EA( p) = 1 (and Eff = 1)

2 = 2T (and Eckold 2 = 0.001)

Transverse tension Transverse compression Transverse compressive failure

YT = XT /10 or XC /10 whichever is the greatest YC = XT /10 or XC /10 whichever is the greatest 2 = YC

Transverse compressive failure

2 = 2C (and Eckold 2 = 0.001)

Inter-fibre failure Mode A (for transverse tension)

冑冉 冊 冉

Inter-fibre failure Mode B (for moderate transverse compression)

1 1 (兹221 + ( p(⊥储 )2)2 + p(⊥储 )2) = 1  S21 1D

21 S21

2

+ 1  p(⊥储+ )

冊冉 冊

YT S21

2

2 YT

2

+ p(⊥储+ )

2 1 =1 S21 1D

Cuntze Hart-Smith(2), Eckold and Bogetti Hart-Smith(3) Hart-Smith(3) Zinoviev, Edge, Eckold, Sun(L) and Huang Hart-Smith(2), Eckold and Bogetti Puck

Puck

Table 3 Continued. Mode of failure

Failure criterion

Inter-fibre failure Mode C (for large transverse compression)

冋冉

Wedge failure (IFF3) Inter-fibre failure (IFF2) In-plane shear failure In-plane shear failure Combined transverse tension and shear Combined longitudinal compression and shear Delamination Matrix failure

Theory

冊 冉 冊册

21 () 2(1 + p⊥⊥ )S21

2

+

2 YC

2

YC 1 =1 (–2) 1D

(b⊥  1)(2 + 3) b⊥(2  3)2 + b⊥储 · 221 + = 1 (and Eff = 1) Eff ⊥ · R c⊥ (Eff ⊥储 · R c⊥)2 321 + b⊥储22221 = 1 (and Eff = 1) (Eff ⊥储 · R⊥储)3 S12 = XT/20 or XC/20 whichever is the greatest 12 = S12

冉冊 冉冊 冉 冊冏冏 2 YT

2

+

12 S12

2

Puck

Cuntze Cuntze Hart-Smith(3) Zinoviev, Edge, Hart-Smith(1), Hart-Smith(2), and Sun(L) Edge

=1 Edge

1 12 + =1 H1c S12

12
te > r ± K3mI3m + K4mI4m + 1 where ± K3m = ±

1 1 , K4m = 2 and I3m and I4m are 2 ± 22 2 S 22m + S 33m S 12m

Edge Mayes

matrix stress invariants.

Matrix failure

冉冊 冉冊 2 Y

2

+

12 S12

Chamis and Sun

2

= 1 (Chamis obtains Y and S12 from micromechaincs)

651

652

Table 3 Continued. Mode of failure Matrix failure

Matrix failure

Matrix failure

Theory

Failure criterion

冉 冊 冉冊 冉冊 冋 册 冒冘冋 册 再 Em(1)1 Ym

兰ˆ 1 1 d1 兰u1 1 d1

2

+

2 Y

2

+

m1

i = 1,2,6

12 S12

Rotem

2

=1

兰ˆ i i di 兰ui i di

Wolfe

mi

< 0.1

 , when (2) ≤ 0, where (1) and (2) are matrix [((1))q + ((2))q] , when (6) > 0, 1 < q ≤ & (1)

eq =

1 q

stresses along and transverse to the fibre direction

Huang

653 2.2.5. Micromechanics Overall, approximately half of the theories relied on micro mechanics in their formulation. Chamis, Hart-Smith(1), Puck, Rotem, Tsai, Mayes and Huang, explicitly required certain properties of the individual fibre and matrix constituents. • Unlike most of the other contributors, Chamis’ theoretical approaches use micro mechanics relations to predict the lamina properties from those of the constituents. The relevant equations for calculating ply properties and stress limits can be found in the ICAN code, Ref. [13]. Using ICAN, Chamis adjusted the constituent properties to give lamina properties close to those provided in the exercise and then used the derived lamina properties to predict laminate behaviour. • Hart-Smith(1) theory uses micro mechanics to obtain the failure strain limits needed for establishing the failure envelopes in the strain plane, which can be then converted into an appropriate envelope in the stress plane. Among the properties needed are fibre volume fraction, fibre modulii, fibre Poisson’s ratios and the modulus and Poisson’s ratio of the matrix. • Tsai’s theory uses micro mechanics to carry out progressive post failure analysis. • Rotem’s theory requires the knowledge of the matrix modulus and the tensile and compressive strengths for use in his matrix failure criterion. • Puck’s theory requires the properties of the fibres (strengths, modulii and Poisson’s ratios) and the modulus of the matrix. They are used to compute, for instance, a stress magnification factor that is needed for establishing biaxial failure envelopes. Cuntze’s theory, like Puck’s, requires some of the properties of the fibres. • Huang’s theory, like Chamis’, relies on micro mechanics for determining the unidirectional (UD) lamina properties. Huang used the nonlinear shear stress/strain curves of the matrix to determine the nonlinear behaviour of the lamina. Using a ‘bridging’ model, Huang adjusted the constituent properties to give lamina properties close to those provided in the exercise • Mayes’s theory is also micro-mechanics based. Finite element analysis is employed to predict lamina properties from constituent properties, assuming linear elastic fibres and non-linear elastic matrix behaviour and allowing one constituent to fail while the other remains intact. 2.2.6. Post-initial failure degradation models The entry in the seventh column of Table 2 specifies whether or not the theory used any degradation model to account for post initial failure behaviour. Multidirectional laminates subjected to uniaxial or biaxial stresses may still be capable of carrying load after first ply failure or initial failure has occurred. Modelling post failure behaviour of a laminate requires that certain assumptions be made regarding the properties of the degraded lamina. Table 4 summarises the different post initial failure models adopted. All of these models shared the following common features • All rely on ply-by-ply analysis. • All assume that fibre failure, be it in tension or in compression, constitutes final failure.

654

Table 4 Summary of the post initial failure degradation models used in the theories. Name Eckold Hart-Smith(1) Hart-Smith(2) Hart-Smith(3) Rotem McCartney Puck

Failure mode

Chamis

After final matrix failure Lamina cracking Cracking under tension Mode (A) Cracking under compression Modes (B) and (C) Matrix failure

Edge

Matrix failure

Wolfe Sun (L)

Matrix failure Shear matrix failure Transverse matrix failure Matrix shear failure

Sun (NL)

Tsai

Transverse matrix failure Matrix failure (2 > 0) Matrix failure (2 0)

Properties degraded No post failure No post failure No post failure No post failure E2 = 0.0, G12 = 0.0, E1 = E01 exp(–k 1 ) , k is a large constant Detailed mathematical analysis for reducing stiffness. E2 =  E02 , G12 =  G012, 12 =  012,  is a parameter which varies with stress G12 = 1 G012, 12 = 012 1 is smaller than  Em is replaced by a negligible value and E2, G12 , 12 and E1 are computed from micromechanics E2 = #1 E02, G12 = #2 G012, 12 = #3 012 Where #1 , #2 and #3 are empirical parameters that decrease with increasing strain E2 = 0.0, G12 = 0.0, 12 = 0.0 E2 = 0.0, G12 = 0.0 E2 = 0.0 E2 = E02 exp(–E) and G12 = G012 exp(–G), E and G are constants,  is normalised crack density E2 = E02 exp(–E) Em = 0.15 E0m, 12 = 0.15 012, E2 and G12 are computed from micromechanics E2 = 0.01 E02, G12 = 0.01 G012, 12 = 0.01 012, E1 = 0.01 E01

Table 4 Continued. Name

Failure mode

Properties degraded

Zinoviev

Open Cracks 2 > 0

For | 12 | < | *12 | (a) when 2 < *2: E2 = 2 E02 , G12 = 3 G012 where  is a function of strain (b) when 2 = *2: E2 = 0.0, G12 = 3 G012 For | 12 | = | *12 | (a) when 2 < *2: E2 = 2 E02 , G12 = 0.0 (b) when 2 > 0: E2 = 0.0, G12 = 0.0. For 2 < 0: (a) when | 12 | < | *12 |: G12 = 3 G012 (b) when  | 12 | > 0 : G12 = 0.0 G12 = 0.0 E2 = 0.0 Curves describing ‘softening’ behaviour Em = Ef = 0.0 resulting in E2 = 0.0, G12 = 0.0, 12 = 0.0, E1 = 0.0 Em = 0.01 E0m, E2 and G12 are computed from micromechanics

Bogetti Bogetti Cuntze Huang Mayes

Closed cracks 2 < 0 Shear failure Transverse strain failure IFF1, IFF2 and IFF3 Matrix or fibre Failure Matrix failure

655

656 • Almost all models distinguish between failures under transverse tension and that under transverse compression. Here the term ‘transverse’ is used to refer to the direction perpendicular to the fibres in a unidirectional fibre reinforced lamina. The post failure methods employed do differ and, for the sake of simplicity, they can be classified into three main groups: (i) Models employing no post failure analysis. These include Eckold and Hart-Smith. (ii) Models employing sudden reduction in the properties of the failed lamina. These are utilised by Tsai, Wolfe, Sun(L), Chamis, Bogetti, Huang and Mayes. (iii) Models employing a gradual drop in the properties of the failed lamina. These were utilised by Cuntze, Puck, Edge, Rotem, Zinoviev, McCartney, and Sun(NL). Full details of each model are given in the contributor’s papers, Refs [13–24,26–30], see also Chapter 3. 2.2.7. Failure criterion Column 8 of Table 2 identifies the origin, nature or the name of the failure theory used. The governing equations that constitute the various failure criteria are of course described, by the participants in their own papers (see also Table 3 for a brief summary). The criterion used by Sun in his linear analysis (Sun(L)) is attributed to Hashin and Rotem but was actually slightly different from that employed by Rotem himself in his contribution to the exercise. Zinoviev’s theory uses a Maximum Stress criterion with further development to cater for various post failure scenarios. It should be noted that the exercise features two variants of the Maximum Strain Theory. Hart-Smith, Ref. [17], provided his 2-dimensional interpretation as used in the aircraft industry, without incorporating any initial failure prediction or progressive failure analysis. The additional paper by Bogetti, Ref. [27], applied the theory in its original 3-dimensional form together with a progressive failure model. 2.2.8. Title of computer program used In view of the complexity of many of the theories, most of the participants have developed their own computer code and the entry in the last column of Table 2 lists the name of the computer program used by the participants. The participants have either coupled their analysis into a commercial code (Chamis, Mayes, Sun(NL)), or written their own (Tsai, Zinoviev, Puck, McCartney, Rotem, Edge, Sun(L), Wolfe, Cuntze, Huang and Bogetti) or used simple calculations without a computer code (Hart-Smith(1), Hart-Smith(2), Hart-Smith(3) and Eckold).

2.3. Breadth of cases analysed by each participant The participants were set 14 Test Cases for analysis and these are summarised in Table 5. The composite systems, laminate configurations and load combinations for the Test Cases have been described in detail in Ref. [12]. Table 6 shows the Test Cases attempted by each

Table 5 Details of the laminates and loading (Test Cases). Laminate lay-up

Material

Test Case

Description (a wide range of biaxial stress ratios unless otherwise indicated)

0°

(90°/ ± 30°/90°)

E-glass/LY556/HT907/DY063 T300/BSL914C E-glass/MY750/HY917/DY063 E-glass/LY556/HT907/DY063

(0°/ ± 45°/90°)

AS4/3501-6

± 55°

E-glass/MY750/HY917/DY063

(0°/90°) ± 45°

E-glass/MY750/HY917/DY063 E-glass/MY750/HY917/DY063

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Biaxial failure stress envelope under transverse and shear loading (y versus xy) Biaxial failure stress envelope under longitudinal and shear loading (x versus xy) Biaxial failure stress envelope under longitudinal and transverse loading (y versus x) Biaxial failure stress envelope (y versus x) Biaxial failure stress envelope (x versus xy) Biaxial failure stress envelope (y versus x) Stress–strain curves under uniaxial tensile loading in y direction (y : x = 1 : 0) Stress–strain curves for y : x = 2 : 1 Biaxial failure stress envelope (y versus x) Stress–strain curves under uniaxial tensile loading for y : x = 1 : 0 Stress–strain curves for y : x = 2 : 1 Stress–strain curve under uniaxial tensile loading for y : x = 0 : 1 Stress–strain curves for y : x = 1 : 1 Stress–strain curves for y : x = 1 : –1

657

658

Table 6 Summary of the Test Cases analysed by the participants. Theory

1 McCartney* Sun(NL) Hart-Smith(2) Hart-Smith(1) Eckold Hart-Smith(3) Chamis Edge Puck Rotem Sun(L) Tsai Wolfe Zinoviev Cuntze Bogetti Huang Mayes

Remarks

Loading Cases analysed (see Table 2 for details of these Cases)

X X 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐

2 X X 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏

X X 䊐 䊐 X 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐

3 X X 䊏 䊏 X 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏

X X 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐

4 X X 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏

X X X X 䊐 X 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐

5 X X 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏

X X X X X X 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐

6 X X 䊏 䊏 X 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏

X X X X X X 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐

7 X X 䊏 䊏 X 䊏 X 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏

X 䊐 X X X X 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐

8 X 䊊 X X X 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏

X 䊐 X X X X 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐

9 X 䊊 X X X 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏

X X X X 䊐 X 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐

10 X X 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏

X 䊐 X X 䊐 X 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐

11 X 䊊 X X X 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏

X 䊐 X X 䊐 X 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐

12 X 䊊 X X X 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏

X Case not analysed. 䊊 Final failure not reached. 䊐 Initial failure predicted only. 䊏 Final failure predicted only. * See McCartney, Ref. [37], and Chapter 5 for other Tests Cases solved subsequently using newly modified approach.

䊐 䊐 X X 䊐 X 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐

13 䊊 䊊 X X X 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏

䊐 䊐 X X 䊐 X 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐

14 䊊 䊊 X X X 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏

X 䊐 X X 䊐 X 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐 䊐

X 䊏 X X X 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏

No final failure No stress–strain curves No stress–strain curves No carbon fibre results

659 contributor. The majority of the participants were able to analyse all of the cases. However, there were some notable shortfalls: • Chamis did not present a final failure envelope for the quasi-isotropic laminate (Test Case 6). • Eckold did not analyse the carbon fibre composites because his simple design method was developed specifically for glass fibre reinforced pressure vessels. He also chose not to analyse shear loading of laminates (Test Case 5) because that type of loading is not covered by the pressure vessel design code. • Hart-Smith did not provide solutions for the seven test Cases involving stress–strain curves. Also he did not predict initial failures in either of the two theories Hart-Smith(1) and HartSmith(2), although that would be possible for some laminates using other interpretations of the Maximum Strain theory (see for example Ref. [27]). He did provide final failure predictions for the stress–strain curves using the 10% rule (Hart-Smith(3)). • The analytical method used by McCartney is in its infancy and at the time of writing his first contribution [16] he was able to apply it to only two cases, (Test Case 12) a 0°/90° laminate under uniaxial tension (SR = y : x = 0 : 1) and (Test Case 13) a ± 45° laminate under SR = 1 : 1 which is equivalent to a 0°/90° laminate under biaxial tension. He did not attempt to predict final failure loads for those cases. It should be noted here that, at a later stage in the exercise, McCartney subsequently modified his model, in his Part B submission, so that other Test Cases could be solved. He then provided stress–strain curve predictions for ± 55° and quasi isotropic laminates (Test Cases 7, 8, 10 and 11), see Ref. [37] and Chapter 6, Ref. [38].

3. Comparing the predictions The task of comparing the predictions of the 19 theories for each of the 14 Test Cases is undoubtedly complex due to the large number of possible differences and similarities between the magnitudes, shapes and the nature of the predicted values. However, in order to facilitate the task of comparing the predictions, the following approach was taken: • The 14 Test Cases can be broadly grouped into three classes (a) biaxial failure of unidirectional laminae (Test Cases 1–3), (b) biaxial failure envelopes of multidirectional laminates (Test Cases 4–6, 9) and (c) stress–strain curves of laminates under uniaxial and biaxial loading (Test Cases 7, 8, 10–14). • Graphs were produced containing the superimposed predictions of all the various theories for each Test Case. • In addition, bar charts of strengths (and, where appropriate, strains) predicted by all of the theories were constructed at specific conditions of loading within the 14 Test Cases. • The largest differences between the predictions of all the theories for some of the loading combinations are recorded in Tables 7 and 8. Brief observations are made on the differences between the predictions for each Test Case before proceeding to discuss more general issues in Section 4 and draw conclusions in Section 5.

660 3.1. Biaxial failure envelopes for unidirectional laminae (Test Cases 1, 2 and 3) The first three Test Cases were to predict failure envelopes for isolated unidirectional plies under combined loading. One of the objectives of studying these Test Cases was to demonstrate the robustness (or otherwise) of the assumptions employed in each theory in the simplest of situations (i.e. a lamina) before considering their merits in more complex laminates. Figs 1a, 2a and 3a compare the failure stresses predicted by each theory for (i) E-glass/LY556 lamina under combined shear and direct loading perpendicular to the fibres (y and xy), (ii) T300/914C lamina under combined shear and direct stresses parallel to the fibres (x and xy) and (iii) E-glass/MY750 lamina subjected to biaxial stresses applied in directions parallel and perpendicular to the fibres (x and y). The bar charts in Figs 1b, 2b and 3b show the failure strengths for each theory at the selected stress ratios indicated in Figs 1a, 2a and 3a. Due to the differences between lamina failure criteria employed, the predicted biaxial failure envelopes described a variety of shapes. The following comments can be made regarding the shape of the envelopes and magnitude of the predicted stresses. • In all of the theories used, excepting those of Eckold, Hart-Smith(2), Huang, Bogetti and Hart-Smith(3), the predicted failure envelopes passed through the values of lamina strengths under uniaxial loading which were provided as part of the data pack (Ref. [12]) to all participants. • Some of the theories, for example the Maximum Stress theory used by Zinoviev, assumed that the strength in one direction was independent of the magnitude of applied stress in the Table 7 Ratios of the highest : lowest predicted lamina strengths for selected stress ratios in Test Cases 1–3. No. Laminate studied Stress ratio

Final failure prediction Highest

1 2

E-glass/LY556 (Test Case 1)

3 4

T300/914C

5

(Test Case 2)

6 E-glass/MY750 7 (Test Case 3) 8 9 10 11 12

Highest : lowest ratio predictions Lowest

y : xy = –1.58 : 1 Edge, Hart-Smith(2), Zinoviev Eckold y : xy = 1 : 2.06 Hart-Smith(2), Zinoviev, Tsai Eckold Huang Eckold y : xy = 0 : 1

3.22 1.54

x : xy = –12.5 : 1 Zinoviev, Sun, Hart-Smith(1), Edge Hart-Smith(2) x : xy = 18.75 : 1 Edge, Zinoviev, Hart-Smith(1) Chamis (2), Sun

1.64

x : y = 8.83 : 1 x : y = –32 : 1 x : y = 7.8 : 1 x : y = 0 : 1 x : y = –1 : 0 x : y = –3.15 : –1 x : y = –14 : –1

Edge, Zinoviev and Sun Eckold Hart-Smith(2), Bogetti Hart-Smith(3) Eckold Bogetti Tsai

Eckold Wolfe Rotem Eckold Wolfe Eckold Eckold

3.6

1.414

3.6 4.77 4.4 3.2 3.8 6.9 2.72

661 Table 8 Summary of theoretical results showing the range of initial and final failure predictions in Test Cases 4–14. No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 23a 23b 24 25 26

Test Case

4

5

6

9

13 14 12

Stress ratio

Final failure prediction

Largest final/initial prediction

Initial failure prediction

Highest

Lowest

Ratio

Highest

Lowest

Ratio

Name

Ratio

y :  x = 1 : 3 y : x = 1 : –1 y : x = –1 : –3 y : x = 1 : 1 y : x = –1 : –1 y :  x = 1 : 0 y : x = –4.26 : 1

Hart-Smith(1) Hart-Smith(1) Eckold Hart-Smith(2) Edge Hart-Smith(1) Zinoviev

Rotem Chamis(2) Zinoviev Wolfe Tsai Wolfe Chamis(2)

6.2 8.42 3.1 7 1.66 2.42 2.11

Huang Huang Huang Huang Huang Huang Zinoviev

Chamis(2) Chamis(2) Bogetti Chamis(2) Eckold Chamis(2) Chamis(2)

13.3 4.7 3.2 9.19 4.55 5.2 7.46

Edge Eckold Eckold Chamis(2) Eckold Edge Eckold

34 5.7 3.74 19 3.66 7.5 5

x : xy = –2.35 : 1 x : xy = 1 : 1 x : xy = 0 : 1 x : xy = –1 : 0 x : xy = 1 : 0

Puck Hart-Smith(1) Puck Eckold Eckold

Chamis(2) Wolfe Chamis(2) Tsai Chamis(2)

3.8 5.21 4.71 3.58 3.13

Huang Huang Huang Huang Huang

Chamis(2) Chamis(2) Chamis(2) Eckold Chamis(2)

4.5 11.6 7.8 7.3 15.5

Puck Edge Puck Eckold Eckold

1.99 12.8 6 12 24

y : x = 0 : –1 y : x = 1.5 : –1 y : x = –1 : –1 y : x = 2 : 1 y :  x = 1 : 0 y : x = 1 : 1

Huang Zinoviev Tsai Sun Zinoviev Zinoviev

Wolfe Wolfe Wolfe Huang Chamis(2) Huang

1.88 2.71 1.72 3 2 4.45

Sun Huang Tsai Bogetti Huang Bogetti

Chamis(2) Chamis(2) Bogetti Chamis(2) Chamis(2) Chamis(2)

9.23 22.6 3.05 17.6 17.6 21.2

Huang Edge Bogetti Chamis(2) Chamis(2) Edge

1.54 15.7 2.15 51 24 28

y : x = 0 : –1 y : x = –1 : 0 y : x = –2 : –1 y : x = 2 : 1 y :  x = 1 : 0 y : x = 2.728 : 1 y : x = 1.33 : 1

Zinoviev Eckold Eckold Hart-Smith(1) Eckold Sun Hart-Smith(2)

Eckold Chamis(2) Chamis(2) Wolfe Chamis(2) Rotem Wolfe

2.53 3.61 4.13 8.7 4.47 7.05 13

Edge Sun Huang Huang Puck Huang

Eckold Eckold Bogetti Chamis(2) Eckold Chamis(2)

3.64 4.95 3.8 5.7 4.88 15.53

Eckold Eckold Eckold Chamis(2) Eckold

1.47 9.6 5 19 9.55

y :  x = 1 : 1 y : x = 1 : –1 y : x = 0 : 1

Chamis 2 Edge Puck

Wolfe Chamis(2) Wolfe

9.7 5.8 2.28

Huang* Huang Huang**

Cuntze Chamis(2) Edge

3.72 2.47 3.85

Puck Edge Puck

13 4 12

* Eckold terminated his curves at a slightly higher stress than that of Huang. ** If Huang’s results without thermal stresses are considered then the ratio would be 5.5.

other directions. These theories produced rectangular failure envelopes in Figs 1a, 2a, 3a and the highest predictions of strength at some stress ratios (see for example Fig. 1b). • Eckold’s method gave similar results to Zinoviev’s, except that the compressive strengths were set to be equal to the tensile strengths. For the isolated lamina, the effects of equating the compressive strength to the tensile strength were that the theory under-predicted the transverse compressive strengths in Figs 1 and 3 by a factor of around 3 and over-predicted the longitudinal compressive strength in Fig. 3. The latter is unconservative, which is usually considered undesirable in design methods. Note, however, that Eckold shows an inner envelope that differs from the final envelope. His additional inner envelopes correspond to a strain of 0.1% in the direction perpendicular to the fibres and 0.4% in the fibre direction.

662

Fig. 1a. Biaxial failure stress envelope for 0° unidirectional E-glass/LY556 epoxy lamina under transverse and shear loading (y versus xy). Test Case (1).

• In Fig. 3, Bogetti does not give a unique prediction for the uniaxial compression strength parallel to the fibres. He supplied two values for the uniaxial compression strength that were both similar to those by predicted by Wolfe, Ref. [23], i.e. respectively, equal to and about one half of the uniaxial compressive strengths for that lamina. Note also that Wolfe produced a discontinuous envelope as the stresses approach uniaxial tension as well as uniaxial compression along the fibre direction (i.e. at x : y = 1 : 0 and –1 : 0). • Bogetti shows an inner envelope that differs from the outer envelope for Test Cases No. 1 and No.3 in which the initial failure occurs due to exceeding the ultimate strain in the through thickness direction.

663

Fig. 1b. Bar charts showing the biaxial failure stresses for a unidirectional E-glass/LY556 epoxy lamina under y : xy = –1.58 : 1, 1 : 2.08 and 0 : 1. Test Case (1).

• Some theories predicted envelopes in which the effective strength in one direction is influenced by the applied stress in the other direction. (i.e. they employed ‘interactive’ failure criteria). These theories include Tsai, Rotem, Sun, Wolfe, Cuntze, Mayes, Bogetti and Huang. • Huang predicts an unusual sharp peak in the compression quadrant of the failure envelope shown in Fig. 1. • Tsai’s theory predicted an elliptical failure envelope with a large enhancement in the longitudinal compressive strength in the y  x space (Fig. 3). It is a feature of his equation that the predicted biaxial compressive strength increases when the value of the uniaxial transverse tensile strength is reduced. • Hart-Smith(2) and Bogetti predict a significant enhancement in the strength under certain ranges of biaxial tension-tension and compression-compression stress states in y  x space (Fig. 3). Indeed, Test Case 3 (i.e. Fig. 3) highlights the largest differences between the various theoretical predictions for lamina strength. Table 7 shows that at stress ratios around x : y = –3.15 : –1, the transverse compressive failure strength (y) predicted by Bogetti’s theory is more than 6 times higher than that predicted by Eckold. This is partly due to the low transverse compressive strength assumed by Eckold. In the tension-tension quadrant, the largest difference between the strength values predicted occurs at stress ratio of x : y = 7.8 : 1. At this stress ratio, the transverse tensile failure strength, y , predicted by Bogetti (and Hart-Smith(2)) is more than 4.4 fold higher than that predicted by almost all of the other theories (see Fig. 3b).

664

Fig. 2a. Biaxial failure stress envelope for a unidirectional T300/BSL914C lamina under longitudinal and shear loading (x versus xy). Test Case (2).

• One of the most common forms of interaction assumed that combined stresses reduced the strength of a lamina to a value lower than its strength under uniaxial loading. Figures 1, 2 and 3 show many examples of this such as Rotem, Sun, Huang, Mayes, Cuntze and Wolfe. In Fig. 3, Wolfe’s theory predicts strengths under combined transverse tension and longitudinal compression that are about one third of the uniaxial strengths for that lamina. • In the tension-tension quadrant, Huang predicts stress interaction that suppresses the strength by more than the other theories. In can be seen from Figs 1, 2 and 3 that although there were some similarities, no two theories gave identical shaped envelopes for the three Tests Cases.

665

Fig. 2b. Bar charts showing the biaxial failure stresses for a unidirectional T300/914C lamina under x : xy = –12.5 : 1 and 18.75 : 1. Test Case (2).

3.2. Failure envelopes for multi-angled laminates (Test Cases 4, 5, 6 and 9) In order to examine the performance of the failure theories at the laminate level, four Test Cases were chosen covering a wide range of materials, lay-ups and biaxial loading conditions. 3.2.1. Failure envelopes for the (90°/ ± 30°)s E-glass/LY556 laminate under biaxial loads, (y versus x), (Test Case 4) As described in Ref. [12], 82.8% of the thickness of this (90°/ ± 30°)s E-glass/LY556 laminate is made of ± 30° plies and only 17.2% of the thickness made of 90° plies. Thus, the laminate is not quasi-isotropic. Consequently, the failure envelopes are expected to lack symmetry, in relation to the material principal loading directions, and also to contain a number of failure modes depending upon the loading direction. Initial failure envelopes predicted by all the theories are shown in Fig. 4a and the final failure envelopes are shown in Fig. 4b. These figures show that the contributors predicted a wide range of different strengths for this laminate. Bar charts in Fig. 5 compare the initial and final failure stresses predicted by all the theories at specific ratios of applied biaxial loads y : x. Table 8 shows the range of predicted values at selected load ratios, including those in the bar charts. With the exception of Bogetti, Eckold, Chamis(2) and Wolfe, the initial failure envelopes shown in Fig. 4a are approximately diamond shaped, but the magnitudes of the predicted initial failure stresses vary considerably from theory to theory. Huang’s envelope was the

666

Fig. 3a. Biaxial failure stress envelope for 0° unidirectional lamina made of E-glass/MY750 epoxy under longitudinal and transverse loading (y versus x). Test Case (3).

largest of all, due to his high predicted lamina strengths (as indicated in Fig. 1). The initial failure predictions were influenced, among other factors, by whether or not residual thermal stresses had been taken into account in a given theoretical method. For instance, the inclusion of thermal stresses by Cuntze, Edge and Chamis(2) led to the lowest predicted initial failure stresses for most of the tensile quadrants of Fig. 4a. The largest deviation between the theoretical predictions (more than a factor of 13 (Huang: Chamis)) occurred at a biaxial tensile stress ratio of y : x = 1 : 3 (see Table 8). Eckold’s results were higher than nearly all the other initial failure stress predictions at that stress ratio (see Fig. 4a).

667

Fig. 3b. Bar charts showing the biaxial failure stresses of E-glass/MY750 lamina under y : x = 7.8 : 1, –32 : 1, –14 : –1, –3.75 : –1 and 8.83 : 1. Test Case (3).

Although no through thickness loading was applied Bogetti predicted low initial failure stresses in the biaxial compression quadrant due to through thickness strains generated by Poisson’s ratio effects that exceed the allowable through-thickness tensile failure strain. Examination of Fig. 4b (and the results for SR = 1 : 1 in Fig. 5) shows that Rotem, Wolfe and Huang’s final failure predictions are very low in the biaxial tension quadrant, compared with those of other theories due to their methods of post failure analysis. The bar charts in Fig. 5 show that at certain stress ratios (e.g. for biaxial compression y : x = –1 : –1) most of the theories, except that of Bogetti, predicted final failure stresses of similar magnitude to the initial failure stresses, whilst at other stress ratios (e.g. y : x = 1 : 0 and 1 : 1) the final stresses are predicted to be much greater than the initial failure stresses. In

668

Fig. 4a. Initial biaxial failure stress envelope for (90°/ ± 30°/90°) laminate made of E-Glass/LY556 epoxy under combined loading (y versus x). Test Case (4).

669

Fig. 4b. Final biaxial failure stress envelope for (90°/ ± 30°/90°) laminate made of E-Glass/LY556 epoxy under combined loading (y versus x). Test Case (4).

670

Fig. 5. Bar charts showing the biaxial failure stresses of (90°/ ± 30°/90°) E-glass/LY556 laminate under y : x = –1 : –3, 1 : 1, –1 : –1 and 1 : 0. Test Case (4).

Table 8 the largest ratio of final : initial failure loads predicted by any one author for this laminate was 34 : 1, by Edge at y : x = 1 : 3. Table 8 shows that the highest ratio of maximum : minimum final failure stresses was 8.4 : 1 (Hart-Smith(1) : Chamis(2)) at stress ratio y : x = 1 : –1. Examination of Fig. 4b shows that Hart-Smith(1) and Hart-Smith(2) theories gave similar final failure envelopes. Eckold predicts the highest (x) compressive strengths in the biaxial compression quadrant. This is as a result of his starting assumption to use a value for the longitudinal lamina compressive strength equal to the longitudinal lamina tensile strength and hence a much larger longitudinal compressive strength than the other participants. 3.2.2. Combined direct and shear loading (x versus xy) of the (90°/ ± 30°)s E-glass/LY556 epoxy laminate (Test Case 5) Figures 6a and 6b show, respectively, the initial and final failure envelopes predicted by the theories for the (90°/ ± 30°)s E-glass/LY556 laminate under combined direct x and shear xy

671

Fig. 6a. Initial biaxial failure stress envelope for (90°/ ± 30°/90°) E-glass/LY556 laminate under combined direct and shear loading (x versus xy). Test Case (5).

loads. Failure stresses for all of the theories at selected stress ratios are shown in the bar charts in Fig. 7. The magnitude of the predicted shear strengths of the laminate varied greatly from one theory to the next. Huang’s initial failure envelope for this test case (Fig. 6a) was again by far the largest predicted by any theory in the exercise (see also Fig. 7 and Table 8). The Tsai, Rotem, Wolfe, and Chamis final failure envelopes tended to be smaller than all of the others (see Fig. 6b). Generally, the majority of theories, showed an enhancement in the final shear strength under the application of moderate tensile loads. Figure 7 shows that the predicted final failure loads were always larger than the initial ones for this type of loading. The biggest difference between initial and final failure loads occurred under uniaxial tension x : xy = 1 : 0. A difference of a factor of 18 was predicted by Chamis, and a factor of approximately 15 by Puck and Edge.

672

Fig. 6b. Final biaxial failure stress envelope for (90°/ ± 30°/90°) E-glass/LY556 laminate under combined direct and shear loading (x versus xy). Test Case (5).

There were also differences between the theories in predicting modes of failure for the same stress ratio. For final failure, the predicted modes of failure at SR = 1 : 1 were transverse tension in the + 30° plies (Wolfe), transverse compression in the + 30° plies (Zinoviev),

673

Fig. 7. Bar charts showing the biaxial failure stresses of (90°/ ± 30°/90°) E-glass/LY556 laminate under combined direct and shear loading x : xy = –2.35 : 1, 1 : 1, 0 : 1 and 1 : 0. Test Case (5).

delamination and local buckling in –30° plies (Puck) and longitudinal tension in the + 30° plies (Hart-Smith(1), Hart-Smith(2) and Sun(L)). The failure load predicted by Hart-Smith(1) was more than 5 times higher than that by Wolfe. 3.2.3. Biaxial envelope for (0°/ ± 45°/90°)s AS4/3501-6 carbon/epoxy quasi-isotropic laminate under combined y and x (Test Case 6) This family of laminates is typical in aircraft structures and is frequently known as ‘black aluminium’. The initial failure envelopes for this quasi-isotropic laminate are presented in Fig. 8a and the final failure envelopes are shown in Fig. 8b. Comparison between the initial and final stresses at selected stress ratios is shown in Fig. 9, for all of the theories. 3.2.3.1. Initial failure envelopes As one might expect, all of the envelopes are symmetric about the 1 : 1 diagonal. The strength under equal biaxial compression is similar for all of the theories except that for Tsai’s interactive theory, which predicts higher biaxial compression strength than the other theories

674

Fig. 8a. Initial biaxial failure stress envelope for (0°/ ± 45°/90°) AS4/3501-6 laminate under combined loading (y versus x). Test Case (6).

(see the bar charts for y : x = –1 : –1 in Fig. 9). Bogetti predicts a lower strength than the others due to through-thickness failure (once again caused by a through-thickness Poisson’s strain effect) at this stress ratio. There were large differences in magnitude of initial failure strengths predicted in the other quadrants of the failure envelope by the different theories. The biggest ratio of maximum : minimum predicted initial failure strengths shown in Table 8 was 22.6 (Huang : Chamis(2)). Some of these differences may be attributed to the effect of thermal residual stresses, see Table 9. 3.2.3.2. Final failure envelopes The predicted final failure envelopes for the 0°/ ± 45°/90° laminate (Figs 8b) fall broadly into two groups. The majority of theories are in the first group and predict diamond shaped failure envelopes. In the second group (Tsai, Wolfe, Rotem, and Huang), each theory predicts an

675

Fig. 8b. Final biaxial failure stress envelope for (0°/ ± 45°/90°) AS4/3501-6 laminate under combined loading (y versus x). Test Case (6).

676

Fig. 9. Bar charts showing the biaxial failure stresses for (0°/ ± 45°/90°) AS4/3501-6 laminate under y : x = 1 : 0, 1.5 : –1, 2 : 1, –1 : –1 and –1 : 0. Test Case (6).

envelope that is unique in shape and in magnitude. Huang and Rotem predict lower final failure stresses than the other theories in the biaxial tension quadrant and Huang and Tsai

677 Table 9 Examples of the effect of thermal stresses on the initial failure stresses of some laminates. Laminate

0°/90° GRP ± 55° GRP 0°/ ± 45°/90° CFRP 0°/ ± 45°/90° CFRP (a) (b) (c) (d) (e)

SR

1:0 2:1 1:1 1.5 : –1

Mode of failure

Transverse tension Transverse tension Matrix failure Matrix failure

Initial failure stresses (MPa) without thermal stresses

with thermal stresses

78 (a) 112 (a) 318 (d) 276.5 (e)

55.4 (b) 68.6 (b) 35 (b) 12.24 (c)

Zinoviev. Edge. Chamis(2). Bogetti. Huang.

predict larger strengths than any of the other theories in different parts of the biaxial compression quadrant. The largest difference between the theoretical predictions of final failure strength (Zinoviev : Huang = 4.45 : 1) occurred under biaxial tensile loading (y : x = 1 : 1). The bar charts in Fig. 9 show the variation between predictions for other stress ratios. Almost all of the theories predicted the initial failure strength to be the same as the final failure strength (i.e. a single, catastrophic failure) over the whole of the biaxial compression quadrant of the failure envelope. The notable exceptions were Bogetti and Huang’s theories. All of the theories predicted final failures that were different from the initial failures when tensile loads were applied (except for Hart-Smith, who did not predict initial failure). The greatest ratio of final : initial failure loads shown in the whole of Table 8 was 51 : 1, predicted by Chamis(2) for this quasi-isotropic laminate loaded at y : x = 2 : 1. That result was taken from the stress–strain data (Test Case 8 presented below) as Chamis did not provide data for the final failure envelope for this laminate. 3.2.4. Biaxial envelope for ( ± 55)s E-glass/MY750 epoxy laminate under combined y and x (Test Case 9) The ± 55° angle-ply laminate is commonly employed in pipes and pressure vessels. The initial failure envelopes are shown in Figs 12 and the final envelopes in Fig. 13. The bar charts in Fig. 14 compare initial and final failure stresses at selected loading ratios. The various theories gave a range of results for the initial failure loads. Edge, Rotem, Chamis(2), Puck and Cuntze all predict very low initial failure strengths in biaxial tension. Huang’s envelope was the largest of all (see Fig. 12). Bogetti again predicted low initial failure stresses under biaxial compression due to a through-thickness Poisson’s strain effect. Eckold’s design approach is seen to be conservative in comparison with the other theories over a wide range of stress ratios, but predicts higher initial strengths than the others for a small range of biaxial tensile loads (e.g. compare with Puck).

678 The biggest difference between predicted initial failure envelopes was at the stress ratio of 2.73 : 1 where the ratios of predicted initial failure strengths was 15.5 : 1 (Huang : Chamis(2)), see Table 8. An equally striking range of predictions was evident for the final failure envelopes for the ± 55° angle ply laminate, although the stress ratios at which the largest deviations took place are different. Figure 13 shows a wide variety of shapes of final failure envelopes with Mayes, Rotem and Wolfe giving much lower final failure strengths than the other theories in the biaxial tension quadrant (see for instance the bar chart in Fig. 14 for y : x = 2 : 1). The biggest discrepancy was at y : x = 1.33 : 1 where the ratio of predicted final failure stresses was 13 : 1 (Hart-Smith(2) : Wolfe, Table 8). With the exception of Sun, Rotem, Edge, Puck and Bogetti, all of the theories predicted initial and final failures as being coincident events in the biaxial compression quadrant, see Fig. 14. However, many of the theories predict very large differences between the initial and final failure loads in the tension-tension quadrant. The largest difference between initial and final failure shown in Table 8 is at y : x = 2 : 1 where most of the theories predict matrix tension failure at low stress and final failure due to tensile fracture along the lamina fibre direction at high stress. The initial and final strengths differ by a factor of up to 19 (Chamis(2)) in this instance.

3.3. Stress–strain curves (Test Cases 7, 8, 10–14) The participants were asked to predict the shape of stress–strain curves for a number of laminates and loading conditions. One objective was to demonstrate the predicted effects that initial failure and any subsequent change in laminate behaviour had on the stress–strain curves. 3.3.1. Stress–strain curves for (0°/ ± 45°/90°)s AS4/3501-6 carbon/epoxy quasi-isotropic laminate under uniaxial tension y : x = 1 : 0 and biaxial tension y : x = 2 : 1. (Test Cases 7 and 8) The stress–strain curves for these loading cases are shown in Figs 10 and 11. Under uniaxial loading (Fig. 10), all of the curves are very similar in shape except for those of Chamis and Rotem. In Rotem’s analysis, there is an abrupt increase in strain associated with initial failure and in Chamis’s curve, there is a large increase in the strain associated with final failure. Most predictions showed only a small reduction in stiffness after initial failure. The initial failure stress was in the range of 15 MPa (Chamis(2)) to 265 MPa (Huang) whilst the final failure stress was in the range of 385 (Chamis(2)) to 728 MPa (Zinoviev). Chamis(2), Huang and Wolfe’s final failure stress predictions were lower than the others (see Fig. 9). Failure was predicted to take place in one stage by all of Hart-Smith’s theories, two stages by Edge and Puck, three stages by Bogetti, Huang, Mayes, Cuntze, Wolfe, Zinoviev and Tsai and in four stages by Sun. The first failure was usually predicted as due to transverse tension in the plies with fibres perpendicular to the applied load and the final stage by tension along the fibres in the plies parallel to the loading direction. For Test Case 8 (Fig. 11), the stress–strain curves for the quasi-isotropic laminate under biaxial tension (SR = y : x = 2 : 1) predicted by nearly all of the various theories were also remarkably similar to one another, with a very small change in slope after initial failure, as

679

Fig. 10. Stress–strain curves for (0°/ ± 45°/90°) AS4/3501-6 laminate under uniaxial tensile loading in y direction (y : x = 1 : 0). Test Case (7).

680

Fig. 11. Stress–strain curves for (0°/ ± 45°/90°) AS4/3501-6 laminate under biaxial tensile loading (y : x = 2 : 1). Test Case (8).

681

Fig. 12. Initial biaxial failure stress envelope for angle ply ± 55° E-glass/MY750 epoxy laminate under combined loading (y versus x). Test Case (9).

682

Fig. 13. Final biaxial failure stress envelope for angle ply ± 55° E-glass/MY750 epoxy laminate under combined loading (y versus x). Test Case (9).

683

Fig. 14. Bar charts showing the biaxial failure stresses for ± 55° E-glass/MY750 laminate under y : x = 1 : 0, 2 : 1, –2 : –1 and –1 : 0. Test Case (9).

can be seen from Fig. 11. Huang and Rotem were exceptional. They showed a step in the y and x curves and their stress–strain curves were truncated at very low stresses, a characteristic arising from their methods of post failure analysis. The initial and final failure stresses predicted by all of the theories are compared in Fig. 9, for SR = y : x = 2 : 1. The initial failure stress ranged from 15 MPa (Chamis(2)) to 264 MPa (Bogetti) while the final stress ranged from 280 MPa (Huang) to 840 MPa (Sun) but the majority of theories gave values of final failure stresses which were close to one another. Stages of failure were similar to those shown for SR = 1 : 0, but the number of stages predicted by some theories increased by one over that described above. Cuntze, Wolfe, Sun, Puck, Tsai, and Zinoviev showed four stages of failure, all predicting initial failure due to transverse tension (matrix failure or Inter-fibre Mode A failure) in the 0° plies, which were perpendicular to the loading direction in this case, and final failure by longitudinal tension in the 90° plies. Failure in the second and third stages occurred in the ± 45° plies and 90° plies respectively with the same mode of failure as that in the 0° plies.

684 3.3.2. Stress–strain curves for ( ± 55)s E-glass/MY750 epoxy laminate under uniaxial tension y : x = 1 : 0 and biaxial tension y : x = 2 : 1. (Cases 10 and 11) For uniaxial tension of the ± 55° laminate (Test Case 10, SR = 1 : 0), the stress–strain curves are shown in Fig. 15, bar charts of predicted failure stresses for all theories in Fig. 14 and for predicted failure strains in Fig. 17. In this case the laminae are, according to linear analysis, subjected to high shear and low tensile stresses in directions perpendicular to the fibres. The transverse stress component changes during loading from being tensile to being compressive when nonlinear analysis is carried out. The initial Young’s modulus is identical in all predictions, except for Hart-Smith(3) who did not provide a prediction of the shape of the stress–strain curve. A large number of the nonlinear theories predicted curves with failure strains in the region of 2–3% except for Huang who predicted a larger strain at final failure (x = –5.33%). Edge and Chamis predicted very different curves from the rest, with very large strains reaching up to 12%, which was the specified limit of the graph provided by the organisers. The predicted final failure stresses for all of the theories, shown in the bar chart in Fig. 14, differ by a maximum factor of 4.5 between Eckold (640 MPa) and Chamis(2) (140 MPa). Note that Eckold’s results were taken from the envelope shown in Fig. 10 of Ref. [14] as his curve shown here in Fig. 15 was terminated at an arbitrary low strain. Nearly all of the participants predicted that initial and final failures in this case were coincident events and that failure was dominated by in-plane shear. Only Zinoviev predicted separate failure loads with the initial failure due to shear and final failure due to compression perpendicular to the fibres. Figs 16 shows the stress–strain curves for biaxial tension Test Case 11 (for the ± 55° angle ply laminate at y : x = 2 : 1). The participants had predicted a very wide range of results for this case, with a particularly large variation in the strains predicted in the x direction. Fig. 16 shows that Rotem, Wolfe, Huang and Mayes produced curves that were truncated at very low stresses and strains. Some of the other curves were bilinear and some (Edge and Sun(NL)) highly nonlinear. In Sun’s linear analysis, the x strain curve intersected the y strain curve at around y = 1% while Zinoviev predicted an intersection point near final failure, i.e. at y = 2.8%. The initial and final failure strains in the x and y directions predicted by all of the theories are shown as bar charts in Fig. 17. All predicted failure strains were tensile, except one final failure strain calculated by Edge. In Edge’s analysis, the strain x started as a positive value and became negative at high strains, see Fig. 16, while the strain y was comparable with those predicted by others. The range of predicted initial failure stresses was 45–276 MPa (see Fig. 14) with general agreement that the mode of failure (where identified) at the lamina level was due to transverse tension (i.e. tensile stresses perpendicular to the fibres). Some of these differences may again be attributed to the effect of thermal residual stresses, see Table 9. The predicted final failure stress ranged from 112 to 993 MPa, again a very wide range.

685

Fig. 15. Stress–strain curves for ± 55° E-glass/MY750 under uniaxial tensile loading with y : x = 1 : 0. Test Case (10).

686

Fig. 16. Stress–strain curves for ± 55° E-glass/MY750 laminate under biaxial tensile loading with y : x = 2 : 1. Test Case (11).

687

Fig. 17. Bar charts showing failure strains for (a) Test Case (11): ± 55° E-glass/MY750 under y : x = 2 : 1 (strain in y direction), (b) Test Case (10): ± 55° E-glass/MY750 under y : x = 2 : 1 (strain in x direction) and (c) Test Case (14): ± 45° E-glass/MY750 under y : x = 1 : –1 (strain in y direction).

3.3.3. Stress–strain curves for (0°/90°)s E-glass/MY750 epoxy laminate under uniaxial tension y : x = 0 : 1. (Test Case 12) The stress–strain curves for uniaxial tensile loading of a 0°/90° laminate predicted by all the contributors are shown in Fig. 18 and a comparison between the predicted failure stresses for

688

Fig. 18. Stress–strain curves for (0°/90°) E-glass/MY750 laminate under uniaxial tensile loading with y : x = 0 : 1. Test Case (12).

689 all of the theories is shown in Fig. 19c. All of the predictions in Fig. 18 are superficially rather similar but a close examination of the results does reveal some differences. For example, the y failure strain ranged from + 0.05% (Sun (L)) to-0.776% (Rotem). McCartney’s stress– strain curve in Fig. 18 resembles the others but he did not attempt to predict final failure and terminated the curve at an arbitrary point. Eckold truncated his curve at a much lower arbitrary strain. The final failure stress was in the range of 293–714 MPa with most of the failure theories (Zinoviev, Sun(L), Rotem, Puck, Edge, Chamis, Cuntze, Bogetti, Mayes and Hart-Smith(3)) predicting final failure by tensile fracture of the fibres in the 0° layers at a laminate stress of around 640 MPa. If it is assumed that the failed 90° plies carry no load, this equates to a failure stress level in the 0° laminae of 1280 MPa (i.e. the value for the uniaxial tensile strength of the unidirectional ply as provided by the organisers, Ref. [2]) . The final failure loads predicted by Huang, Tsai and Wolfe were lower than the others (see Fig. 19c). It was noted from the work of Huang, Ref. [30], that his final strength prediction increased from 488 MPa to 690 MPa when the thermal stresses are neglected. In all cases where predictions of both initial and final failure stress were provided by participants the final stresses were much larger than the initial stresses, the largest ratio of final : initial stresses being more than 11 : 1 for Puck, Edge and Cuntze. 3.3.4. Stress–strain curves for ( ± 45)s E-glass/MY750 epoxy laminate under equal biaxial tension y : x = 1 : 1. (Test Case 13) This Test Case, where equal biaxial tension is applied to a ± 45° laminate, is equivalent to a 0°/90° laminate loaded under equi-biaxial tension. It differs from Test Case 12 in that the x and y strains are expected to be of the same sign and of equal magnitude, such that cracking would be expected to occur in all of the layers at an intermediate stress level well below final failure. The stress–strain curves are shown in Fig. 20 and bar charts comparing the predicted initial and final failure stresses for all theories are presented in Fig. 19a. McCartney showed separate curves for cracking of the inner and outer layers but he stopped his calculation before final failure and did not attempt to predict final failure. The initial and secondary slopes predicted by all the theories were remarkably similar to one another, McCartney’s being the exception. His secondary stiffness was noticeably higher than the rest. Chamis predicted a reduction in stiffness and a large strain (approximately 17%) at final failure (see Fig. 20). In general, the predictions indicated that the laminate would fail initially, in all layers, by transverse tension failure, at a stress of 49–175 MPa. Final failure would occur either by fibre failure (Chamis, Edge, Puck, Sun, Zinoviev, Cuntze, Bogetti, Hart-Smith(3)) at a laminate stress up to 714 MPa or by matrix failure (Mayes, Huang, Rotem and Wolfe) at a laminate stress below 175 MPa due to the post failure models they employed. Tsai predicted final failure at an intermediate level due to the failure criterion adopted. 3.3.5. Stress–strain curves for ( ± 45)s E-glass/MY750 epoxy laminate under biaxial tensioncompression y : x = 1 : –1. (Test Case 14) This case is equivalent to a 0°/90° laminate subjected to pure in-plane shear. The stress–strain curves are superimposed in Fig. 21. The predicted failure stresses and strains are shown in the bar charts in Figs 19b and 17c, respectively. Some of the contributors (Tsai, Sun (L), Eckold,

690

Fig. 19. Bar charts showing failure stresses for (a) Test Case (13): ± 45° E-glass/MY750 under y : x = 1 : 1, (b) Test Case (14): ± 45° E-glass/MY750 under y : x = 1 : –1 and (c) Test Case (12): 0°/90° E-glass/MY750 under y : x = 1 : 0.

Chamis and Hart-Smith(3)) predicted linear stress–strain curves and failure at very low strains. Puck, Mayes, Wolfe, Rotem, Sun(NL), Cuntze, Bogetti, predicted failure in shear at about 2–3% strain but Huang showed an extended stress–strain curve up to 10% strain at failure. Zinoviev showed unusual behaviour after initial failure, predicting final failure strains up to 8%, Fig. 21. The initial failure was by shear whereas the final failure was by transverse compression. Edge’s curves are very peculiar and different from the others. His predicted failure strains reached 15%, far higher than any other strain predicted by the rest of the participants. Wolfe, Tsai, Sun, Huang, Bogetti, Cuntze, Mayes, Hart-Smith(3) and Chamis predicted final failure stresses to be the same as the initial failure stresses but Rotem, Zinoviev, Puck and

691

Fig. 20. Stress–strain curves for ± 45° E-glass/MY750 laminate under biaxial tensile loading with y : x = 1 : 1. Test Case (13).

692

Fig. 21. Stress–strain curves for ± 45° E-glass/MY750 laminate under biaxial tensile loading with y : x = 1 : –1. Test Case (14).

693 Edge predicted final failure stresses that were higher than the initial failure stresses (see the bar chart in Fig. 19b).

4. Discussion Theoretical results, predicted by the participants in the failure exercise, have been presented in the form of superimposed failure envelopes, bar charts, and superimposed stress–strain curves for 0°, (90°/ ± 30°)s, (0°/ ± 45°/90°)s, ( ± 55°)s, ( ± 45°)s and (0°/90°)s laminates. The differences between the predictions of all the theories for each of the cases analysed have been described. Among the factors that have affected the predicted failure envelopes and stress– strain curves for the laminates are: • Differences in the failure criteria applied at the lamina or constituent level. • Allowances made in some theories for the differences in the lamina strength when tested in isolation and when embedded in a laminate. • Inclusion of residual thermal stresses as a result of the curing process. • Type of analysis (linear or nonlinear). • Differences between post failure models and mechanisms. These factors are discussed in more detail below.

4.1. Differences between lamina failure criteria Some of the characteristics of the different failure criteria, which were observed in the failure envelopes for unidirectional fibre reinforced laminae, are reflected in the failure envelopes for the multidirectional laminates. The high failure strength of the quasi-isotropic (0°/ ± 45°/90°)s AS4/3501-6 carbon/epoxy laminate under equal biaxial compressive load predicted by Tsai in Fig. 8 results from the enhanced strength that he predicts for unidirectional laminae under biaxial compression (e.g. Fig. 3). The enhancement of the lamina biaxial transverse strength, predicted by Hart-Smith(2), is not shown in the results for multidirectional laminates presented here. That is primarily due to the fact Hart-Smith chose to omit the constant transverse strain failure lines from his laminate failure envelopes and did not predict initial failures for reasons stated in his papers [17,18]. However, Bogetti’s theory [27], which also represents the maximum strain theory, does show the effects of the predicted enhancement in the lamina biaxial tension strength on the initial failure of multi-directional laminate, see for instance the initial failure envelope of ( ± 55°) GRP laminates, Fig. 12. Bogetti [27] predicted initial failures in the unidirectional fibre reinforced laminae Fig. 1 and Fig. 3 and in quasi-isotropic carbon fibre (Fig. 8) and ± 55° angle ply glass fibre (Fig. 12) laminates at lower stresses than the other theories when they were subjected to in-plane compression loads. In all cases this was predicted due to the increase in thickness exceeding the allowable strain in that direction, despite the fact that there were no applied throughthickness stresses.

694 In Eckold’s predictions, the effects of equating longitudinal compressive and tensile strengths of a lamina are reflected in the failure envelopes for the multi-directional laminates. For example, the failure strength of (90°/ ± 30°) and ± 55° (Figs 4b and 13) and laminates was higher than that predicted by the other participants in the compression-compression quadrant. BS4994 [33], which uses Eckold’s method, does allow for different tensile and compressive strengths to be employed if measured values are available. Using the compressive strength values provided in the exercise would remove some of the anomalies from Eckold’s results for laminae and laminates. Huang’s higher predictions for transverse tensile and shear strengths of unidirectional fibre laminae (Fig. 1) contributed to high predictions of initial failure strengths for multi-directional laminates (see Figs 4a, 6a, 12 and 21). A number of theories showed that the biaxial lamina strength under compressive load in one direction combined with tension in the other direction is smaller than that under uniaxial load, see Fig. 3. This is reflected for example in the low initial failure stresses predicted by Tsai compared with those of Zinoviev in the following areas: (i) (0°/ ± 45°/90°)s AS4/3501-6 carbon/epoxy laminate under equal biaxial compressivetension loading of y : x = 1 : –1, see Fig. 8b, (ii) ( ± 30°/90°)s glass/epoxy laminate under y : x = –2.35 : 1, Fig. 4b, (iii) ( ± 30°/90°)s glass/epoxy laminate under x : xy = –4.25 : 1, Fig. 6b. In the last two cases, using the same input data, Tsai’s theory predicted initial failure due to a combination of longitudinal compression and transverse tension within the laminae whereas Zinoviev’s theory predicted failure due to in-plane lamina shear.

4.2. Properties of an embedded lamina Some of the contributors chose to differentiate between the behaviour of an isolated lamina and that of a lamina embedded within a laminate. They assumed that the strength (occasionally the stiffness) of a lamina embedded in a laminate is higher than that of an isolated lamina. For example, in his analysis of multidirectional laminates, Sun, after studying work by Bailey et al. [34] and Flaggs and Kural [35], assumed new values for the shear and transverse tensile strengths of the laminae, which were 50% higher than those provided by the organisers and employed by other participants. Following the findings of Rotem and Hashin [36], Rotem assumed that, for all of the laminae in the exercise, the shear and transverse strengths, as well as the corresponding stiffness of the embedded laminae, increased by 20% above their values measured on isolated laminae. He took the measured isolated lamina strengths as being the cause of initial cracking in the constrained lamina and the modified strengths as controlling the onset of final failure. Thus, Rotem predicted, for instance, that ± 45° glass/epoxy laminate, whose behaviour is shear dominated under SR = 1 : –1, finally failed at an applied stress of 87.6 MPa, which is 20% higher than the initial failure stress of 73 MPa. He also predicted that the ± 45° glass/ epoxy laminate, subjected to equal biaxial tension loading, would fail initially at a stress corresponding to a lamina transverse tension strength of 40 MPa and that the final failure was also by transverse tension but at a load corresponding to 48 MPa transverse lamina stress, i.e. 20% higher than that at initial failure.

695 On the basis of his experience with carbon/epoxy materials and their use in the aircraft industry, Hart-Smith assumed theoretical limits to the transverse strengths of embedded laminae, see Refs [17,18], which were much higher than those suggested by Rotem and Sun. The clearest consequence of the high embedded lamina strength can be seen in the results of the Hart-Smith(1) and Hart-Smith(2) predictions for the biaxial failure envelopes of the ± 55° glass/epoxy, (90°/ ± 30°)s glass/epoxy and (0°/ ± 45°/90°)s carbon/epoxy laminates. In each of these envelopes predicted by Hart-Smith, the transverse lamina strengths were raised to such an extent that the matrix tensile cracking (initial failure) mode never occurred. Those theories that make the assumption of a gradual decrease in lamina stiffness after initial damage (see Table 4) are actually treating the embedded lamina behaviour as being different from that of the isolated lamina. Clearly an isolated lamina would immediately lose all stiffness at the onset of initial damage (i.e. first crack) because it would then be in several pieces! However, it is not unreasonable to assume a somewhat different response when embedded within a laminate, since the adjacent plies may serve to transfer load around the cracked regions. The impact of these various post initial failure modelling assumptions is discussed further in Section 4.4.

4.3. Residual thermal stresses due to curing Another major factor that affected the strength prediction of the laminates was thermal stresses. All the participants were aware that residual thermal stresses could arise in composite laminates that were made at elevated temperatures and then cooled to the operating temperature. However, only some of the contributors (Chamis, Edge, McCartney, Puck, Sun, Cuntze and Huang) attempted to consider the effects of thermal stresses on their predictions. Puck, for instance, opted for taking into account only 50% of the thermal stresses on the grounds that relaxation with time, due to moisture absorption and the accompanying swelling, leads to reduction of the thermal residual stresses. In this exercise the moisture content of the composites was not specified and was not requested by any participant. Tsai introduced 0.5% moisture content in his calculation and assumed a temperature difference of –100°C in all the laminates he analysed. These two factors tended to cancel each other out, Ref. [22]. Chamis, Ref. [13], was the only one who demonstrated the extent to which the thermal residual stresses affect the biaxial failure envelopes. In his Fig. 11, Chamis plotted the biaxial stress failure envelope for (0°/ ± 45°/90°)s carbon/epoxy laminate with and without thermal stresses. The highest influence was in the tension-tension quadrant of the envelope where the initial failure was dominated by transverse tension failure. Chamis reported that the failure stress at SR = 1 : 1 when the thermal stresses were neglected, was tenfold higher than that when the thermal stresses were included. Table 9 compares the initial failure stresses predicted by Edge and Zinoviev, for three of the laminates loaded in different ways, where the initial modes of failure were all transverse tension (and shear stresses were small, to eliminate other differences between the theories). The large differences between the results of Edge who took thermal stresses into account and Zinoviev, who did not, support Chamis’ findings. The initial shear strength of the (90°/ ± 30°) laminate (Fig. 6a) was influenced by thermal stresses but shear loading of that laminate produced initial failure of the ± 30° layers by transverse tension rather than by shear.

696 Those contributors who took thermal stresses into account used conventional laminate theory to calculate the effects of differential thermal contraction between layers. Only Huang allowed for differential contraction between fibres and matrix within a layer. The experimental measured properties of the unidirectional laminae used as input data would include some effects of residual stresses arising during curing but the issue of residual stresses has not been thoroughly addressed in this exercise.

4.4. Post initial failure models Having briefly described the post failure models used by the participants in Table 4, it is possible to show examples of the effect of post failure modelling on the predicted behaviour of the laminates. The theories that reduced suddenly and simultaneously two or more ply properties, once failure occurs, usually gave lower predictions of final failure loads than other theories. When E2 and G12 are reduced to zero in angle ply laminates, both the initial and final failure stresses are coincident, due to the singularity in the stiffness matrix. For example, Wolfe, Rotem, Mayes and Huang predicted low final failure stresses compared with many other theories for ± 55° GRP laminate loaded under biaxial tension (see Figs 13 and 14b) and for biaxial loading of ± 45° GRP laminates under SR = 1 : 1 (Figs 19a and 20). The mode of final failure in those two cases was by matrix failure rather than the fibre tension predicted by others. After initial failure in transverse tension, in addition to reducing E2 and G12, Rotem also reduced the Modulus (E1) parallel to the fibres in the failed lamina. In the quasi isotropic laminate loaded at SR = 1 : 1, where all layers experience the same stress after initial failure, the value of E1 will be reduced in all the laminae simultaneously and the laminate cannot carry further loads. On the other hand, in the 0°/90° laminate loaded in uniaxial tension (SR = 0 : 1), only the 90° layer fails initially and the 0° layer can still carry load. The effects of different post failure models can also be observed in the stress–strain curves. One case which showed striking differences between the theories was the behaviour of the ± 55° laminate under y : x = 2 : 1 (Fig. 16). Theories which employ a gradual nonlinear reduction in lamina properties after the initial failure (e.g. those of Sun(NL) and Edge) predicted nonlinear behaviour in both the x and the y curves, although, the degree of nonlinearity was more pronounced in the x curve as this is matrix dominated. In the case of Edge’s prediction, which was based on reducing gradually and simultaneously the transverse, shear moduli and Poisson’s ratio, once transverse tension failure occurs, the slope of the x curve changed gradually and eventually the strain switched from being tensile to being compressive. It was also observed that the final portions of Edge’s stress–strain curves in Figs 15 and 21 do not appear to conform to and hence are not consistent with his post failure model described in Table 4. The simultaneous drop to zero of both the transverse and shear modulii in the case of Wolfe, Rotem, Mayes and Huang, as described above, truncated the stress–strain curve. The two models proposed by Sun(NL) and McCartney, which relied on a crack density formulation for predicting the stiffness degradation after initial failure, gave very similar shaped stress–strain curves to nearly all the other theories for the 0°/90° laminate under uniaxial tension, Fig. 18. There was one noticeable difference between McCartney and Sun’s models and that was the transverse strain predicted by Sun(NL), which was larger than that

697 predicted by McCartney. However, when the same two models (Sun(NL) and McCartney) were used to predict the stress–strain curve for the ± 45° laminate under equal biaxial tension, they gave different results. McCartney’s results were stiffer than the others and he did not predict final failure.

4.5. Nonlinear behaviour The main source of nonlinearity in the behaviour of the laminates preceding initial failure was the decrease in in-plane lamina shear stiffness with increasing shear strain, Ref. [12]. The ± 45° laminate loaded at SR = 1 : –1 (Test Case 14) was equivalent to a 0°/90° cross ply laminate loaded in pure shear so the stress–strain curves in Fig. 21 should reflect the given nonlinear shear curve for the unidirectional lamina supplied in Ref. [12]. For that case the initial failure strains in the nonlinear analyses (Puck, Edge, Rotem, Wolfe, Cuntze, Bogetti, Mayes, and Sun(NL)) were around 2%, which is, as expected from a simple Mohr’s Circle analysis, half the shear strain to failure of an isolated lamina. That predicted strain was up to 3.2 fold higher than the initial failure strains in the linear analyses (Zinoviev, Chamis, Sun(L) and Tsai), which produce truncated stress–strain curves. In the laminates whose behaviour is not affected by in-plane lamina shear, linear analysis should give identical answers to nonlinear analysis for initial failure prediction. The behaviour of the ± 45° laminate under SR = 1 : 1 (Test Case 13) and the 0°/90° laminate under SR = 0 : 1 (Test Case 12) are clear examples of stress–strain curves where shear nonlinearity is absent. Zinoviev introduced an additional geometric nonlinearity by allowing for the change of ply angle as deformation progressed. It is this geometric nonlinearity, which results in the slight increase in stiffness at large strains in his stress–strain curves for the ± 45° laminate loaded in ‘pure shear’, Fig. 21. Certain theories (Edge, Cuntze, Sun(NL) and Puck) introduced additional material nonlinearity by assuming that the transverse and shear moduli decreased gradually after initial failure. It is the combined nonlinearities which result in the nonlinear divergence of the strains x in the stress–strain curves predicted by Sun(NL) and Edge for the ± 55° laminate loaded at SR = 2 : 1 (Fig. 16). The inclusion of any of these forms of nonlinear behaviour into the laminate analysis usually requires the introduction of iterative numerical methods of solution.

4.6. Large and unbounded deformations Several of the participants encountered problems in deciding when to terminate numerical solutions that predicted ever-increasing deformations. For example, Edge produced stress– strain curves for ± 55° laminates under SR = y : x = 1 : 0 and SR = 2 : 1 and for the ± 45° laminate under SR = 1 : –1 in which the predicted failure strains were very large compared with those obtained by others. In Chamis’ results, large failure strains were shown for the (0°/ ± 45°/90°)s AS4/3501-6 quasi-isotropic laminate under SR = 1 : 0, in the ± 45° laminate under SR = 1 : 1 and in the ± 55° laminates under SR = 1 : 0. Chamis, Ref. [13], pointed out that the large increase in the strains with very little increase in the stress would happen only in monotonic loading and any

698 fluctuation in the applied load would lead to imminent failure at the onset of that large deformation. There were other cases where large deformations led to open (discontinuous) failure envelopes (i.e. failure envelopes with indeterminate regions). Puck reported that both the initial and final biaxial failure envelopes for the ± 55° laminate were open where large shear strains, combined with transverse tension stress, were induced in the plies. The initial failure envelope was open in the range 7 : 1 > y : x > 2.5 : 1. The final envelope was also open at the same range of stress ratios and in another area between 0 : 1 < y : x < 2 : 1. Zinoviev also reported the occurrence of an open envelope for the final biaxial failure stresses between stress ratios 0 : 1 < y : x < 2 : 1, which is the same as the second range reported by Puck.

5. Conclusions 1. Leading theories have been employed, by their originators, to predict the strength and deformation response in fourteen Test Cases, involving six different fibre reinforced polymer (FRP) laminates (0°, (90°/ ± 30°)s, (0°/ ± 45°/90°)s, ( ± 55°)s, ( ± 45°)s and (0°/90°)s laminates) subjected to a range of in-plane biaxial loading conditions. 2. The exercise revealed that some theories lack flexibility and others require further development. One contributor who adopted a ‘damage mechanics’ approach solved only two of the fourteen test Cases in the first instance. 3. There were many similarities between the predictions of various theories, but also some striking differences. 4. A number of the test cases (Test Cases 1 to 3) were chosen to compare the theories at the most simplistic level, i.e. that of a unidirectional fibre reinforced lamina. On a lamina level, each theory generated a unique biaxial failure envelope for the three Test Cases considered and differences as great as 690% were observed in the strength predictions. Some theories (Huang, Bogetti, Hart-Smith(2), Hart-Smith(3), Eckold and Wolfe) predicted uniaxial strength data of the lamina that were different from those measured and actually provided as input data. The largest deviation between measured and predicted uniaxial strength values (transverse tensile strength) was a factor of 3.2. This highlights some very significant problems within some of these theories. 5. At the next level of complexity, certain Test Cases were chosen by the organisers because they represented material/lay-up/loading situations which were believed to be well understood by the community, and as a consequence, one might have expected some unanimity in the predictions. Test Case number 6 considered a carbon fibre quasi-isotropic laminate, which is the fundamental building block used in aircraft structures, world wide. Test Case number 13 considered a simple ( ± 45°) GRP cross ply, which has been studied extensively by researchers for the last 25 years. Test Case number 11 considered a ± 55° GRP laminate loaded at a stress ratio SR of y : x = 2:1, which is featured in piping and pressure vessels currently in-service throughout the world. The exercise revealed that even for these very familiar cases, the spread in the final failure strengths predicted by the participants was surprisingly large. Ratios of highest : lowest predicted final strengths as great as 450% were observed for the quasi-isotropic aircraft laminate test case, 970% for the cross ply ( ± 45°) laminate test case and 870% for the ± 55° GRP piping/pressure vessel laminate loaded at SR = 2:1 (and 1300% at SR = 1.33:1)

699 6. A number of major issues emerged in predicting the laminate responses: • There was little unanimity between the participants in how to account for the residual thermal stresses resulting from elevated temperature curing of the laminates, particularly when moisture content was not specified. For example, as can be seen from Table 9, predictions of initial failure loads for the quasi-isotropic carbon/epoxy laminate varied (maximum : minimum) by 2260% in one of the worst instances. • For many of the test cases the most extreme predictions of initial failure strength came from two contributors. The contribution from Huang contained the highest predictions, whereas Chamis gave the lowest (possibly because he made full allowance for thermal stresses). • The participants used a variety of methods to predict laminate behaviour after initial failure. The type of post failure modelling employed by Rotem, Wolfe, Mayes and Huang produced much lower final failure strength values than the other contributors in many cases. • Some participants made a distinction between the behaviour of an isolated lamina and its behaviour when embedded in a laminate. Thus, when asked to analyse the Test Cases containing laminates, certain contributors applied correction factors to the lamina properties provided by the organisers (particularly the transverse tensile and shear strengths). They argued that the embedded properties of a lamina would be higher due to the constraint effect of the surrounding layers. This approach leads to significantly higher values of initial failure stress for laminates. • In many instances, the theories differed in the mode (and occasionally the location) of failure each predicted, and this was associated with different predictions of failure strengths. In some cases, the mode of failure was not clearly identified. • Many of the theories utilised some form of nonlinear analysis, to account for inherent material nonlinearity and that associated with post-initial failure. In some instances, there appeared to be evidence of instability in the nonlinear algorithms, leading to the generation of open failure envelopes or unexpectedly large failure strains. By comparison, the alternative linear-elastic analytical approach employed in several theories, led to the prediction of stress–strain curves that appeared to be prematurely truncated, in some instances. A variety of shapes of stress–strain curves were predicted by the linear and nonlinear theories. Differences between the predicted failure strains for some laminates reached as high as 25:1. • Micromechanics featured in several of the theoretical approaches. Three participating groups Chamis, Huang and Mayes, relied on the properties of the fibres and matrices to compute the linear elastic constants, strengths and nonlinear behaviour of the composite laminae. Others required properties of the fibres and/or the matrix to establish failure conditions and to carry out post-initial failure modelling. 7. In this article we have concentrated on drawing attention to differences between the various theories and particularly to differences between their predictions for the selected laminates and biaxial loading cases. The fact that a theory predicts different results from all of the other theories does not necessarily mean that it is wrong. Any judgements as to which theoretical approach is the best should be suspended until the comparison with experimental results is conducted and commented upon. This forms the subject matter of Chapters 5 and 6 of the book and also Ref. [32].

700 References 1 Chamis C C, ‘Failure criteria for filamentary composites’, in Testing and Design, ASTM STP 460, pp 336–351, 1969. 2 Sandhu R S, ‘A survey of failure theories of isotropic and anisotropic materials’, AFFDL-TR72–71, AD756889, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio, USA, 1972. 3 Owen M J and Rice D I, ‘Biaxial strength behaviour of glass-reinforced polyester resins’ in Composite materials: testing and design, 6th Conf, ASTM STP 787, (Edited by I M Daniel), pp 124–144, 1982. 4 Soni S R, ‘A comparative study of failure envelopes in composite laminates’, J Reinf. Plast. Compos., V2, pp 34–42, 1983. 5 Tsai S W, ‘A survey of macroscopic failure criteria for composites materials’ J Reinf Plast Compos, V3, pp 40–62, 1984. 6 Tsai S W, ‘Composite design’, 4th edition, Think Composites, Dayton, Ohio, 1988. 7 Rowlands R E, ‘Strength (failure) theories and their experimental correlation’, in Handbook of composites, Vol 3, failure mechanics of composites, Edited by G C Sih and A M Skudra, Elsevier Science Publishers B V, pp 71–125, 1985. 8 Nahas M N, ‘Survey of failure and post failure theories of laminated fibre reinforced composites’, J Compos Techn Res, V8, pp 138–153, 1986. 9 Chen A S and Matthews F L, ‘A review of multiaxial/biaxial loading tests for composite materials’, Composites, V24, pp 395–406, 1993. 10 Daniel I M and Ishai O, ‘Engineering Mechanics of Composite Materials’, Oxford University Press, Oxford, UK, 1994. 11 Hinton M J, Kaddour A S and Soden P D, ‘The World-Wide Failure Exercise : Its Origin, Concept And Content’, see Chapter 1 in this book. 12 Soden P D, M J Hinton and A S Kaddour, ‘Lamina properties and lay-up configurations and loading conditions of a range fibre reinforced composite laminates’, Compos Sci Tech., V58, pp 1011–1022, 1998, see also Chapter 2.1 in this book. 13 Gotsis P K, C C Chamis and L Minnetyan, ‘Prediction of composite laminate fracture: micromechanics and progressive fracture’, Compos Sci Tech., V58, pp 1137–1150, 1998, see also Chapter 3.1 in this book. 14 Eckold G C, ‘Failure criteria for use in the design environment’, Compos Sci Tech., V58, pp 1095–1106, 1998, see also Chapter 3.2 in this book. 15 Edge E C, ‘Stress based Grant-Sanders method for predicting failure of composite laminates’, Compos Sci Tech., V58, pp 1043–1044, 1998, see also Chapter 3.3 in this book. 16 McCartney L N, ‘Predicting transverse crack formation in cross-ply laminate’, Compos Sci Tech., V58, pp 1069–1082, 1998, see also Chapter 3.4 in this book. 17 Hart-Smith L J, ‘Predictions of the original and truncated maximum strain failure models for certain fibrous composite laminates’, Compos Sci Tech., V58, pp 1151–1178, 1998, see also Chapter 3.5 in this book. 18 Hart-Smith L J, ‘Predictions of a generalised maximum-shear-stress failure criterion for certain fibrous composite laminates’, Compos Sci Tech., V58, pp 1179–1208, 1998, see also Chapter 3.6 in this book. 19 Puck A and H Schürmann, ‘Failure analysis of FRP laminates by means of physically based phenomenological models’, Compos Sci Tech., V58, pp 1045–1068, 1998, see also Chapter 3.7 in this book. 20 Rotem A, ‘Prediction of laminate failure with Rotem failure criterion’, Compos Sci Tech., V58, pp 1083–1094, 1998, see also Chapter 3.8 in this book. 21 Sun C T and J X Tao, ‘Prediction of failure envelopes and stress–strain behaviours of composite laminates’, Compos Sci Tech., V58, pp 1125–1136, 1998, see also Chapter 3.9 in this book.

701 22 Liu K-S and S W Tsai, ‘A progressive quadratic failure criterion of a laminate’, Compos Sci Tech., V58, pp 1023–1032, 1998, see also Chapter 3.10 in this book. 23 Wolfe W E and T S Butalia, ‘A strain energy based failure criterion for nonlinear analysis of composite laminates subjected to biaxial loading’, Compos Sci Tech., V58, pp 1107–1124, 1998, see also Chapter 3.11 in this book. 24 Zinoviev P, S V Grigoriev, O V Labedeva and L R Tairova, ‘Strength of multilayered composites under plane stress state’, Compos Sci Tech., V58, pp 1209–1224, 1998, see also Chapter 3.12 in this book. 25 Soden P D, M J Hinton, and A S Kaddour, ‘Comparison of the predictive capabilities of current failure theories for composite laminates’, Compos Sci Tech., V58, pp 1225–1254, 1998. 26 Hart-Smith L J, ‘Expanding the capabilities of the ten-percent rule for predicting the strength of fibre-polymer composites’, Compos Sci Tech., V62, pp 1515–1544, 2002, see also Chapter 3.17 in this book. 27 Bogetti T A, C P R Hoppel, V M Harik, J F Newill and B P Burns, ‘Predicting the nonlinear response and progressive failure of composite laminates’, Compos Sci Tech., V64, pp 329–342, 2004, see also Chapter 3.13 in this book. 28 Cuntze R G and A Freund, ‘The predictive capability of failure mode concept based strength criteria for multidirectional laminates’, Compos Sci Tech., V64, pp 343–377, 2004, see also Chapter 3.14 in this book. 29 Mayes S J and A C Hansen, ‘Composite laminate failure analysis using multicontinuum theory’, Compos Sci Tech., V64, pp 379–394, 2004, see also Chapter 3.15 in this book. 30 Huang Zheng-Ming, ‘A bridging model prediction of the tensile strength of composite laminates subjected to biaxial loads’, Compos Sci Tech., V64, pp 395–448, 2004, see also Chapter 3.16 in this book. 31 Kaddour A S, M J Hinton and P D Soden, ‘Comparison of the predictive capabilities of current failure theories for composite laminates: Additional contributions’, Compos Sci Techno, V64, pp 449–476, 2004 32 Hinton M J, A S Kaddour and P D Soden, ‘A Further assessment of the predictive capabilities of current failure theories for composite laminates: Comparison with experimental evidence’. Compos Sci Tech., V64, pp 549–588, 2004. 33 BS4994: Specification For Design And Construction Of Vessels And Tanks In Reinforced Plastics, BSI, London, UK, 1987 34 Bailey J E, P T Curtis and A Parvisi, ‘On the transverse cracking and longitudinal splitting of glass and carbon fibre reinforced epoxy cross ply laminates and the effect of Poisson and thermally generated strain’, Proc R Soc Lond, A366, pp 599–623, 1979. 35 Flaggs D L and M H Kural, ‘Experimental determination of the in-situ transverse laminate strength in graphite epoxy laminates’, J Compos Mater, V16, pp 103–116, 1982. 36 Rotem A and Z Hashin, ‘Failure modes of angle ply laminates’, J Compos Mater, V9, pp 191–206, 1975. 37 McCartney L N, ‘Predicting ply crack formation and failure in laminates’, Compos Sci Techno, V62, pp 1619–1632, 2002, see also Chapter 5.5 in this book. 38 Kaddour A S, M J Hinton and P D Soden, ‘Predictive capabilities of nineteen failure theories and design methodologies for polymer composite laminates: Part B: Comparison with experiments’, see Chapter 6 in this book.

Section 5 Comparison Between the Individual Theoretical Predictions and Experimental Results

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

703

CHAPTER 5.1

Application of progressive fracture analysis for predicting failure envelopes and stress–strain behaviors of composite laminates: a comparison with experimental results P. K. Gotsisa, C. C. Chamisa,* and L. Minnetyanb a

NASA, Glenn Research Centre, 21000 Brookpark Rd, Cleveland, OH 44135, USA b Clarkson University, PO Box 5710. Postdam, NY 13699, USA

Abstract The theoretical predictions, published in Part A of the failure exercise, are compared with experimental results provided by the organizers of the exercise. Two computer codes, ICAN and CODSTRAN, developed at NASA (Glenn Research Center at Lewis Field), were applied to predict the damage initiation, damage growth and global structural fracture in a wide range of multidirectional laminates. CODSTRAN was employed to predict (I) seven biaxial failure envelopes of [0°] unidirectional and [0°/ ± 45°/90°]s, [ ± 0°/90°]s and [ ± 55°]s multi-layered composite laminates and (II) seven stress–strain curves for [0°/ ± 45°/90°]s, [ ± 55°]s, [0°/90°]s and [ ± 45°]s laminates under uniaxial and biaxial loadings. In general, CODSTRAN gave reasonable predictions for cases where final failure was dominated by fibre fracture. There was, however, large discrepancy between the predicted and measured failure strengths and strains for cases where failure was dominated by matrix failure. Some of the discrepancy is attributed to (a) the effect of residual matrix stiffness that is discounted in simulations and (b) sensitivity of the specimens to the presence of biaxial stress state in certain cases. Keywords: Composites; C. Computational simulation; C. Laminates

1. Introduction In Part A of the failure exercise, Gotsis et al. [1], presented theoretical predictions for a total of 14 problems involving seven biaxial failure envelopes of unidirectional and multidirectional composite laminates and also for seven stress–strain curves for a number of laminates under uniaxial and biaxial stresses. Two computer codes ICAN (Integrated Composite Analyzer) and CODSTRAN (Composite Durability Structural Analyzer), were * Corresponding author. Tel.: + 1-216-433-3252; fax: + 1-216-977-7051. E-mail address: [email protected] (C. C. Chamis).

704 used to generate the biaxial failure envelopes. Only CODSTRAN was used to obtain the stress–strain curves. Details were provided in Ref. [2] describing the laminates analyzed including unidirectional material properties, loading configurations, and the format of presenting the predicted theoretical results. The theoretical results presented by Gotsis et al. [1] are compared in Ref. [3] with those of other models used by ten different groups who participated in the failure exercise. All these groups used the same input data to analyze the same 14 test problems. In Part B of the failure exercise, the participants were supplied with experimental results for the 14 problems and these were fully described by Soden et al. [4]. With the details of the test results, the participants were supplied with graphs in which their theoretical predictions were superimposed on the experimental results. The participants were requested to comment on the correlation between the predicted and measured data and to describe how their model can be used in the design situations. Almost all the experimental tests were carried out on tubular specimens whereas the theoretical predictions were made for flat laminates. In some of the test results, initial failure in the form of cracks or leakage was described and, in other test cases, only final stresses were provided. The present paper aims at comparing the theoretical prediction with the experimental results and at attempting to suggest how modifications can be introduced to obtain better fit between the theoretical and experimental results, wherever possible.

2. Summary of the theoretical modeling Two codes were used to predict the behavior of the laminates. They are ICAN and CODSTRAN. ICAN is only capable of predicting the initiation of damage in a lamina or a laminate while CODSTRAN is capable of predicting the progression of damage in any composite structure. In ICAN the initiation of damage is detected using modified distortion energy (MDE) failure criterion [5], which is a second order polynomial given by

F=1

冋冉 冊 冉 冊 冉 冊冉 冊 冉 冊 册 L11a SL11a

2

+

L22b SL22b

2

 KL12

L11a SL11a

L22b L12s + SL22b SL12s

2

where the directional interaction factor KL12 is given in terms of the lamina elastic constants by KL12 =

(1 + 4vL12  vL13)EL22 + (1  vL23)EL11 兹EL11EL22(2 + vL12 + vL13)(2 + vL21 + vL23)

and • L11 and L22, are the lamina stresses along and perpendicular to the fibres and L12S is the shear stress • SL11 and SL22 are the lamina strengths along and perpendicular to the fibres and SL12S is the shear strength. If the prevailing stress in the fibre direction is tensile (L11a 0) then the strength SL11 is the tensile longitudinal strength SL11T and if the prevailing stress in the fibre

705 direction is compressive (L11a < 0) then the strength SL11 is the compressive longitudinal strength SL11C . The same is applied for the transverse (perpendicular to the fibres) stresses. • EL11 and EL22 are the lamina moduli along and perpendicular to the fibres • vL12, vL13 and vL23 are the lamina Poisson’s ratios Failure is deemed to take place when F = 0 and failure modes can be either fibre or matrix failure depending upon the dominant terms in the MDE equation. In the results presented in Part A [1,5], fibre failure is assigned when one of the following equations is satisfied:

冉 冊冉 冊 L11a SL11a

2

≥

L22b SL22b

2

or

冉 冊冉 冊 L11a SL11a

2

≥

L12s SL12s

2

The stress limits SL11T , SL11C , and SL12S in the MDE failure criterion are calculated by ICAN based on the constituent fiber and matrix strengths and micromechanics equations. The lamina elastic properties used in the directional interaction factor KL12 are also computed by ICAN from the fiber and matrix elastic properties via micromechanics equations. In the CODSTRAN code, in addition to the MDE criterion, maximum stress criteria are used with the positive and negative limits of all six ply stress components, namely, L11, L22, L12, L33, L13, L23. If any of these ply stresses exceed their computed limits, even if MDE criterion does not predict failure, damage is assigned. If L11 exceeds its limit both fiber and matrix are assumed to have failed. If any of the other ply stresses exceed their limits, only matrix failure is assumed. The use of maximum stress failure criteria in CODSTRAN along with the MDE criterion means that the combined stress effects are only taken into account when they reduce ply strength. If fibre failure takes place in a ply, the fiber (E1f) and matrix (Em) longitudinal moduli are replaced by negligible values so that the ply effectively does not carry any more load. If a matrix failure takes place in a ply, the matrix is assumed to have failed and the matrix modulus is replaced with a negligible value, i.e. Em 0. We should note that the drop in Em takes place suddenly, i.e. the modulus does not decay gradually after failure occurs. Micromechanics relations [5], are used to compute revised properties (longitudinal modulus EL11, transverse modulus EL22, shear modulus GL12, and Poisson’s ratios vL12, vL13 and vL23) of the failed lamina. The revised properties are then used to carry out the progressive analysis of the damage. The procedure is repeated until all the plies have failed and the laminate cannot carry any more load. It is noted that in an angle ply laminate, once two elastic properties are dropped to zero, the stiffness matrix becomes singular. Therefore, to prevent the stiffness matrix from becoming singular, matrix stiffness after failure is reduced to 6.895 kPa (1.0 psi) rather than zero. Accordingly, the simulation computes (i) the transverse modulus to be a negligible value, as computed by micromechanics with unit matrix stiffness, when transverse failure takes place and (ii) the shear modulus drops to a negligible value when shear failure takes place.

706 When all plies of a laminate sustain fiber fractures, the laminate is considered to have fractured at that position (i.e. at that node in a finite element model). The fractured node is deleted from the computational model and new disjoint nodes are created at the location of the fractured node. The number of new nodes created to replace the fractured node equals the number of finite elements with connectivity to the deleted node. Each new node is assigned connectivity to one of the elements that were associated with the fractured node. This simulation strategy allows for the computational tracking of fracture progression. Global structural failure takes place when the structure is broken into two pieces by a fracture line. This is determined by checking the overall connectivity of the finite element model. If a part of the structure represented by a group of elements becomes disconnected from the remainder of the structure then the structure is considered to have globally failed. This definition of global structural failure is independent of the type of loading such as monotonic, cyclic, impact, etc. The codes can take into account the thermal stresses in a laminate and in some of the laminates analyzed in Part A [1], the thermal residual stresses were included. However, the codes did not use the lamina nonlinear stress–strain curves in their analysis and hence the laminae were assumed to behave in a linear manner. The scissoring (change in angle) effect was monitored but was not taken into account for damage evolution. The delamination criterion was activated in most of the ply longitudinal compressive failure cases.

3. Comparison between theory and experiments 3.1. Biaxial failure envelopes for unidirectional fibre reinforced composites Figures 1–3 show the failure envelopes for unidirectional laminae under various types of loading. Figure 1 presents a comparison between test results and theoretical predictions for an E-glass/LY556 epoxy lamina under combined transverse and shear loading (2  12). The correlation is good for tensile 2 stresses. However, for compressive 2 stresses, the theory predicts a quadratic envelope with no enhancement in the shear strength due to application of compressive stress while the test results showed a degree of enhancement, especially for moderate values of 2 stresses. This is because the MDE criterion does not contain any terms that would predict the increase in shear strength due to transverse compression. The experimental uniaxial transverse compressive strength [4], is greater than that provided for use in the analysis [1]. If one is to re-run the analysis taking into account the actual strengths (transverse compressive strength SL22C = –140 MPa and shear strength SL12S = 62 MPa) rather than those provided (SL22C = –114 MPa and SL12S = 72 MPa), the fit would remain poor for moderate values of compressive stresses. We note that the MDE criterion would predict the increase in shear strength in the presence of a small biaxial stress state. If it is certain that the experimental conditions excluded the possibility of a small biaxial compression stress state, an additional term may be required in the MDE failure criterion to take into account the enhancement of shear strength by the presence of uniaxial transverse compression. Before modifying the MDE criterion more tests are needed to verify that the transverse compressive stresses are truly uniaxial and there are no longitudinal compressive stresses. Another consideration is that if an interaction term considering the beneficial effects of transverse compression on shear strength were to be added, it would reduce the shear strength under

707

Fig. 1. Biaxial failure stresses for 0° lamina made of GRP material. Material Type: E-Glass/LY556/ HT907/DY063.

transverse tension. Current results are in agreement with test data in the transverse tensionshear quadrant. Therefore, any corrective term may have to be limited to the compression-shear quadrant. Figure 2 presents the theoretical and experimental results for T300/914C carbon/epoxy lamina under combined longitudinal and shear loading (1  12). A wide scatter is observed in the test results, especially in the values of uniaxial shear strength. In the test results, the data suggest (a) an enhancement in the shear stress above the uniaxial shear strength when the

Fig. 2. Biaxial failure stresses for 0° lamina made of CFRP material. Material: T3000/914C.

708

Fig. 3. Biaxial failure envelope of 0• GRP lamina under combined x and y stresses. Material: E-glass/ MY750 epoxy.

longitudinal tensile stress is around 900–1000 MPa and (b) a reduction in the shear strength when a compressive stress is applied. The theory employed under-predicted the test results for combined shear and longitudinal tensile stresses. The difference between theory and test results may be due to two possible reasons: (1) the MDE criterion does not consider a (1  2) combined stress term. (2) If a biaxial normal stress state (1  2) were present, it would have increased the shear failure stress according to the MDE criterion. It is possible that a small transverse tension was present during the test that was not taken into account in ICAN computations. Figure 3 shows a comparison between test results and theoretical prediction for E-glass/ MY750 epoxy lamina under combined longitudinal and transverse loading (1  2). The test results were obtained from tests on ± 85° winding angle tubes subjected to combined internal pressure and axial load. The theoretical curve showed an interaction between the stresses especially in the first quadrant. In the tension-tension quadrant, the biaxial strength was predicted to exceed the uniaxial transverse tensile strength by 10% at the stress ratios 1 /2 = 13.68/1 and to exceed the uniaxial longitudinal stress by a similar amount at 1 /2 = 81/1. The test results available did not show such trend as no tests were conducted near the 1 /2 = 13.68/1 stress ratio. It is worth mentioning that the degree of increase in the biaxial strength depends upon the value of KL12. As the value of KL12 increases the biaxial strength increases and vice versa. However, the practical range of KL12 is around 0.8 for the four materials studied in the failure exercise. In the tension compression quadrants, the theoretical results showed that the biaxial strength is always lower than the uniaxial strength. The test results in the longitudinal tensiontransverse compression quadrant also suggest a degree of interaction between the stresses, similar to that predicted by the theory.

709 In the compression compression quadrant, the MDE criterion predicted also an enhancement in the biaxial strength by around 10%. The biaxial transverse compressive strength reached a maximum at 1 /2 = –2.53/  1 and the biaxial longitudinal compressive strength reached a maximum at 1 /2 = –14.4/–1. It is, however, not possible to make a full comparison between the theory and experiments as no test results were available in the second and third quadrants. 3.2. Biaxial failure envelopes for multidirectional composites Figures 4a and b, and 5 show the experimental and theoretical failure envelopes of the [ ± 30°/90°]s laminate made of E-glass/LY556/HT907/DY063 epoxy subjected to combined direct stresses (x  y) and combined axial stress and shear loading (x  xy). The test results were obtained from tests on tubes and the test data available described only the final failure stresses, i.e. no initial failure stresses were supplied for the laminates. Hence, it is not possible to compare the theoretical predictions obtained from ICAN for initial failure with test data. The theoretical first-ply failure stresses under x  y stresses (Fig. 4a), are conservative in almost all the quadrants. For uniaxial tensile loadings, the experimental results were higher than the theoretical results by a factor of 1.4, at y : x = 1 : 0, a factor of 2.7 at y : x = 0 : 1. For uniaxial compression loadings, the theory and test results are relatively close. For biaxial compressive loading the test results were lower than theoretical predictions due to buckling of the cylindrical shell during tests. For other biaxial loading, the first ply-failure theory underestimated the test results by a factor of up to 7 at y : x = 1 : + 1 and by a factor of around 3 at y : x ≈ 1 : –1. CODSTRAN results depicted in Fig. 4b are in better agreement with the experiments than the ICAN first ply failure predictions for the tension-tension biaxial stresses. Nevertheless, the test results in some cases still exceed the predictions by more than twice. The implications of these findings are that (i) the composite laminate has significant damage tolerance, especially for tension along the x axis and (ii) the tested tube specimen behaves differently compared to the simulated flat specimen for this laminate. In the case of the tube specimen the 90-degree hoop plies constrain the diameter of the tube and therefore do not allow the relative rotation of ± 30° plies, preventing in-plane shear failures of the plies. On the other hand, for the flat specimen, the edges shrink due to Poisson’s ratio effects and the ± 30 plies are more likely to fail due to in-plane shear failures. This argument is verified by the fact that the simulated laminate fractures were always initiated at the edges of the flat specimen. When the laminate is subjected to combined axial stress and shear loading (x  xy) (Fig. 5), the first-ply failure theory still underestimated the test results for uniaxial and biaxial loadings. For pure shear, the theoretical prediction gave results that were almost 4.5 times lower than the failure loads measured on tubes. The implication of this comparison is that the laminate structure has a very high level of damage tolerance, especially under tensile loads. Figure 6a and b shows the experimental and theoretical results of the biaxial failure envelopes of [ ± 55°]s laminates made of E-glass/MY750 epoxy under combined x  y biaxial loading. The test results were obtained by testing tubes under combined internal (or external) pressure and axial loadings. The tubes used in the internal pressure loadings were tested with and without the use of plastic liner and the strengths of ‘lined’ and ‘unlined’ specimens were measured and quoted in Ref. [4]. The theoretical results in Fig. 6a obtained from ICAN for initial failure were lower than the test results on lined tubes for all the measured points.

710 Fig. 4. (a) Biaxial failure envelope for (90°/ ± 30°) E-glass/LY556 laminate based on first ply failure (predicted by ICAN Code). (b) Biaxial failure envelope for (90°/ ± 30°) laminate under combined ax and ay stresses. Material: E-glass/LY556 epoxy.

711

Fig. 5. Biaxial failure stresses for (90°/ ± 30°) laminate under rxy and ax stresses using CODSTRAN.

The first ply failure theory under-predicted the experimental strength in a number of areas. In the compression compression quadrant at stress ratio SR = y : x = –2 : –1, the theory gave first ply failure values which were lower than the measured failure loads by a factor of around 4. Also, the theoretical predictions of first ply failure were lower than the experimental results of ultimate strength at tensile stress ratios in the region of SR = 1/0. For instance, at SR = 10 : 1, the hoop strength y was almost five-fold higher than the predicted first ply failure strength. The theoretical final strength in Fig. 6b computed by CODSTRAN overpredicted the experimental leakage strength at stress ratios between SR = 1 : 1 and 2 : 1. and the first-ply failure strength was considerably below the experimental leakage strength. In the compression compression quadrant CODSTRAN predicted that the composite had no damage tolerance. In other words, the ultimate fracture was predicted to occur immediately after first ply failure due to longitudinal compressive stresses. These predictions were approximately three times lower than the test results at a stress ratio of approximately –2 : –1 . This may be partly attributed to (a) the high fibre volume fraction of the test specimens used in this quadrant [4], (b) the use of thick tubes in the experiments while the theory uses thin

712

Fig. 6. (a) Biaxial failure stresses for ( ± 55°) E-glass/MY750 laminates using ICAN. (b) Biaxial failure stresses for ( ± 55°) E-glass/MY750 laminates using CODSTRAIN.

flat plate, (c) simulations neglected the residual matrix stiffness of the failed plies and that may have under predicted the ultimate failure load under compressive stresses.

713 Biaxial first-ply failure envelopes for [0°/ ± 45°/90°]s laminate under x  y biaxial loading are plotted in Fig. 7. The correlation between the first-ply failure theory and test results is poor. This is partly attributed to the following factors: (a) The theoretical results were for initial failure, i.e. no final failure prediction was made in Part A paper, except for two loading cases taken from the stress–strain curves. In particular, the test results which were generated taking into account the thermal stresses, in the tension-tension quadrant, i.e. when the tubular test specimens are subjected to combined internal pressure and axial tension, indicated practically zero strength for (y /x = 2/1). The implication is that residual stresses from the fabrication process approached the tensile strength of the matrix. Therefore, the theoretical matrix damage occurred as soon as tensile loading was applied. (b) The test results in the compression-compression quadrant, i.e. when the tubular specimens are subjected to combined external pressure and axial compression, are influenced by shell buckling, [4], and hence no comparison can be made at the moment. However, the theory predicts the initial and final failures to be coincident. This is due to immediate fracture propagation from initial ply longitudinal compressive fractures in the laminate. In other words the subject laminate has no damage tolerance under compressive loading. (c) For the cases where the tubes were subjected to internal pressure and axial compression, i.e. in the tension-compression quadrant, the fit between the test results and the first ply failure theory is reasonable for low tensile stresses because of the dominance of

Fig. 7. Biaxial failure stresses for (0°/ ± 45°/90°) laminate of CFRP material (with thermal and without thermal stresses).

714 compressive x stress effects. However, the tested ultimate strengths in the tension-tension quadrant are as much as three times higher than the predicted first-ply failure strengths due to the ability of the fibers to carry biaxial tension after matrix damage.

3.3. Stress–strain curves The stress–strain curves were predicted using CODSTRAN models of flat coupons. Figure 8 shows the theoretical and experimental stress–strain curves for [0°/90°/90°/0°] laminate under uniaxial loading (y /x = 0/1), where the loading direction coincides with the 0° direction of the laminate. In the experiments the coupon test specimens started cracking at 0.375% strain (137.5 MPa stress) leading to a gentle drop in the modulus. Longitudinal splitting in the 0° plies took place at a strain 1.3% which led to a kink in the stress–strain curve, not shown on the figure. Final failure took place by fiber fracture in the 0° plies at a strain of x = 2.69% and y = –0.125% and a stress of 609 MPa. The theoretical curves are linear up to failure, at over 2% strain. The theory predicts damage initiation (initial failure) in the form of transverse tensile failure in the 90° plies at a stress of 76 MPa. The occurrence of initial damage did not lead to a change in the slope of

Fig. 8. Stress–strain curves for 0°/90° E-glass/MY750 laminate underuniaxial tension (y = 0).

715 the stress–strain curve. The final failure was predicted to be fibre fracture in the 0° plies at a stress of 636.5 MPa and strains x = 2.073% and y = –0.3006%. The experimental results indicated that splitting in the 0° plies (the plies oriented along the loading direction) took place at around 330 MPa stress. CODSTRAN code also predicts transverse tensile failure (matrix failure) in the 0° plies but at a stress of 430 MPa. However, this failure did not affect the slope of the stress–strain curves because the lost matrix stiffness was negligible compared to fiber stiffness. It is noted that the reduction of matrix stiffness would reduce the lamina transverse stiffness to a negligible value. Therefore, the slightly nonlinear stress–strain relationship obtained from experiments appears very reasonable. One possible explanation why CODSTRAN simulation was not able to represent the nonlinearity may be that the simulated rectangular model was overconstrained at the end boundaries. Another possible reason is that matrix stiffness is neglected in CODSTRAN in computing the lamina longitudinal stiffness. Nevertheless, the tested strength was only 4.3% below the predicted strength. Figure 9 shows the theoretical and experimental results for the [ ± 45°]s laminate under equal-biaxial tensile loading (y /x = 1/1). The predicted initial failure stress was 68 MPa that is similar to that 70 MPa observed in the test, which corresponded to initial cracking. The progressive damage starts at 68 MPa stress and the predicted stress–strain curves were almost linear up to a strain value of 2.6% and strength of 636.5 MPa. Beyond this point, the strain values exhibited a rapid increase up to 18% with little increase in the stresses (from 636.5 to 728.9 MPa). The reason for the large strains above 2.6% is the propagation and coalescence of the simulated local fractures in the laminate. Also, to be able to represent the biaxial stress state a square finite element model was used and tractions were applied to all four sides. In a CODSTRAN simulation nodes where forces or boundary conditions are applied are not checked for damage. Therefore, the model was not able to develop a fracture line that would break it into two pieces. Accordingly, simulated strains above 2.6% should not be taken into account. The predicted final failure stress was 636.5 MPa and this is larger than the strength

Fig. 9. Stress–strain curves for ± 45° E-glass/MY750 laminate under y /x = 1/1.

716 observed in the tests which was 210 MPa at leakage for tests on tubes tested without a liner and 502 MPa at rupture for tests on tubes with liner [4]. The main reason for the difference between the simulation and test results is believed to be the sensitivity of the failure load to the stress ratio. If the stress ratio were 0.99/1 or 1.01/1 instead of 1/1, the simulated failure load would also be reduced. The theory predicted the strains x and y to be identical. However, the test results showed that the hoop strain y to be larger than the axial strain x. Furthermore, at any given stress the measured strains were larger than those predicted. This could be due to problems with the test specimens. Considering the difference in the measured x and y strains, it is reasonable to conclude that the stress ratio y /x in the tested case was somewhat greater than 1/1. Figure 10 shows the predicted and measured stress–strain curves for [ ± 45°]s laminate subjected to equal biaxial tension-compression loading with y /x = 1/–1. The theory predicted a linear stress–strain behavior with a failure stress of y = 51 MPa at a failure strain of y = 0.439%. Based on a simple thermo-elastic analysis of a balanced and symmetric angle ply flat laminate, the thermal residual stresses in the 45° laminate are –18 and 18 MPa in directions parallel to and perpendicular to the fibre direction of the ply. The failure stress is predicted to be 73 MPa, without including thermal stresses, and 56 MPa, when the thermal stresses are included. The CODSTRAN prediction of failure stress of 51 MPa, taking into account the thermal stresses, is slightly lower than the predicted strength based on the simple analysis. The predicted ultimate failure stresses are very similar to the test stresses corresponding to strain propagation or the commencement of rapid increases of the test strains. In the tests, after the strain propagation stage the ultimate stresses rose approximately to 95 Mpa at the expense of very large strains. The experimental strains which were around 10% and were obtained from testing tubes under combined internal pressure and axial compression were

Fig. 10. Stress–strain curves for ± 45° E-glass/MY750 laminate (y /x = 1/–1).

717

Fig. 11. Stress–strain curves for ± 55° GRP laminate under uniaxial tension (y /x = 1/0).

much higher than the predicted final failure strains. The CODSTRAN simulation used a flat square coupon model and therefore could not have predicted the large post-failure strains that were developed due to the stabilizing effects of the internal pressure in the tested tubes. Figure 11 shows the predicted and measured stress–strain curves for [ ± 55°]s laminate subjected to uniaxial tensile loading with y /x = 1/0. The measured curves were obtained from internal pressure tests on a thin tube tested with a plastic liner [4]. The correlation between the measured and predicted failure stresses is poor. The predicted stress–strain curves are linear up to a stress of 113.8 MPa (or a strain of 0.487%). It is noted that the measured strains were very close to the predicted strains up to 112.5 MPa. However, beyond 112.5 MPa the measured strains showed a high degree of nonlinearity up to a final failure strain of around y = 9%, corresponding to a stress of around 600 MPa. Beyond y = 0.487% strain, the theoretical y strain increased sharply from y = 0.487% to y = 12.38% as the stress increased from 113.8 to 137.2 MPa. The sharp increase in the strain was reported in Part A as being due to application monotonic loading in the analysis. The program stopped at 12.38% strain because the specimen broke into two pieces. The main difference between predicted and experimental results is that theory predicted very little damage tolerance after the ± 55° composite plies experienced in-plane shear failures, whereas test results indicated significant increase in loading after initial shear failure. It is noted that predictions depicted in Figs 10 and 11 represent similar in-plane shear failures. However, the Fig. 11 prediction indicates substantial strains before ultimate fracture, whereas in Fig. 10 the theoretical stress–strain plots stop abruptly at failure. The main reason for the difference in predictions is the stressstate. If the ± 55° laminate were also subjected to a y /x = 1/–1 stress state its failure prediction would also be sudden. Lateral compression causes a longitudinal compressive stress on plies that cause a more brittle failure. The predicted failure stress in Fig. 11 was 137.2 MPa which is some 4.4 times lower than the measured failure stress in the tests carried out with liner. The predicted failure stress was 2.8 times lower than the measured strength of tubes tested without a liner.

718 In the opinion of the authors the simulated failure stress approximates the true uniaxial failure strength better than the pressurized tube test data. Possible reasons for the discrepancy are (i) the analysis is performed on flat laminates whereas test results were obtained from test on tubes under internal pressure; (ii) the test x may have been greater than zero due to structural geometry and boundary conditions of the test, and (iii) stabilizing effect of internal pressure supporting the tube structure after first ply failure, allowing large changes in the fiber orientations without matrix support. Under these conditions, the tube is not a composite any longer but a net of fibers. To predict this test an analysis of the fiber net in the shape of the cylindrical tube rather than a composite coupon simulation is needed. The behavior of [ ± 55°]s laminate subjected to biaxial tensile loading with y /x = 2/1 is depicted in Fig. 12. The measured curves were obtained from internal pressure tests on a thin tube tested with a plastic liner. When compared with the test results, the following observations can be made from the results in Fig. 12. 1. The predicted shape of the y  y curve agreed generally well with the experimental results. 2. The predicted shape of x  x curve agreed partially with the measured curve. The agreement appears to be good up to a stress of around 200 MPa. It is noted that the theoretical stress–strain curves indicate reduction of stiffness at the 44 MPa damage initiation stress. Above 44 MPa the theoretical curve continued fairly linearly (with a smooth increase in the slope of x) whereas the experimental curve started softening with the slope decreasing considerably as the stress increases. Hence, beyond 200 MPa, the theoretical prediction was always above the measured one. The predicted failure strain was 1.28% which is approximately a third of the measured failure strain for ex shown in Fig. 12.

Fig. 12. Stress–strain curves for a ± 55 laminate made of E-glass/MY750 epoxy material under y /x = 2/1.

719 3. The predicted final strength for y was 849 MPa which was close to the measured average strength of 736 MPa of the lined tubes and was almost three times higher than the failure stress for unlined tubes which failed by leaking. 4. The test results are somewhat puzzling because it is not kinematically consistent that the x  x plot become nonlinear without affecting the y  y plot. In the opinion of the authors the tube specimen was not able to achieve the required y /x = 2/1 true biaxial stress state. The premature failure of the tube specimen was probably due to flexural stresses near the regions of transition from the built-up end sections to the middle test section. Figures 13 and 14 show the stress–strain curves of quasiisotropic [0°/ ± 45°/90°]s AS4/3501-6 carbon/epoxy laminates under uniaxial (y /x = 1/0) and biaxial loadings (y /x = 2/1). For the uniaxial loading, (Fig. 13), the theory predicted initial transverse tensile failure in the 0° plies (plies perpendicular to loading direction) by transverse tension at a stress of 130 MPa That was followed by matrix (shear) failure in the ± 45° plies at a stress of 135 MPa. The slope of the theoretical stress–strain curves suffered a decrease due to the occurrence of the shear failure. The theory predicts fibre failure at a stress of 345 MPa and strain of 0.882%. Further increase in the load led to large strain along the loading direction reaching 27.7% as the flat model of the composite specimen was being torn into two pieces at a stress of 385 MPa. Therefore, strains beyond 0.882% are not meaningful and should not be taken into account. Accordingly, the simulated stress–strain curves in Fig. 13 should be terminated at 345 MPa. There is an agreement between the predicted and measured stress–strain curves up to the point of the predicted matrix shear failure in the ± 45° plies. Beyond that point, the predicted curve exhibited softer characteristics. Finally the predicted strength was some 50% lower than

Fig. 13. Stress–strain curves for (0°/ ± °/90°) laminate under uniaxial tension (y /x = 1/0).

720

Fig. 14. Stress–strain curves for (0°/ ± 45°/90°) laminate made of CFRP material (y /x = 2/1) (Results predicted by CODTRAN code).

the measured final strength. The main reason for the differences in failure strength was that the tests were conducted on a cylindrical specimen where the actual stress ratio was y /x = 20/1 [4], rather than truly uniaxial. The sensitivity of quasiisotropic laminate tensile strength to the presence of biaxial stresses is well known. From the simulations perspective, the large difference between the failure loads of y /x = 1/0 and y /x = 20/1 is that in the 90° plies, after matrix stiffness is reduced to a negligible value due to transverse tensile failures, fiber fracture is sensitive the presence of longitudinal compressive stresses. When y /x = 1/0 longitudinal compressive stresses develop in the 90° plies due to Poisson’s ratio effects. On the other hand, if y /x = 20/1, longitudinal stresses in the 90° plies would remain tensile and the correct test strength would have been predicted. Predicted fracture load in Fig. 13 would have increased significantly if a small percentage (3%) of matrix stiffness were retained after transverse tensile failures. However, there is no rational theoretical basis to determine the appropriate percentage of matrix stiffness that should be retained after ply transverse tensile failure. For the biaxial loading at y /x = 2/1, Fig. 14, the correlation between the measured and predicted behaviours are better than those under the nominal uniaxial loading. For the biaxial loading, the theory predicted matrix failure in the 90° plies in the form of transverse tension at stress of 230 MPa. The theory then predicted a matrix failure in the 45° plies in the form of shear failure at a stress of 580 MPa. The occurrence of this type of failure did not lead to a change in the slope of the stress–strain curves, as was the case in the uniaxial loading case. Finally the final failure was predicted to be by fibre tension in the 0 plies at a stress of 772 MPa.

721 The correlation between experimental and the theoretical results is in general good, especially in terms of the final failure predictions where the predicted strength was only 5% lower than the measured value, 808 MPa. 4. Discussion There are a number of lessons that were learnt and gaps that were identified from the failure exercise concerning the application of the computer codes to solve the 14 problems imposed by the organizers. These involve the following: (a) The correlation between first-ply failure strengths predicted by the MDE failure criterion and test results on unidirectional lamina were good in general. We note that the MDE criterion is sensitive to biaxial normal stresses. The MDE term associated with the directional interaction coefficient KL12 makes a small biaxial normal stress component very significant in determining the failure stress. (b) The damage progression modeling embedded in CODSTRAN is suitable to predict initial damage and progression of failure if the actual stress state and the structural geometry and boundary conditions are accurately specified. (c) Computational simulation via CODSTRAN is able to simulate material nonlinearities by reducing the stiffness of failed constituents locally. Accounting for the geometric nonlinearity requires the exact shape of the structure to be simulated with large deformation option. (d) Ability of CODSTRAN to predict the strength and deformation in multidirectional laminates subjected to biaxial loading is sensitive to the accuracy of the specified biaxial stress ratio. (e) The effect of thermal residual stresses may be significant, especially when predicting first ply failure due to tensile stresses. (f) The use of the computer codes in the design of structures requires knowledge of the actual stress state. Structural geometry and boundary conditions should be accurately modeled. For design with composites, the sensitivity of the failure strength to the changes in the stress state should be evaluated. (g) Computational simulations by CODSTRAN did not take into account the confining effects of the liners used in tests with pressurized cylindrical tubes. In particular, the large changes in fiber orientation after matrix failure due to in-plane shear stresses and the resulting large strains afforded by the liners were not considered. Additionally, the changes in diameter of the tubular composite specimens caused significant variation of the hoop stresses that were not considered in simulations. (h) Future work on the theoretical and experimental fronts to improve the fit between the theory and the experimental results are: (1) exact geometry, loading, and boundary conditions of the structure for both the test and calibration cases should be known for the application of theory correctly. (2) Changes in the stress state during tests should be monitored. (3) Sensitivity of the failure stress to the variations in the stress state should be evaluated by theory. (i) An important lesson that was learned/confirmed by the failure exercise was that composite failure strength is often sensitive to the stress state. The presence of a small biaxial stress component can significantly change the apparent composite strength.

722

(j)

(k)

(l)

(m)

(n)

(o)

(p)

(q)

Accordingly, computational simulations for design need to consider all possible small changes in the loading and boundary conditions. One way to do this is by probabilistic simulations in which geometry, loading, boundary conditions, material properties, and fabrication parameters can be defined as uncertain random variables. The failure load then would be computed as a response variable. The sensitivity of failure to the different random variables would be determined from probabilistic analysis results. The main shortfall of the progressive damage and fracture simulation model appears to be neglecting the residual matrix stiffness after ply transverse tensile or ply shear failures. It has been observed in computational simulation tests that retaining a small percentage of matrix stiffness (3–5%) after initial damage makes a very significant difference in the failure load, especially for cases where ply longitudinal compressive failure is predicted. However, because of the absence of a rational method for computing the true residual matrix strength after initial damage, the simulations neglected matrix residual stiffness. The theoretical model used herein has been used successfully for designing composite structures. In general, simulations have been very successful for complex structures such as stiffened aircraft wing panels and entire flight structures [6]. In such cases simulations that were carried out prior to testing predicted the failure load within 5% on the average [6]. Simulation results have been less successful for the prediction of failure for coupon level specimens subjected to relatively simple stress states. The authors regard the general current understanding of failure in composites as mature. Individual stages of failure are well understood. However, the overall composite strength depends on the sequence of damage processes that can only be identified and evaluated by computational simulation. Computational simulation models can be further improved by tracking the damage progression details at the constituent materials microstress level. It would enable the computation of supply level fiber/matrix damage accumulation that would improve the effects of neglecting the matrix residual stiffness after initial damage. The authors are convinced that, in general, initial failure predictions and the simulation of damage growth/accumulation characteristics are essential for the determination of the correct final load. Further efforts to understand certain modes of failures? e.g. leakage failure, large deformation, delamination, etc., should be studied in association with specific design cases. It is not productive to study specific failure modes in the general sense as variations of the stress state modify and change the failure characteristics even if the observed failure mode may be the same. The authors suggest additional test cases that represent more complex stress states such as stiffened composite panels with discrete defects or damage, subjected to tension, compression. and flexure. In any additional test or exercise the exact loading and boundary conditions should be disclosed to the theoretical investigators. One way to prevent the possibility of theoretical investigators discovering the test results prior to their predictions is to delay the physical experiments until after the theoretical predictions are turned in. Thermal stresses should be taken into account since they can become significant in certain laminates under certain stress states. Some researchers argue that residual thermal stresses may be neglected due to matrix relaxation. The authors do not believe that thermal stresses can be neglected in general unless justified by special circumstances for

723 a specific composite. Additionally, neglecting thermal stresses is not conservative for design investigations. (r) In situ fiber/matrix constituent properties should be taken into account. In situ properties are usually determined by calibrating the fiber/matrix properties with composite ply and coupon level laminate test data. The following tests are recommended to identify the in situ constituent material properties: (i) transverse tensile test, (ii) longitudinal compressive test, (iii) longitudinal tensile test, (iv) 10° off axis tensile test, (v) [0/90/ ± 45]s quasiisotropic laminate coupon tensile test, and (vi) quasiisotropic laminate open-hole tension test. (s) Micromechanics is an important feature of the presented simulation method. The main reason for emphasis on micromechanics is that it allows the constituent fiber and matrix level damage tracking. The ICAN micromechanics module used in the current analysis assumes unidirectional lamina properties to be linear or piecewise linear. The authors consider linear composite mechanics to be sufficient for the simulation of structures made of PMC materials, although the prediction of the failure exercise cases produced large discrepancies when failure was matrix dominated. Composite mechanics modules with nonlinear lamina analysis capability, i.e. METCAN [7], and CEMCAN [8], are available for high-temperature metal matrix and ceramic matrix composites.

5. Conclusions A comparison was made between the theoretical predictions, first-ply strengths obtained from ICAN and progressive fracture simulations to ultimate strength obtained from CODSTRAN codes, and test results of the biaxial failure envelopes and stress–strain curves of unidirectional and multidirectional laminates. The results show the following features: 1. Based upon the test data on unidirectional lamina subjected to combined loading of either longitudinal and transverse stresses, longitudinal and shear stress or transverse and shear stresses, use of the Chamis failure criterion (modified distortion energy or MDE criterion), the following can be concluded: (a) Good correlation with test results when the lamina is under combined transverse tension and shear and poor correlation when the lamina is under combined transverse compression and shear. In the latter case, the theory did not predict the enhancement in the shear stress due to application of moderate compressive stresses. It is noted that if there had been a small longitudinal compressive stress, the MDE criterion would have predicted the increase in shear strength with transverse compression. Therefore, before attempting to improve on the MDE criterion, it is necessary to verify that there was no longitudinal compression in the test cases. (b) Fair agreement is observed when the lamina is under combined longitudinal tension and transverse compression. (c) A lack of test results in some of the quadrants meant that the failure criterion used cannot be checked and further testing is needed. 2. When CODSTRAN is applied to predict the biaxial envelopes of multidirectional laminates, it was found that:

724 (a) The results of CODSTRAN have in general underestimated the test results for final failure strengths for [ ± 30°/90°]s glass/epoxy laminates under combined direct stresses and under combined direct and shear stresses. In the worst case, the theory using a flat laminate model underpredicted the final strength of the tube by a factor of 7. Reasons for the differences between the response of a tube specimen and a flat laminate were discussed. (b) It was not possible to compare the experimental results with the prediction of CODSTRAN for the final failure envelope of quasiisotropic [0°/ ± 45°/90°]s AS4/3501-6 carbon/epoxy laminates because of the very limited theoretical data. (c) It was only in a limited number of segments of the biaxial failure envelope for [ ± 55°]s glass/epoxy laminate that an acceptable correlation was observed with test results. CODSTRAN under-predicted the final strength in the vast majority of the envelope segments. The reasons for under-predictions were attributed to (i) the high fibre volume fraction of the test specimens used in this quadrant, Ref [4], (ii) the use of thick tubes in the experiments while the theory uses thin flat plate, (iii) simulations neglected the residual matrix stiffness of the failed plies that may have under predicted the ultimate failure load under compressive stresses. 3. When CODSTRAN is applied to predict the stress–strain curves of multidirectional laminates under uniaxial and biaxial stresses, the following were observed: (a) A good agreement with the final strength of laminates whose failure is largely dominated by the fibre tensile strength, such as the E-glass/epoxy [ ± 45°]s laminate under equal biaxial tension, [0°/90°]s glass/epoxy laminate under uniaxial tension, [ ± 55°]s glass/ epoxy laminate under SR = 2/1 and [0°/ ± 45°/90°]s carbon/epoxy laminates under SR = 2/1. However, CODSTRAN underpredicted the uniaxial tensile strength of [0°/ ± 45°/90°]s carbon/epoxy laminates under SR = 1/0. Nevertheless, it was determined that the true stress state was not uniaxial during testing and CODSTRAN simulations are sensitive to the presence of biaxial tension. (b) There was a large discrepancy between experimental and theoretical failure strains that are dominated by transverse and shear matrix failure. These include both branches of the failure strains of [ ± 45°]s under SR = 1/–1, both branches of [ ± 55°]s glass/epoxy laminate under SR = 1/0, and the x strain of [ ± 55°]s glass/epoxy laminate under SR = 2/1. The excessively large strains measured during tests were attributed to the stabilizing effects of the internal pressure in the tested tubes. The CODSTRAN simulation used a flat square coupon model and therefore could not have predicted the large post-failure strains that were developed in the tubes. 4. Although there was a lack of test data in a number of key loading cases of unidirectional and multidirectional laminates, the available data have verified the ability of the codes to solve a number of problems provided that the state of stress is accurately known.

References [1] Gotsis PK, Chamis CC, Minnetyan L. Prediction of composite laminate fracture: Micromechanics and progressive fracture. Composites Science and Technology, V58, pp 1137–1150, 1998, see also Chapter 3.1 in this book.

725 [2] Soden PD, Hinton MJ, Kaddour AS. Lamina properties and lay-up configurations and loading conditions of a range fibre reinforced composite laminates. Composites Science and Technology, V58, pp 1011–1022, 1998, see also Chapter 2.1 in this book. [3] Soden PD, Hinton MJ, Kaddour AS. Comparison of the predictive capabilities of current failure theories for composite laminates. Composites Science and Technology, V58, pp 1225–1254, 1998, see also Chapter 4.1 in this book. [4] Soden PD, Hinton MJ, Kaddour AS. Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates: failure exercise benchmarking data. Composites Science and Technology, V62, pp 1489–1514, 2002, see also Chapter 2.2 in this book. [5] Murthy PLN, Chamis CC. Integrated composite analyzer (ICAN): users and programmers manual. NASA Technical Paper 2515, March 1986. [6] Huang D, Minnetyan L. Postbuckling response and progressive fracture of stiffened and stitched composite sheer panels. Journal of Reinforced Plastics and Composites, 2001;20:1617–1632. [7] Chamis CC, Hopkins DA. Thermoviscoplastic nonlinear constitutive relationships for structural analysis of high temperature metal matrix composites. NASA-TM 87291, November 1985. [8] Murthy PLN, Chamis CC, Mital S. Computational simulation of continuous fiber-reinforced ceramic matrix composites behavior. NASA Technical Paper 3602, July 1996.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

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CHAPTER 5.2

Failure criteria for use in the design environment G. C. Eckold AEA Technology, Culham Science Centre, Oxfordshire OX14 3ED, UK Abstract This paper is concerned with the comparison of composite failure data with the design criteria used for the design of GRP process equipment. This industry sector makes extensive use of composite materials and employs a pragmatic approach to design. Full details were discussed in Part A of this exercise. Examination of the test data yields a number of issues associated with what must be included within a predictive capability, such as the correct interpretation of stress–strain behaviour, the use of appropriate material constants, and a thorough understanding of the test methods used to measure material and component responses. Not all of these points are relevant to the design of structures for long-term service as design strains are limited to low values and not all combinations of load apply. Given the relative simplicity of the design methods used it is considered that the correlation between calculation and measurement is satisfactory. Some improvements are possible and these are the subject of consideration in the on-going revision of standards documentation. Keywords: B. Mechanical properties; C. Failure criterion; C. Laminates; B. Modelling; B. Strength

1. Introduction In the corresponding paper [1] of the first part of this exercise the characteristics of failure criteria for use in the design environment were discussed. It was suggested that in these circumstances absolute predictive capability was not the only attribute necessary for success. Simplicity, ease of use and auditability were considered to be essential if the calculation method was to be used by industry at large. Generally design and manufacture organisations need approaches that can deliver results quickly using easily obtained material property data. Determination of ‘fitness for purpose’ is the primary requirement. A fundamental aspect of this latter concept is that the term failure cannot be considered as a unique material parameter – it is inextricably linked to the duty of the component or structure of concern. This is especially true when working with composite materials as they have a far greater range of damage mechanisms that may (or may not) lead to failure in particular circumstances. In contrast to other contributors to this exercise the purpose of the first paper [1] was not to advocate one particular failure theory, but to put forward the industrial perspective. The E-mail address: [email protected]

727 example used to illustrate the various issues was the process plant industry. This sector uses composites in a technically and commercially demanding environment and the availability of reliable design methods is paramount. To be effective there needs to be a compromise between the appropriate levels of conservatism to ensure integrity, especially when handling hazardous materials, and the ability to design effectively in a competitive market place. Care needs to be taken when assessing the design method in the context of experimental data or the work of other contributors to this exercise. Specifically, the calculated design envelopes should not be misinterpreted as being predictions of ‘initial’ failure. Design allowables should reflect what is required for ‘fitness for purpose’ and would normally incorporate a margin or safety factor to ensure that ‘failure’ is not reached during service life. The calculated design envelopes in this and the corresponding Part A paper should be reviewed with this perspective. 2. Design method The design method employed in the analysis was that given in BS4994.1987 [2] which is the prevailing code in the UK for the design of glass-fibre-reinforced plastic (GRP) process equipment. For some products that are manufactured on a commodity basis, such as pipework, alternative procedures are used based on performance testing [3], but this approach is not considered in this paper. The BS4994 approach is essentially a strain-based criterion and this is used to determine long-term design allowables. Of necessity the values obtained are well below ultimate properties in order to reflect performance over product lifetime which, typically, could involve sustained loading with chemically aggressive environments for 20 years or so. The current strain limitation used in design is specified as 0.2%. Full details are given in the original paper. As the BS4994 process is focused towards design, as opposed to the prediction of ultimate performance, there was some difficulty in its application for this failure exercise. The way forward was described in Part A, but is repeated here as it is relevant for the interpretation of the results: • Material properties. Tabulated material properties in BS4994 are intended primarily for resin systems of the polyester/vinylester type. Reference is also made to epoxy systems, but these are rarely used in the process sector. This is largely due to cost and the fact that styrenated resins have better chemical resistance characteristics, especially in the more aggressive environments. In the development of the standard epoxy systems were included for completeness and tabulated properties understate the mechanical performance such laminates could offer. However, the standard does allow alternative values to be used provided they are based on test data. In this exercise the as supplied property values for unidirectional laminae were used together with simple laminate analysis to give off-axis information. A difficulty also arises in the specification of an allowable maximum strain. For polyesters and vinylesters this is currently 0.2% (transverse tension is limited to 0.1%), but this may be considered low for epoxy systems. The results from performance based testing on epoxy pipe [3] indicates that values in the range 0.3 0.4% are satisfactory. A limit of 0.4% was used in all of the calculations described in the following sections for longitudinal and shear strains. For transverse tension the 0.1% limit was maintained. The selection of 0.4% was, in a sense arbitrary, but it does provide a basis for this exercise.

728 • Prediction of ultimate strength. The design code is not concerned with the prediction of ultimate performance, and, furthermore, there is no mechanism for doing so. For the purposes of this exercise the method adopted was that which would be taken by an engineer given the as supplied properties and the general approach to laminate design described in BS4994. The given ultimate strains were used as a basis for the assessment. For example, the calculated strength value for the stress ratio y : x = 1 : 0 in Fig. 4 was determined using the quoted value of ultimate shear strain. Calculations were performed at selected points and straight lines drawn between them. • Thermal stresses. Most GRP process vessels are fabricated using room temperature cured materials. Sheer size effectively precludes systems, which need elevated temperatures during processing, and as a result the effects of thermal stresses are not relevant. However, in the calculations for biaxial envelopes a simple manual correction was applied using given thermal expansivities and the stated stress free temperatures. The significance of this effect changes with the laminate construction. For example, with the single layer, unidirectional material there would be no residual stresses, whereas in the more complex laminates, where there is a thermal expansivity mis-match between individual layers there will be some affect. However, in the context of the uncertainties of this exercise this was not a first order consideration. • Compressive properties. BS4994 does not provide tabulated data for properties in compression and where compressive loading does feature measured values are required for the calculation. However, this is rarely carried out as such testing is costly and the results can be difficult to interpret. Furthermore, for most applications covered by BS4994 where there is significant compressive load, e.g. vacuum, structural instability will inevitably govern the design. This is especially true for glass reinforced composites where modulus values are relatively low. In this exercise no attempt was made to allow for differences in strengths in the two modes of loading. • Non-linear properties. Non-linearity, either through a material characteristic or ply discount, does not feature in the BS4994 calculation method as allowables are generally based on low strains prior to any form of initial failure. No attempt was made to accommodate this in the following predictions. Calculated stress–strain curves were limited (arbitrarily) to 0.5% strain (for both tension and compression). • Carbon fibre materials. CFRP does not yet feature in the catalogue of material systems employed by the process industry. A basic premise behind the arguments expressed in this and the previous paper is that design methods should be appropriate for the industry and materials for which they are intended. More advanced materials (and their associated cost) require, in principle, more sophisticated design methods if the advantages of their use are to be realised. On this basis no attempt was made to offer predictions for CFRP laminates.

3. Correlation between design method and experimental data Comparison of the results from the BS4994 design method and experimental data [4] is a valuable exercise as it is a good test of whether or not the degree of conservatism embodied within the methods is consistent and appropriate. The comparison with measured ultimate strengths is also of interest because it can quantify margins within which non-recurring upset conditions can be safely accommodated without excessive design penalty.

729 It should be noted that on each of the figures showing the comparison between calculated and measured data two curves are shown; one representing design allowables and the other providing an assessment of ultimate performance. The first of these must not be confused with the prediction of initial failure, which will feature in the work of other contributors to this exercise. Broadly speaking the intention is for the design level to be below initial failure in as much as this was defined by the weeping of fluid through the wall of the test specimen. This is a valid basis for design as, returning to the point made above, a process vessel, which cannot contain its contents, is not ‘fit for purpose’. 3.1. Unidirectional laminate The calculations and data for unidirectional material are shown in Figs. 1 and 2. Figure 1 shows the effect of combined in plane shear and transverse load and as anticipated in the previous paper the calculations for ultimate load in the tension quadrant are high with respect to the data. An important consideration here may be the complexity of lamina/laminate behaviour in shear, e.g. non-linearly in the shear stress–strain response would result in a higher strain (an important failure issue) for a given load than would be the case in a simple linear calculation. In the compression quadrant no attempt was made to take into account the higher strength in this mode of failure which explains the discrepancy. This comment applies to all of the failure envelope correlations. In terms of component design this issue is not relevant as most structures consist of thin shells where instability considerations will govern response. With regard to the design allowable envelope the factor between it and the test results changes as the load combination varies. The minimum ratio between the design value and the test data is approximately 1.5, although it should be remembered that the 0.4% strain limit used in the calculations. Whilst not entirely arbitrary this value should not be construed

Fig. 1. Biaxial failure stresses for 0° lamina made of GRP material E-Glass/LYS56/HT907/DY063.

730

Fig. 2. Biaxial failure envelope of 0° GRP lamina under combined ox and ay stresses. Material: E-glass/ MY750 epoxy.

as the established design limit for epoxy systems. Reference should be made to Part A for further information on this point. In general, the results are satisfactory. Figure 2 shows the effect of combined transverse and longitudinal load. The outcome of the exercise is as expected. Issues associated with compression are as per the previous example although in this case the longitudinal compression strength is less than its tensile counterpart. The relationship between design and ultimate is as required by the current version of the code (BS4994). A point to consider with the data is the spread of values for the combined load in the tension/tension quadrant which are nominally at the same load combination. It would be interesting to speculate whether or not lower and upper bound envelopes derived from a corresponding range of uniaxial properties would have captured the experimental results.

3.2. (90/ ± 30°/90) laminate The data for the 90°/ ± 30° laminate are shown in Fig. 3. Again, where there is a significant compression element the correlation is not good for the reasons already given. Also, in the description of test data [4] buckling was reported to have occurred. Arguably these points should be separately identified on the data plot as they refer to a completely different form of structural response linked to geometry and stiffness, and less to material strength. For the combinations of tensile load, correlation at low stress ratios (y : x) is good as is the shape of the overall final failure envelope. However, where the application of circumferential load through internal pressure is significant the comparison is less good. For example, the calculated value of the final hoop strength is 200 MPa compared with measured values in the range 267–312 MPa. This poses an interesting issue as the calculation was based on the simplistic, and often accepted view, that for a laminate such as 90°/ ± 30° the circumferential properties would be dominated by the 90° layers. The 200 MPa value was

731

Fig. 3. Biaxial failure envelope for (90°/ ± 30°) laminate under combined x and y stresses. Matenal: E-glass/LY556 epoxy.

derived from the as supplied unidirectional strength, 1140 MPa, and the quoted volume fraction of the hoop layers in the laminate as a whole, 17.2%. There could be a number of reasons for this disparity, viz: • the ply discount approach, where the contribution of the ± 30° layers to the laminate stiffness at these high levels of stress is reduced to zero, is too severe. An approximate calculation suggests that an effective circumferential modulus of around 5 GPa for these layers, as opposed to a fully discounted value of zero, would result in a stress of 280 MPa at the quoted failure strain of the hoop material (2.13%). Given that the effects of nonlinearity in shear response in the angle ply will be significant, this suggests that this is a viable explanation for at least some of the discrepancy; • the calculated failure envelope is based on unidirectional strengths presumably measured by conventional means. Fibre failures observed during such tensile testing of flat plate specimens are usually preceded by longitudinal splitting [5]. Specimen failure tends to be catastrophic and detection of the failure point is without ambiguity. In a lined tubular specimen with other layers present detection of ‘failure’ may be more difficult. Consideration of the design envelope is more difficult as there are no initial failure point data with which it can be compared. With respect to the final failure data the ratio between the two is satisfactory where circumferential loads dominate. However, where the load is primarily axial the design values appear to be very conservative. This was anticipated in the Part A and can be attributed to the low value of strain allowed for transverse tension in the 90° layers.

732

Fig. 4. Biaxial failure stresses for ( ± 55°) E-glass/MY750 laminates.

3.3. Angle ply ( ± 55°) laminate Figure 4 shows the data for the angle ply ( ± 55°) laminate. In terms of the final failure envelope many of the comments made above for the 90/ ± 30° material apply. Correlation in the tension/tension quadrant is good except where internal pressure loadings are at their maximum, although it should be noted that the predicted envelope in this region is simply a linear interpolation between the 2 : 1 and 1 : 0 stress ratios. In the summary paper [4] describing the test data reference is made to high levels of deformation and the non-linear effects associated with this, for example fibre rotation, may be significant. For the design envelope initial failure point data are available for comparison. The relationship between design and initial failure is satisfactory up to the 2 : 1 stress ratio. At the 0 : 1 stress ratio (axial tension) the ratio between design and measured values is approximately 1.5 which may be considered to be low, but it would be unusual in practice to severely load such a laminate in this mode in this application. Beyond the 2 : 1 load combination the design envelope appears to be very conservative. The reasons for the large margin between calculation and measurement are not clear. It does lead to a number of issues, which are fundamental to this failure exercise and more generally for composite design. As discussed briefly in the Introduction and in Part A, failure processes in composites can be complex and this can impact on structural performance in a major way. Reviewing the original source of the experimental data [6] indicates that the observed mode of failure varied markedly with the applied stress ratio. This was not only the case for ± 55°, but for other angles as well. Data from tests on ± 45° tubes provided surprising results. It was found that the apparent strength in the circumferential direction was 1.4 times its equivalent value in the axial direction – clearly a situation which requires further explanation as superficially strengths should be the

733

Fig. 5. Stress–strain curves for ± 55° GRP laminate under uniaxial tension (y /x = 1/0).

same. A further point to note was that the measured envelopes for initial and final failure coincided at stress ratio 1 : 0 (hoop) but not at stress ratio 0 : 1 (axial) (whilst the initial failure data at 0 : 1 were not presented this statement can be inferred from values at stress ratios close to 0 : 1). A factor which differentiates these two test results was the way in which load was applied. Both methods provided uniaxial tension, but the circumferential value involved internal pressure, which would have generated an additional through thickness stress. The value is small and would be expected to be insignificant in terms of a stress based criteria of failure, but it may be important in other ways. Other work has shown that through thickness compression can have an effect on delamination, which was an important damage mode reported from these tests. A rationale could be put forward by which the action of internal pressure suppressed delamination and hence the stress concentration which results from this type of damage. This would have caused an increase in apparent strength and a coming together of initial and final failures. The latter assumes that the onset of delamination is a necessary element for the generation of a through thickness leak path for the test fluid. This does not mean that the measured hoop strength value is incorrect – it is just different. It relates to an alternative set of circumstances and is valid in its own context. A similar logic can be applied to the ± 55° curve where at high pressures measured strengths are high. The key point that arises from this analysis is that the data are fundamentally affected by the details of the test method. Simple stress or strain based criteria cannot cater for these effects. Stress–strain curves for the ± 55° example are shown in Figs. 5 and 6. Design calculations were only carried out up to 0.5% strain and no attempt was made to accommodate nonlinearity. Again, note that the 0.5% strain used in the calculation was an arbitrary limit and should not be construed as a prediction. The agreement is good with respect to initial modulus. This is as expected given that it is simply a test of laminate theory. There is also a reasonable margin between the strain at which the onset of leakage was reported during the experimental measurements and typical design values. Definitive conclusions are difficult to

734

Fig. 6. Stress–strain curves for a ± 55° laminate made of E-glass/MY750 epoxy under y /x = 2/1.

develop, not least due to the marked difference in measured leakage strains (approx. 4 and 1% for the 1 : 0 and 2 : 1 stress ratios, respectively). The effect of stress ratio on the point of leakage is discussed above.

3.4. Angle ply ( ± 45°) laminate Figures 7 and 8 show the stress–strain characteristics for ± 45° laminates. The comments made on the ± 55° curves apply. The measured strain at which first leakage was observed for the ± 45° laminate is similar to that for the ± 55° laminate (2 : 1 stress ratio), albeit there is a greater degree of non-linearity with the ± 45° material. Fig. 7 also shows the measured value of strain at which first cracks were observed approx. 0.25%. This may suggest that, for this load condition at least, the use of a design strain for epoxy laminates greater than that for polyester/vinylester materials may be questionable. Although the nature of the observed cracking and the fact that there are no similar data for other examples makes a judgement difficult.

3.5. Cross ply laminate Figure 9 shows a stress–strain curve for the cross ply laminate subjected to uniaxial loading. Again it is linear and no attempt was made to identify the ‘knee’ which occurs due to transverse cracking. The calculated strain transverse to the loading direction was curtailed

735

Fig. 7. Stress–strain curves for ± 45° E-glass/MY750 laminate under y /x = 1/1.

Fig. 8. Stress–strain curves for ± 45° E-glass/MY750 laminate under y /x = 1/–1.

736

Fig. 9. Stress–strain curves for 0°/90° E-glass/MY750 laminate under uniaxial tension (y = 0).

below the (arbitrary) 0.5% used for the other stress–strain curves as the low Poisson’s ratio would have meant extending the data to high and unrealistic load values. 4. Design issues There are a number of design issues that arise from the comparison between the BS4994 approach and the measured data. These can be split into those which affect the design envelope, and those which concern the estimation of final failure. As has already been discussed the design code does not really address ultimate performance as its primary function is to provide a means by which long-term material allowables can be determined. However, there are many circumstances where rarely occurring upset conditions could dominate a design if they were treated as a standard load case. The ability to be able to cater for these less conservatively, but without prejudicing the structure as a whole, would represent a significant way forward. Even if this resulted in some local damage, which needed to be the subject of remedial action, there would be benefit. As a result the ability to predict the final failure would be an asset. For the final envelopes where the applied load features tension/tension the maximum strain approach using the as supplied data produced envelopes in not unreasonable agreement with

737 the measured data, particularly given lack of sophistication in the approach. Progress here will require consideration of a whole range of other effects and it is probable that the benefit of this compared with the cost of the added complexity will not be deemed to be worthwhile in the context of the design of process plant. Where compression is an important loading there is clearly a need to incorporate realistic property values in the calculation if the predictive capability is to be improved. However, this is unlikely to be relevant to the vast majority of situations for process plant due the probability of instability being the dominant design issue. The problem associated with shear stresses is most likely because the assumption of linear stress–strain response does not reflect material behaviour. As strains used in design are low action here is also not a priority issue. The calculations for the design envelopes were all conservative and broadly to the intent of the prevailing code for the stress ratios of most practical interest. This matter is being developed at the present time as all such documents are being reviewed within the CEN (Comitee Europeen de Normalisation) harmonisation process. It is probable that allowable design strains will be increased for all reinforcement types, not just those produced by filament winding. A further enhancement, which is going forward, is the treatment of combined loading so that design envelopes can be developed more easily [7]. A specific point also arises with transverse tension where the present proposal is to retain a low allowable. The ability to take into account the effect of thickness on the crack initiation strain may be advantageous and this has been put forward. However, on the whole, use of significantly higher values is dependent on whether or not the industry is prepared to accept an element of cracking in these reinforcement layers. In principle, this will be satisfactory provided that the gel coat which acts as the corrosion barrier remains sound. Whether or not some of the very high measured values of apparent strength for certain angles and stress ratios, for example, with the 55° laminate in the 2 : 1 stress ratio region, should be used for design purposes is unclear. Certainly they appear to be real effects for short-term data, but whether or not they will be so marked when loads are extended over long periods and damage accumulation is through time dependent processes remains to be seen. It is well known, for example, that modes of failure can change significantly with time [7]. The cause of the very high values, i.e. are they manifestations of the test method or the result of some other mechanism, will need to be understood before this can be taken further. It will be interesting to examine the results from the other contributors to this exercise in this area. As to ‘pushing the limits’ for the upset condition some benefit will be gained through use of the undoubted load carrying capability between design and final failure through a proposal to allow a doubling of design strain for what are defined as rarely occurring events [8]. This will still be conservative in the context of short-term ultimate performance, but could make a large difference to the economics of a particular structure. More generally it is likely that the simple approach to design as highlighted here would be of benefit to other industrial sectors, although the direct application of established design codes such as BS4994 for other materials and applications should only be done with care. The scopes of such documents are carefully crafted and limitations should be noted as they will be linked to basic assumptions underpinning the design processes. For example, glass reinforcements are only modestly anisotropic with a ratio of longitudinal to transverse modulus in the region of 2–3. Carbon composites on the other hand have degrees of anisotropy an order of magnitude greater. This means design formulae may not be adequate and in these cases designers must resort to other sources of guidance. Similarly, use for other

738 applications should not be undertaken as important issues may not be addressed. For example, a wide range of tubular products are used in the offshore industry, but these tend to be thick walled (relative to diameter) and the service conditions can be dominated by non-pressure related loads. These factors introduce a number of technical issues that are simply not considered (as they are not relevant) in BS4994 and similar documentation. This does not mean that procedures using simple design rules are not possible for other materials or applications. The key point is that they need to be tailored to meet specific requirements as with the introduction of simplicity there will be a loss of generality.

5. Conclusions This paper considers the comparison of measured failure data with the calculations methods currently embodied in the prevailing design code for GRP process equipment (BS4994). It has always been acknowledged that the design method in this document contains a number of very basic assumptions, some of which would not find favour in, for example, the academic community. However, this is balanced by a simplicity of approach, which all organisations in the supply chain can understand and interpret. In the future more sophisticated, computer based methods may become adopted, but this is not the current need at the moment. Bearing in mind the nature of the calculation method the results of the exercise are considered to be satisfactory. A number of improvements are possible, which have the scope to reduce some of the conservatism without compromising integrity or the benefits of the presentation method. The results of this exercise support this view. The process of revision is ongoing through proposals to the standards forming infrastructure and it will be of interest to see if the other contributions to this exercise can lead to additional inputs to the arising documentation.

References [1] Eckold GC. Failure criteria for use in the design environment – part A. Composite Science and Technology 1998;58(7). [2] BS4994: Specification for design of vessels and tanks in reinforced plastics, BSI, 1984. [3] Eckold GC. A performance based design methodology for glass reinforced plastic pipework and fittings. Proc Instn Mech Engrs, vol 1995; 209. [4] Soden PD, Hinton MJ, Kaddour AS. Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates: Failure exercise benchmark data: V62, pp 1489–1514, 2002, see also Chapter 2.2 in this book. [5] Phillips LN. Design with advanced composite materials. London: Springer-Verlag, 1989. [6] Soden PD, Kitching R, Tse PC, Tsavalas Y. Infiuence of winding angle on the strength and deformation of filament-wound composite tubes subjected to uniaxial and biaxial loads. Composites Science and Technology 1993;46:363–78. [7] Provost ET. Extending the GRP option: a performance based purchase specification for GRP pipes and fittings. Glasgow: OMAE, 1993. [8] Eckold GC. A design method for filament wound tanks and pipework – 2. Proc Institute of Mechanical Engineering [in press].

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

739

CHAPTER 5.3

A comparison of theory and experiment for the stress-based Grant-Sanders method E. C. Edge British Aerospace, Military Aircraft Division, Structures Unit W427C, Warton Aerodrome, Preston, Lancashire PR4 1AX, UK Abstract In general the predictions given in the part A paper [Compos. Sci. Technol. (1998) 1033] agree well with test results, which were supplied by the organisers of the world-wide failure exercise, where the laminate contained fibres in the directions of the applied loadings. The agreement with experiment is more variable for situations where the matrix-dominated properties assume more significance. This is most pronounced in the comparison with experimental stress–strain curves for ± 45° and ± 55° glass/ epoxy laminates, where the test-theory discrepancies were often large, although where these specimens achieved higher stress and sometimes appreciably higher strain levels than those at which final failure might have been expected, this behaviour was predicted with some success. Closer examination of these discrepancies has revealed that they mainly occurred where the ply transverse and shear strains exceeded the values for which stress v. strain data had been provided. For strength prediction the main shortcoming in the model used appears to be in the application of the shear-compression interaction in glass/epoxy laminates, where agreement with experimental results was improved by dropping this criterion, although results indicate a weaker interaction, rather than none at all. Agreement was also improved by dropping a delamination criterion which led to some under-predictions in the two cases involving (90 ± 30)° glass-epoxy laminates. Keywords: Strength and stress v. strain predictions

1. Introduction This Part B paper contains, for each of the 14 cases studied in this exercise, a graphical comparison of the original predictions given in the part A paper [1] with the experimental results. The experimental results, which were supplied by the organisers of the exercise, are fully described in [2]. These figures are numbered 1 to 14 to correlate with the test-theory comparison given to the participants (where Fig. 7 originally contained diagrams of test rigs used to obtain some of the experimental results). Where modifications have been made in the

E-mail address: [email protected] (E. C. Edge).

740 light of the experimental data these revised graphs are given by the suffix a, e.g. Fig. 3a compares an amended prediction for case 3 with the experimental results for case 3, and Fig. 10 compares an amended prediction for case 10. Section 2 of this paper comments on the test-theory correlation for each case. Subsequent sections discuss proposed refinements to the model and changes to the input data and the reasoning behind these refinements and other improvements which might be attempted in the light of the experience gained in this exercise. These raise such issues such as embedded lamina, treating the lamina as homogenous, the modelling of matrix behaviour, thermal stresses and the relation between initial and final failure. The stress-based Grant-Sanders method employed in part A, full details of which can be found in [1], is based on a ply-by-ply analysis with discrete failure criteria, the only interactions being shear-tension for matrix failure and shear-compression for fibre failure. Non-linear stiffnesses are accommodated by inputting curves of secant modulus versus strain. Shear and transverse tension stiffnesses have been taken to reduce gradually as strain increases above a threshold value. There is no capability to predict in situ values of transverse tension strength and other matrix dominated properties and values obtained from unidirectional testing have been used in all cases. Matrix failure through longitudinal splitting parallel to the fibres is not considered. Formulations to predict matrix failure parallel and circumferential to the fibre, and also interfacial shear failure have been derived at Warton, but never published nor validated against test or compared with similar work elsewhere. To have introduced them to possibly improve the predictions where longitudinal splitting or leakage of unlined tubes have been observed on test is not considered to be justified. The effect of these shortcomings on the test-theory correlation is discussed in this paper. Test cases 1–3 focussed on biaxial strength prediction of unidirectional (100% 0°) material. Cases 4–6 focussed on biaxial strength prediction of (90 ± 30)° and ( ± 55)° E-glass/epoxy tubes, while in case 9 participants were asked to predict a biaxial failure envelope for a quasiisotropic carbon-epoxy tube. Cases 7, 8 and 12–14 focussed on stress v. strain predictions for E-glass/epoxy tubes under various loading ratios, the lay-ups including (0/90)°, ( ± 45)° and ( ± 55)°. In the remaining cases (10 and 11) stress v. strain predictions were requested for the quasi-isotropic tube under two loading ratios. The model performed well in most of the strength predictions on multiangular laminates, where behaviour was mainly fibre dominated. It performed somewhat less well predicting the matrix dominated strengths of the unidirectional laminates. In the stress v. strain modelling, the use of unidirectional rather than in situ properties meant that initial failure was generally underpredicted by a considerable margin. However where under monotonic loading stress and more particularly strain levels were attained which exceeded, sometimes by a considerable margin, the levels at which failure would have been expected, a degree of success was achieved in predicting this behaviour. In this the author was assisted by past research work at Warton where similar behaviour was observed. The theory had no capability for longitudinal splitting prediction, which might have been useful in predicting leakage in unlined tubes, although the increase in crack density following initial failure can be expected to lead to leakage at some point even in the absence of longitudinal splitting. The most serious shortcoming was however observed when individual lamina strains exceeded the highest values for which stress v. strain data had been supplied. The theory lacks any capability to predict stress v. strain behaviour in such areas and attempts to improvise were generally unsuccessful.

741

Fig. 1. Biaxial failure stresses for 0° lamina made of GRP material; material type: E-Glass/LY556/ HT907/DY063.

2. Specific comments on test v. theory comparison 2.1. Case 1, Fig. 1 – biaxial failure stresses of 0° GRP lamina under y v xy The material type in these tests was E-glass/LY556/HT907/DY063. The interaction between tensile y and xy was predicted reasonably well; agreement would have been very good if the UD shear strength had been taken at circa 61 MPa (the test value) instead of the value of 72 MPa provided. The situation with negative y is puzzling. Starting with the (0.0; 61.0) test point already cited, the test results show an apparent increase in the shear strength due to the application of moderate transverse compressive stress. There are test results at roughly (–43.7; 81.9) and (–70.2; 96.5). However for higher transverse compressive stress this trend is reversed. From and including the (–70.2; 96.5) point up to the UD value of y of circa (–137; 0.0) (the mean of 3 points), the results are consistent with an interaction in which increasing the compressive stress reduces the apparent shear strength. The model provides no explanation for this apparently anomalous behaviour. As the theory assumes no interaction between the stresses in this quadrant, neither the enhancement of shear strength nor the interaction at higher negative values of y were predicted. 2.2. Case 2, Fig. 2 – biaxial failure stresses (x v. xy) of 0° T300/914C CFRP lamina Apart from the high test values of shear strength at values of x between 700 and 1000 MPa, this laminate has been modelled quite well. For the interaction between compressive x and xy a somewhat higher value of k than 1.5 [see Eq. (3)] may be indicated. Where a single

742

Fig. 2. Biaxial failure stresses for 0° lamina made of CFRP material; material: T300/914C.

specimen, or a very small sample size, is employed the scatter normally encountered within a sample manifests itself in inconsistency between test points. This seems to have been the case here, particularly in pure shear. 2.3. Case 3, Fig. 3 – biaxial failure stresses (x v. y) of 0° GRP lamina The material type in these tests was E-glass/MY750/HY917/DY063 and the loading was x v. y. A reduction in tensile y strength with decreasing tensile x is observed but over too

Fig. 3. Biaxial failure envelope of 0° GRP lamina under combined x and y stresses; material: E-glass/ MY750 epoxy.

743

Fig. 3a. Revised biaxial failure envelope of 0° GRP lamina under combined x and y streesses. Material: E-glass/My750 epoxy.

small a range to be regarded as significant. In the quadrant of + ve x and –ve y , there appear to be two populations. Each represents an interaction that the theory does not predict, in which the presence of a tensile x reduces the compressive y strength and vice versa. Whatever interpretation is put on the foregoing, the author considers the observed behaviour to be the result of the absence of fibres in the direction of the y load and does not expect it to be seen in a well balanced laminate. 2.4. Case 4, Fig. 4 – biaxial failure envelope of (90°/ ± 30°) GRP laminate (x v. y) The material type in these tests was E-glass/LY556/HT907/DY063. Agreement is reasonable over most of the envelope, but the compression-compression test results appear to have suffered through buckling. The application of the delamination criterion in the tension-tension quadrant appears questionable, while at the bottom of the curve (–x with small y) the severity of the assumed shear-compression interaction is open to doubt. Some aspects of the calculated envelope are similar to those shown in Fig. 10 (see Section 2.9) for the quasiisotropic AS4/3501-6 carbon-epoxy tubes under x v. y loading. 2.5. Case 5, Fig. 5 – biaxial failure envelope of (90°/ ± 30°) GRP laminate (x v. xy) The material type in these tests was E-glass/LY556/HT907/DY063. Agreement is patchy at best, with the (1 : 0) and (–1 : 0) results considerably underpredicted. The severity of the delamination and the shear-compression interactive criteria are also questionable.

744

Fig. 4. Biaxial failure envelope for (90°/ ± 30°) laminate under combined x and y stresses; material: E-glass/LY556 CpOXy.

2.6. Case 6, Fig. 6 – biaxial failure envelope of ( ± 55°) GRP laminate (x v. y) The material type in these tests was E-glass/MY750/HY917/DY063. The predictions did not take account of the differing tube configurations (i.e. with or without liner, thick or thin). A lack of symmetry is obvious in the test results in tension-tension where the failure envelope bends sharply over to the right. It was impossible to reproduce this bias with the analysis assumptions and the best explanation that can be offered for it is that fibre rotation may have occurred at (y : x) > 2.5 so as to increase the suitability of the laminate to resist these loadings. It is also noted that this bias is largely absent from the compression-compression results. A further point is that for y /x ~ Ey /Ex ~ 2/1 the compressive strength approaches the tensile. This could not have been predicted from the UD values supplied. It has, however, been noted that at (y : x) = 2/1 the Grant-Sanders model predicted a complete annihilation of matrix properties so that the same prediction would be obtained with the netting analogy. 2.7. Case 7, Fig. 7 – stress v. strain curves for ( ± 55°) GRP laminate (y /x = 1/0) The material type in these tests was E-glass/MY750/HY917/DY063 and the loading was uniaxial y. The predictions gives good agreement with test up to y ~ 266 MPa, which is where in plane shear failure is predicted with F12 = 73 MPa. Beyond this point the curves diverge widely. Unfortunately the stress v. strain curve supplied for this material in shear

745

Fig. 4a. Revised biaxial failure envelope of (90/ ± 30) GRP laminates.

understandably does not go beyond this point. A similar situation reigned with respect to transverse compression where the strain had also reached the end of the curve supplied by this point. Some attempt has been made to extend the stress v. strain curves beyond the data supplied by backing information out of the test values. When the model encounters a value of strain higher than any for which input data is available, it simply extrapolates linearly from the two

Fig. 5. Biaxial failure stresses for (90°/ ± 30°) laminate under xy and x stresses.

746

Fig. 5a. Revised biaxial failure stresses for (90°/ ± 30°) laminate under xy and x stresses in the revised curves: delamination criterion and shear-compression interaction dropped.

highest values of strain for which data is supplied. This is clearly liable to significant error, so an attempt was made at high strain values to estimate what value of modulus would give the best agreement with test. This is what is meant by ‘backing information out of the test values’. This, perhaps not surprisingly, proved to be something of a lottery, and it is felt that no useful purpose would be served by giving details of these calculations, except to say that the lamina failure shear strain used to construct the stress–strain curves in Fig. 7 was 4.7 rather than 4% as supplied by the organisers. The extreme non-linearity of the material means that severe convergence problems were encountered beyond y = 266 MPa. Only a methodology which actually predicts the loss of matrix properties with increasing stress–strain would appear to be able to perform this task successfully.

Fig. 6. Biaxial failure envelope of ± 30° GRP laminates.

747

Fig. 6a. Revised biaxial failure envelope of ± 55° GRP laminates.

However there are positive features. It was stated in the part A [1] paper that in-plane shear failure is considered to be final failure because the distortion which usually accompanies this condition is unacceptable from an aircraft structural viewpoint, not because the structure cannot take additional load. QED with a vengeance! Here the test piece survived the predicted shear “failure” stress by a factor of 2.25 to 1 and attained a strain 5.5 times the predicted failure value, which itself was large.

Fig. 7. Stress–strain curves for ± 55° GRP laminate under uniaxial tension (y /x = 1/0).

748 It was because he anticipated and wished to demonstrate this that the author did not terminate the stress v. strain curve of Fig. 7 at 266 MPa. However in the failure predictions of Fig. 6, 266 MPa is given as the final failure value at the (0,1) point. This is because the predicted failure mode is in-plane shear. The theory considers this as final failure because in an aircraft situation, for which the model was developed, the distortion associated with this level of loading is considered to be unacceptably large. It does not mean that the structure cannot take any more loading, as this exercise amply demonstrates. However as far as the failure theory is concerned, the carrying of load beyond 266 MPa in-plane shear stress is not predicted. The sketchy nature of the attempt to extrapolate beyond that point is amply illustrated by the large initial drop in the predicted stress followed by a linear predicted response, neither of which appear in the experimental curve. In the context of pressure vessels, this failure prediction can reasonably be compared with the leakage stress, but should be considered as a lower bound to the leakage stress. Note that at this point the initial and final failure predictions coincide, and it is the initial failure value that should be related to the leakage stress where they are distinct. In addition the exercise demonstrates clearly that it is not acceptable to put matrix stiffnesses to zero following “failure.” The loss of stiffness is large but gradual and the “failed” plies continue to make some contribution well beyond the failure strength measured in a UD test. The final linear portion of the predicted curve is the result of an implicit assumption that while the matrix dominated stiffness, including shear stiffness, can be virtually annihilated, the fibre dominated ones remain constant. Thus behaviour after the annihilation point is predicted to be linear.

2.8. Case 8, Fig. 8 – stress v. strain curves for ( ± 55°) GRP laminate (y /x = 2/1) The material type in these tests was E-glass/MY750/HY917/DY063 and the loading was (y /x = 2/1). The agreement for y is not as good as might appear at first sight as a continuous softening was predicted, whereas the test results, after an initial softening, showed an apparent hardening later. The very large discrepancy in x is due to a similar problem to that encountered in the uniaxial curve, i.e. that the material was operating at higher strain values than those for which stress v. strain data was available. In this case transverse tension rather than transverse compression behaviour is of concern. The author thinks that the only way the model can be adapted to obtain better agreement for x would be to actually model the non-linearities rather than rely on inadequate stress v. strain data as in this exercise, and that this would eliminate the feature where x is predicted to become negative at high stress, in contrast to the actual experimental behaviour where x attained high positive values at high stress. It has been found that if E2 is kept constant but G12 is allowed to vary, that negative x was predicted, so that it appears that the negative x was predicted as a result of the use of the shear stress v. strain data supplied by the organisers in conjunction with the author’s model. The strain x had turned negative just below the maximum strain for which data was available, so that incorrect extrapolation to higher strains was not the root cause of negative x, although the absolute value of predicted sx increased rapidly once extrapolation commenced. It would appear that that this laminate configuration in conjunction with the loading is sufficiently

749

Fig. 8. Stress–strain curves for ± 55° laminate made of E-glass/My750 epoxy material under y / x = 2/1.

different from the conditions under which the data was generated to render any read across unrealistic. This model has been developed in an aircraft research environment, and validated largely against test data for final rather than initial failure. Consequently there is less emphasis on initial failure than might be the case with work on pressure vessels. The author considers his initial failure predictions might be considered a lower bound for leakage stresses, but leakage may not be detected until damage has progressed somewhat from initial failure, and the theory employed cannot model this process. Except where initial and final failure predictions coincide, final failure is not expected to give much guidance to leakage stresses. Final failure stresses could be considerably greater than leakage stresses just as they can be very much greater than initial failure stresses.

2.9. Case 9, Fig. 9 – biaxial failure envelope of (0°, ± 45°, 90°) AS/3501-6 laminate (x v. y) The material type in these tests was (0°, ± 45°, 90°) AS/3501-6 and the loading was (x v. y). In this exercise excellent agreement was achieved in the tension-tension quadrant. An implication of this is that there is little if any interaction between the two loads in this quadrant. Agreement is poor in the compression-compression quadrant, but this is attributed by the experimenters to testing problems, particularly buckling. It is not possible to determine whether the low test results in the tension-compression quadrant are due to fibre shear failures giving a cut-off or again the result of premature failure due to buckling. If compressive failure is mainly by buckling, then if a lower compressive allowable were to be used, agreement may

750

Fig. 9. Biaxial failure stresses for (0°/ ± 45°/90°)AS4/3501-6 laminates.Iaminates.

improve. There are certainly problems with this test set-up in all but the tension-tension loading mode. It has been observed that there are similarities between the results of these tests and those on the (90°/ ± 30°) GRP laminate (see Section 2.4 and Fig. 4). In both cases there is good agreement in the tension-tension quadrant whereas premature failure, presumably due to buckling, occurred in compression compression. The tension compression results are also similar in the two cases. In contrast, the use of thicker specimens appears to have prevented buckling in compression compression in the ( ± 55°) GRP laminate and the full strength appears to have been realised there. However, the different configuration leads to a very different shape of envelope in the ( ± 55°) case.

2.10. Case 10, Fig. 10 – stress v. strain curves for (0°, ± 45°, 90°) laminate (y /x = 1/0) The material type in these tests was (0°, ± 45°, 90°) AS/3501-6 and the loading was y uniaxial. For y both test and theoretical curves show a change of slope due to matrix softening, the theoretical curve showing it at about 0.21% strain and the test curve at about 0.63% strain. The main if not the only reason for this large discrepancy in predictions is that the in situ (embedded) lamina properties within a multiangular laminate are a function of the laminate configuration and loading and can differ significantly from the pristine state as measured by a test on a unidirectional coupon. The theory lacks the ability to predict these in situ lamina properties. Matrix cracking has as a result been predicted to occur at the failure

751

Fig. 9a. Revised biaxial failure stresses for (0°/ ± 45°/90°)AS4/3501-6.

Fig. 10. Stress–strain curves for (0°/ ± 45°/90°) laminate under uniaxial tension (y /x = 1/0).

752 strain of a 90° coupon, this being appreciably less than can be expected in a well balanced laminate. However, it should be noted that the non-linearity in the test curve is just as pronounced as in the theoretical; it has been appreciably delayed but definitely not eliminated. The theory has overestimated the Poisson’s ratio. The author now understands that the actual experimental loading was closer to 20/1 than to 1/0 loading ratio. Performing a calculation at 700 MPa stress, the 20/1 ratio gave a prediction for x 16.3% less than for 1/0, whereas the test value was 22.6% lower than the prediction for a 1/0 ratio. Thus about 72% ( 16.3/22.6) of the test-theory discrepancy appears to have arisen from using an incorrect input loading. The predictions for sy are not significantly affected.

2.11. Case 11, Fig. 11 – stress v. strain curves for (0°, ± 45°, 90°) laminate (y /x = 2/1) The material type in these tests was (0°, ± 45°, 90°) AS/3501-6 and the loading was (y /x = 2/1). In this case the non-linearity in the test curves was much sharper suggesting a much more rapid loss of matrix stiffness beyond initial failure. This behaviour was not well captured in the theoretical curve, nor was the sudden loss of stiffness at high strain, which suggests that some fibre failures may have occurred. However, agreement between test and theory, with the exception of initial failure, is reasonable on the whole, showing how fibre properties dominate the response for this type of laminate. There was again a large discrepancy in the prediction of the change of slope due to matrix softening, which occurred at about 0.75% strain against

Fig. 11. Stress–strain curves for (0°/ ± 45°/90°) laminate under biaxial stress (y /x = 2/1).

753 a theoretical prediction as low as 0.17%. The reason is the same as in the case of the 1/0 load ratio (Section 2.10), i.e. that the model lacks the capability to predict in situ as opposed to pristine lamina properties.

2.12. Case 12, Fig. 12 – stress v. strain curves for ( ± 45°) GRP laminate (y /x = 1/1) The material type in these tests was E-glass/MY750/HY917/DY063 and the loading was (y /x = 1/1). Considering the problems encountered with the ± 55° configuration the agreement, while not looking particularly good on paper, is considered by the author to be satisfactory. However initial cracking was underpredicted at 0.145% as compared with a test value of circa 0.27%. The reason for the test-theory discrepancy in initial failure predictions is basically the same as the even larger discrepancies discussed in Sections 2.10 and 2.11, i.e. that the model lacks the capability to predict in situ properties from the configuration of a multiangular laminate. The theory does not predict leakage of unlined tubes as such. The author does not regard this as a genuinely distinct stage of failure; instead it is postulated that it will occur as a result of the gradual increase in crack density following initial cracking, possibly followed by longitudinal splitting, until a threshold value is attained at which liquid leaks through the tube wall. This leakage was observed at 0.97% strain. Final failure by fibre fracture of lined tubes was underpredicted, occurring at 509 MPa against a predicted value of 641 MPa. This could result from the non-optimum configuration of the specimen, which could cause fibre rotation which allied to matrix damage could cause premature failure. This may also account for the overprediction of the laminate stiffness at strains higher than about 0.5%.

Fig. 12. Stress–strain curves for ± 45° E-glass/MY750 laminate under y /x = 1/1.

754 The effect of thermal stresses on the predicted curves has been investigated by repeating the calculations with this feature removed. For low mechanical strains, where there has been little or no degradation of stiffness, the predicted laminate strains including the thermals were up to 10% lower than the corresponding values without thermals thermals (thermal stresses were calculated for T = –100°C and no moisture). However the free thermal strains have been assumed to be independent of the degree of matrix damage, with the result that as the matrix dominated stiffness properties reduce with increasing mechanical strain, so the thermal stresses reduce. In the limit, at high laminate strain and considerably reduced matrix dominated stiffnesses, the thermal effect becomes negligible.

2.13. Case 13, Fig. 13 – stress v. strain curves for ( ± 45°) GRP laminate (y /x = 1/–1) The material type in these tests was E-glass/MY750/HY917/DY063 and the loading was (y /x = 1/–1). Particular difficulty with numerical convergence was encountered in this exercise, and convergence was only effected for the part A paper [1] by assuming a stress v. strain behaviour pattern much stiffer than that actually observed on test. However, as far as the failure theory is concerned, the carrying of load beyond 73 MPa in-plane shear stress is not predicted. The sketchy nature of the attempt to extrapolate beyond that point is amply illustrated by the large initial drop in the predicted stress followed by a linear predicted response, neither of which appear in the experimental curve.

Fig. 13. Stress–strain curves for ± 45° laminate made of GRP material (y/x = 1/ – 1).

755 In these circumstances the author accepts his inability with his current model to undertake a sensible prediction for this case, and feels unable to comment on the discrepancy between his predictions and the test curve, except to say that the latter is in line with rational expectations. In the table in Section 4.5 a failure strains of + 2.35% is listed as the predicted failure strain for this laminate although this corresponds to in-plane shear strains of + 4.70% in the individual plies. The explanation for this apparent anomaly lies in the non-linear stress v. strain behaviour in shear. Using the in-plane shear stress v. strain data supplied to the participants ([14], Table 6), and the interactive algorithm given in Ref. [1], the value of inplane shear modulus G12 corresponding to a strain of 4.5% is calculated as 1.55 GPa giving a stress of 73 MPa. This value is the failure stress in in-plane shear supplied to the participants, and this value of 73 MPa has been assumed to be independent of the loss of stiffness. Paradoxically the predicted laminate failure stress is in this case independent of the level of thermal residual stress. This is because although a transverse tensile strain and a small longitudinal compressive strain is predicted due to thermal effects, no corresponding residual shear strain is predicted. The effect of the predicted transverse tensile strain, in conjunction with the high level of shear stress, is to reduce the transverse tension modulus, so that a higher laminate strain to failure is predicted in the thermal case, even though there is no corresponding change in the predicted failure stress. The combination of transverse tension and shear is not considered to reduce the final failure stresses in the laminate [1]. A failure strain of + 2.0% is predicted ignoring thermal effects.

2.14. Case 14, Fig. 14 – stress v. strain curves for (0°/90°) GRP laminate (x /y = 1/0) The material type in these tests was E-glass/MY750//HY917/DY063 and the loading was x uniaxial. The test v. theory comparison is very similar to that for the (0°, ± 45°, 90°) CFRP laminate. On test, the non-linearity appears at about 1.55 times the strain at which it occurs in the theoretical curve, a value of x of 0.36% against 0.23%. In the theory this onset of nonlinearity is attributed to transverse tension failure on the 90° plies. Transverse tension in the 0° plies is predicted at a strain x of 1.26% (stress of 294 Mpa). On test longitudinal splitting was observed at a strain x of about 1.62%, about 26% higher than the theoretical figure for transverse tension failure. It is considered that this comparison is legitimate. Otherwise agreement is good. Final failure by fibre tension failure was predicted to occur at a stress of 640 MPa, compared with failure in this mode on test at 611 MPa, an over-prediction of only about 4.7%, which is satisfactory. The reason for the test-theory discrepancy in initial failure predictions is basically the same as the even larger discrepancies discussed in Sections 2.10 and 2.11, i.e. that the model lacks the capability to predict in situ as opposed to pristine lamina properties. As the Poisson’s ratio of a 0°/90° laminate is small y is also small, nevertheless the theory over-predicted y at high values of strain by a margin of about 2 to 1. The theory predicts that the E2t, G12 and v12 values for the 90° ply will be virtually annihilated at a stress of 384 MPa, well below the value at which the test-theory discrepancies became significant. Therefore if E2t, etc. had been decreased even more rapidly this would not have improved the test-theory match at high strain. On the other hand if E2t, etc. had been decreased less rapidly a higher

756

Fig. 14. Stress–strain curves for 0°/90° E-glass/MY750 laminate under uniaxial tension.

laminate Poisson’s ratio would have been predicted at high strain and the test-theory discrepancy would have been even greater. It is concluded that the way E2t, etc. have been decreased in the 90° plies is not the cause of these discrepancies. However, with the 0° plies being under transverse compressive strain, although the transverse stress in these plies is tensile, it has been assumed that they retain their virgin properties right up to final fibre failure. This ignores the effect of matrix stress parallel to the fibres, which can be expected to reduce the matrix stiffness and Poisson’s ratio. However, as in this example we are dealing with uniaxial loading in the x(0°) direction, it is the stiffness of the 0° plies in the fibre direction that is important. As the axial stiffness of the fibres is an order of magnitude greater than the matrix stiffness any loss of matrix stiffness will have a very small effect on the ply and laminate longitudinal stiffness, and the test-theory correlation for sx will not be affected to any significant degree. However, the damage to the matrix may have caused a reduction in ply and laminate Poisson’s ratio. As the test results were obtained from tests on coupons under uniaxial tension it would appear unlikely that these coupons would have been subject to any unintentionally applied external transverse tension loading. However something like this appears to have occurred in case 10 (see Section 2.10). Either effect may go some way to explaining the test-theory discrepancy in this exercise.

757 3. Modifications to the model/data to improve accuracy of predictions Three modifications to the theory were made in an effort to improve the test-theory correlation. The delamination criterion was removed for reasons discussed in Section 3.1, but in this exercise this affected only cases 4 and 5. The shear interaction with longitudinal compression was removed for glass epoxies but retained for carbon epoxies. The reasons and implications of this are discussed in Section 3.2. In case 3, biaxial failure stresses of 0° GRP Lamina E-glass/MY750/HY917/DY063 under x v. y loading, some ad hoc experiments were tried to improve the test-theory correlation in the quadrant of + ve x and –ve y . Because different failure mechanisms can be expected in other quadrants and also in cross-ply, angle-ply and multiangular laminates, more general application of these modifications is not proposed. These attempts are described in Section 3.3.1.

3.1. Delamination criterion The predicted truncations due to delamination in the failure envelopes for the (90°/ ~ 30°) laminate in Figs. 4 and 5 were not supported by the test results. In consequence it has been decided to redraw the theoretical envelope with the criterion eliminated, and it can be seen (Figs. 4a and 5a) that agreement with test is improved thereby. The philosophy behind this criterion [6] was that, while out-of-plane edge stresses may be present, of sufficient value to have caused local layer separation, delamination failure will not propagate across a section until a critical level of interlayer loading is reached away from the edge boundary layer. The criterion (see Section 3.2.5 of Ref. [1]) was made dependent on 12 because it was thought that the interlaminar stresses were proportional to 12. In fact only the 13 branch is proportional to 12, the (23; z) branch being proportional to 2 [7]. Also delamination may be initiated away from the edges by matrix cracking, and more importantly still edge effects are not present in plain tubes. The point that delamination failure will not propagate across a section until a critical level of interlayer loading is reached away from edge boundary layers remains valid. However a decision has been taken to remove the criterion from the model altogether. Its retention in this exercise was an oversight. In practice, the criterion has been invoked only rarely, for thick lamina, serving to warn against excessive ply blocking which is now generally accepted to be bad practice. For example, Ref. [7] states “The magnitude of the edge stresses is increased when, for given laminate thickness, a smaller number of layers is used each containing a large number of plies.”

3.2. Longitudinal compression-shear interaction The interactive formula used is: 1 | 12 | + ≥1 H1c H12

(1)

where, as stated in the part A paper, H1c and H12 may be the same as F1c and F12 respectively, but are not necessarily so, though generally H12 = F12. The criterion was derived following

758 combined loading tests on ± 45° HTS/914C tubes (Ref. [6], see Fig. 15). These results fitted a formula: 1 | 12 | + ≥1 F1c kF12

(2)

with k = 1.5. However results presented by Fleck, Ref. [8], see Fig. 16) for unidirectional graphite-epoxy tubes give excellent agreement with k = 1. In this exercise, for the T300/914C unidirectional lamina under x v. xy , k = 1.5 was used. There is reasonable agreement but a somewhat high value of k is indicated. No such agreement was obtained for the (90°/ ± 30°) and ± 55° glass-epoxy laminates. The argument in favour of this interaction is that it is the support against fibre microbuckling provided by the resin matrix which is critical. The resin shear modulus Gm reduces with increasing strain/stress, hence it seems reasonable to expect a reduction in compression strength when coupled with shear. Work by Wisnom [9] has shown by finite element analysis that very high shear stresses are induced into u/d compression test specimens at the tabs in indirect testing. Recent work at BAe Warton has shown that the same feature is present to a lesser degree in the testing of multiangular laminates. A new test method recently developed [10], where the load is applied to the end of the specimen, has yielded failure strains as high as 1.96% for u/d XAS/914C. The data supplied for this exercise show that the glass fibres have a much greater transverse modulus (by a factor of 5) than carbon fibres (their shear modulus is over double). It is

Fig. 15. Shear-compression interaction for combined loading test on + /–45° HTS/914C tubes [6].

759

Fig. 16. Stress–strain curves for a ± 55 laminate made of E-glass/ MY750 epoxy material under y /x = 2/1.

postulated that in view of this, glass fibres require less support in compression than carbon, and hence are less affected by a reduction in Gm. There may be indirect confirmation of this in Wisnom’s observation [11] on the non-linear stress–strain response of carbon-epoxy laminae in compression, consistent with a zero tangent modulus at roughly –3.5% strain, whereas glass-epoxy laminae exhibit much less non-linear behaviour. The upshot is that longitudinal compression strength itself, and by implication any interaction with shear, is more of in situ rather than a material property. 3.3. Revised predictions for certain cases 3.3.1. Case 3, Fig. 3/3a – biaxial failure stresses (x v. y) of 0° GRP lamina An attempt has been made to obtain a better agreement between test and theory. Figure 3a shows it superimposed on the curve theoretical curve from Ref. [1]. This prediction uses a modified Azzi-Tsai criterion. This criterion which has been cited previously by Norris and McKinson [3] is expressed as:

冉冊冉冊冉冊 1 F1t

2

+

2 F2c

2

+

12 F12

2

≥1

(3)

If an additional term – 12 /(F1t)2 is included in the l.h.s. of (3), the criterion becomes identical to that of Tsai-Azzi [4]. This 12 term has been omitted because its physical justification is not obvious to the author. In this case its effect, while small, marginally worsens the agreement with test. However, the author expects it to lead to incorrect predictions when it is significant. Information supplied by the organisers with the experimental results suggests that a ± 5° rather than a 0° laminate may be a better representation of the laminate configuration. Therefore, calculations using (3) have been performed for both a 0° and a ± 5° laminate, but

760 in the fourth quadrant only (the test results are confined to the first and fourth quadrants), both curves being shown on Fig. 3a. Selective use of (3) in the fourth quadrant only and not in the first, where it would give unrealistic results, is considered to be justified by the different transverse failure modes involved, brittle in tension and ductile in compression/shear, with a ratio of 3.5 to 1 in transverse strength. It will be seen from Fig. 3a that there is a significant difference between the curves based on and ± 5° laminates gives a better fit. For relatively low values of x the ± 5° curve gives an excellent fit, but for high values of x it is the 0° curve which gives the better agreement with test. No explanation is offered for this unusual result. It is accepted that this selective use of (1) does not help the author in promoting his theory, which is in any case not best suited to matrix dominated situations such as this example. It is offered only in an attempt to obtain better agreement in the fourth quadrant and to explore possible explanations of the behaviour observed on test. 3.3.2. Case 4, Fig. 4/4a – biaxial failure envelope of (90°/ ± 30°) GRP laminate (x v. y) The revised envelope shown in Fig. 4a is based on a model from which both the delamination and the compression-shear interaction criteria have been removed. These modifications considerably improve agreement in the tension-tension and compression-compression quadrants respectively, but it appears that a milder shear-compression interaction, rather than none at all, would do better still. 3.3.3. Case 5, Fig. 5/5a – biaxial failure envelope of (90°/ ± 30°) GRP laminate (x v. xy) As in the case of Fig. 4a (Section 2.4) the revised envelope shown in Fig. 5a is based on a model from which both the delamination and the compression-shear interaction criteria have been removed. Some other points in the tension quadrant have also been recalculated, where small errors had occurred in the calculations for part A. The biggest improvements in testtheory agreement can be seen at the (x; xy) = (–1 : 0), (0 : 1) and (1 : 0) although there is still a significant under prediction at (1 : 0). Apart from these points there is still under prediction in compression and over prediction in tension, except for high values of x, but not to the same extent as previously. 3.3.4. Case 6, Fig. 6/6a – biaxial failure envelope of ( ± 55°) GRP laminate (x v. y) It has been noted that at (y; x) = 2/1 the Grant–Sanders model predicted a complete annihilation of matrix properties so that the same prediction would be obtained with the netting analogy. This was followed up with netting predictions at (y; x) = 2.5/1 and (y; x) = 3/1 which gave good agreement with the experimental results. For values of (y; x) > 3/1 the netting analogy over predicts, however the a large majority of the test results in that area have been under predicted by a considerable margin, probably indicating some fibre rotation. The elimination of the shear-compression interaction from the revised Fig. 6a has improved agreement, but as is the case for the (90°/ ± 30°) GRP laminate the optimum appears to be a weaker interaction, rather than none. The netting analysis predictions at (y; x) = 2.5/1 and (y; x) = 3/1 are also shown on Fig. 6a.

761 3.3.5. Case 9, Fig. 9/9a – biaxial failure of (0°, ± 45°, 90°) AS/3501-6 laminate (x v. y) A paper in which the tension-tension and tension-compression results are published (Swanson and Nelson [5]) indicates that the u/d compression strength obtained from a tube test was –1040 MPa as opposed to the –1480 MPa obtained from a flat coupon. On the revised Fig. 6a a revised failure envelope based on F1c = –1040 MPa is superimposed. It will be seen that this revision gives significantly better agreement in the compression-compression and tensioncompression quadrants. Even if buckling problems contributed to this low u/d value, might not similar conditions be expected to prevail in all the compression tests in this series? The three highest uniaxial compression results were obtained with 1.5 inch diameter cylinders, instead of the 4 inch diameter used in the other experiments, these higher diameter tubes being presumably more susceptible to bucking failure.

4. Issues arising from laminate strength prediction exercise 4.1. Embedded laminae The computer programme allows different material properties to be specified for each layer or ply (lamina) in the laminate. In this way temperature and moisture gradients in the throughthickness direction can be analysed. No advantage was taken of this facility in this exercise and in most cases the material properties supplied were used uncritically in all applications. It is not of course questioned that many properties are in situ rather than material properties and that more accurate modelling requires them to be calculated from the lamina position within the laminate. Failing this, better predictions can be obtained by inputting values calculated elsewhere or else based on sound engineering judgement. All matrix properties can be considered as in situ and not material properties and so to a considerable extent is longitudinal (i.e. in the fibre direction) compressive strength. In this exercise using unidirectional properties and ignoring embedded effects appears to have led to a significant under prediction of initial failure in many cases. However apart from problems with the shear-compression interaction in glass-epoxies the effect on the prediction of final failure appears to be less significant, being largely confined to the laminate modulus at final failure being somewhat underestimated. An important distinction is however made between a unidirectional laminate and an embedded lamina, in that matrix failure in any mode is taken to be catastrophic in a u/d laminate, whereas matrix failure (other than in-plane shear failure) in an embedded lamina is not treated as catastrophic.

4.2. Treating the lamina as homogenous It is of course accepted that the lamina is heterogeneous and that more accurate predictions would be obtained by correct modelling of the different responses of the fibre and matrix to the applied loading. It is not felt that the strength predictions in this exercise suffered greatly from treating the lamina as homogenous. For stress v. strain more serious errors have appeared in the more matrix dominated exercises. The author’s model has been shown to be too

762 simplistic in these cases, relying on test data from unidirectional coupons which often cannot attain the strain levels reached in more complex structures. Modelling the lamina as heterogenous would appear to be one of the steps to actually modelling behaviour. The author would promote micromechanical modelling more enthusiastically had the results produced by the contributors been more consistent and better correlated to the test data. It appears that a considerable amount of work is required before the full potential of micromechanical modelling is realised.

4.3. Modelling of matrix behaviour Except when the matrix is under transverse tension, ductile behaviour is assumed. The loss of in-plane shear modulus G12 with increasing strain is taken to be reversible on unloading from up to at least 90% of the failure stress, in line with Warton cycling testing of ± 45° XAS/914C in tension [6]. It would appear intuitive that the same would be true of the transverse compressive modulus E2c, although the author knows of no direct evidence to this effect. However these assumptions have not influenced the predictions as monotonic loading to failure has been assumed in all cases. Failure in transverse tension is assumed to be via matrix cracking. Following transverse tension failure in an embedded lamina, a gradual reduction of the transverse tension modulus E2t is assumed, as illustrated in Fig. 17 for HTS/914C and XAS/914C (different rates of stiffness loss have been assumed for the materials used in this exercise, as shown in Fig. 1 of the part A paper [1], and Fig. 18 of this paper). Associated reductions in G12 and Poisson’s ratio v12 are calculated using the algorithm given in Section 2.5.2 of Ref. [1].

Fig. 17. Variation of secant transverse tensile modulus with transverse tensile strain for HTA/914C and XAS/914C.

763

Fig. 18. Variation of secant transverse modulus with transverse strain for different laminate.

The E2sec v. strain curve given for HTS/914C and XAS/914C in Fig. 18 was derived from a BAe Warton study of the difference between the stress v. strain responses of 0° and 0°/90° HTS/914C and XAS/914C coupons [6]. E2sec is constant until a strain of 0.6%, which is the failure strain of a 90° coupon in these materials. In contrast experimental work by Nensi [12] on (0°/90°)s E-glass/Epikote 828 showed that for 90° plies 0.125 mm thick and a low cured resin, transverse matrix cracking commenced at four times the strain at which a 90° coupon failed, whereas as the thickness of the central ply increased the crack initiation strain reduced, tending to the u/d value in the limit. This ratio of 4 reduced somewhat with a higher degree of cure, but remained large. The Warton results are exceptional in implying a negligible embedded lamina effect, as in the stress v. strain studies in this exercise the non-linearity always appear to commence at a higher strain than the failure strain of a 90° coupon, and it is now generally accepted that transverse tension strength is an in situ property. As it is impossible to directly generate test data to cover these in situ effects the future would appear to lie in modelling them. In contrast to the transverse compression case, the loss of stiffness at high strain is largely irreversible, at least while any residual tension, including thermal effects, remains [12]. Once started, the cracking and associated loss of stiffness appeared to follow the pattern shown in the Warton tests, with a virtually complete loss of matrix stiffness appearing to have occurred in some cases, particularly with the ± 55° laminate.

4.4. Thermal stresses The thermal stresses were modelled in the conventional way by assuming a homogenous material. On the other hand a drop of –100°C (–157°C for the AS4/3501-6 laminates) from

764 the stress free temperature was assumed, room temperature being taken as 20°C. The first assumption is optimistic and modelling the fibre and resin separately gives stresses parallel and circumferential to the fibres greater than those given by the homogenous model. Work on these aspects of thermal effects has been done at Warton [13] but has not yet been included in the computer programme. However swelling due to absorbed moisture will alleviate the stresses due to resin shrinking. For T300/914C, a material included in this exercise, a transverse thermal coefficient of expansion of 26
10–6/°C is quoted [14], and the transverse moisture coefficient of expansion of this material has been measured [15] at 0.003/(% moisture content). Thus a moisture content of about 0.87% by weight will roughly negate a temperature drop of – 100°C. No moisture contents have been supplied, but nor has there been any mention of drying specimens out prior to test. It is very probable that the laminates contained 0.5% or more of absorbed moisture, in which circumstances it is not clear whether or not the thermal effects were underestimated.

4.5. Relationship between initial and final failure In most cases initial failure was under predicted, which is not surprising as unidirectional properties were used with no consideration for embedded lamina effects. It is not possible to directly verify under prediction from the strength prediction exercises, but the following table has been constructed from the stress v. strain predictions. The test values for the strain at initial failure have been read from the supplied experimental curves, readings which may be subject to error and/or misinterpretation. The table shows under predictions of up to 4.5 to 1. While under predictions of this magnitude are bound to have some effect on stress v. strain predictions, it is difficult to pinpoint any instances in this exercise where large differences in final failure predictions resulted. The effect of allowing for initial failure as opposed to assuming undamaged lamina seems to have been no more that the reduction of laminate stiffness resulting from the loss of matrix properties. For a quasi-isotropic laminate, or even a 0°/90, this is effect is not enormous.

5. Strength prediction The strength predictions for the multiangular laminates generally agreed quite well with the experimental results. The discrepancies between test and theory for the (0°/ ± 45°/90°) laminate appear to arise more from difficulties in testing in compression than from any shortcoming in the predictions. Agreement for the (90°/ ± 30°) and ± 55° laminates exceeded expectations. Indeed the x v. y test failure envelope for the (90°/ ± 30°) was not dissimilar to that for the (0°/ ± 45°/90°) laminate, and similar difficulties in compression testing were experienced in both cases. Somewhat closer agreement can be obtained for the (90°/ ± 30°) and ± 55° laminates by revising the shear-compression interaction criteria and in the former case eliminating the delamination criterion (see Sections 3.1 and 3.2). Predictions for two of the 0° laminates were satisfactory. However in the case of the E-glass/MY750 epoxy under x v. y loading, the observed interactions in the tensioncompression quadrant were not predicted (see Sections 2.3 and 3.3.1).

765 Table 1 Comparison between predicted and measured initial failure strains for selected cases. Initial failure strain (%) Laminate

Loading

Test

Predicted

0/90 E-Glass ± 55° E-Glass ± 55° E-Glass (0/ ± 45/90) CFRP (0/ ± 45/90) CFRP ± 45° E-Glass ± 45° E-Glass

(x : y) = (1 : 0) (x : y) = (1 : 0) (x : y) = (2 : 1) (x : y) = (1 : 0) (x : y) = (2 : 1) (x : y) = (1 : 1) (x : y) = (1 : –1)

$x = 0.36 ($y : $x) = (4.0 : – 4.3)a ($y : $x) = (1.0 : 0.48)a ($y : $x) = (0.63 : –0.13) ($y : $x) = (0.75 : 0.13) ($y : $x) = (0.27 : 0.27) Not detected

$x = 0.23 ($y : $x) = (2.08 : –2.08)b ($y : $x) = (0.27 : 0.073) ($y : $y) = (0.21 : –0.063) ($y : $x) = (0.17 : 0.033) ($y : $x) = (0.145 : –0.145) ($y : $x) = (2.35 : –2.35)b

a

Leakage of unlined specimens observed at this strain. In-plane shear failure predicted. This would be regarded as final failure in an aircraft structure, but under monotonic loading (at least) the tubular specimens may be expected to take some further load beyond this point. b

6. Stress v. strain predictions Agreement for fibre dominated loading is considered to be satisfactory. It might have been improved had stress v. strain data been available in transverse tension. In the absence of such information, curves were used which had been read across from XAS/914C data. This has been illustrated in Fig. 1 of the part A paper [1] and Fig. 18 of this paper. This read across was inevitably of an arbitrary nature. However, the failure to make proper allowance for embedded lamina was almost certainly more significant. The point must be made that for the stress v. strain curve of a quasi-isotropic laminate or even a 0°/90° to exhibit even a dinkum nonlinearity in laminate behaviour requires extreme non-linearity in the matrix stiffnesses. Agreement for matrix dominated loading was less good, largely because individual laminae operated at strains well above those for which unidirectional properties can be obtained from test, and there is no facility within the computer programme to calculate the response under such conditions. However early experience with the ± 45° tension test had shown the author and his colleagues how this type of laminate can continue to take monotonic load (at least) up to well beyond the nominal failure level. Consequently the calculations were continued up to very high strains, ignoring apparent failure conditions.

7. Discussion and conclusions 7.1. Specific conclusions (a) Predictions generally agree well with test results where the laminate contains fibres in the directions of the applied loadings. However, the shear-compression interaction appears to be appreciably weaker with glass than with carbon/epoxies (b) The agreement with experiment is more variable where the matrix dominated properties are significant. Predictions of failure remain reasonable on the whole but the theoretical stress v. strain curves for the ± 45° and ± 55° laminates lacked realism at high strains although it

766 was correctly anticipated that very high strains would be attained despite failure conditions apparently having existed at much lower load levels. (c) The Grant-Sanders contention that in cross-ply and multiangular laminates the matrix dominated ply stiffnesses are not annihilated on initial failure, but neither are the nonlinearities completely eliminated by the support of fibres of different orientations, is considered to be justified by the experimental stress v. strain results. The loss of stiffness is gradual over a range of increasing strain. However in some cases at least, particularly with the ± 55° glass-epoxy laminate, it appears probable that before failure the matrix stiffnesses had been annihilated and the laminate was operating as a net. (d) The problems predicting the stress v. strain behaviour of the matrix dominated ± 45° and ± 55° laminates appear to be largely due to the individual plies operating at higher strain levels than those for which u/d data was supplied, see Table 1. It is not possible to obtain the required data from unidirectional testing.

7.2. More general observations (i) The model has generally worked well for strength predictions on the cross-ply and multiangular laminates, even the ± 55° and (90°/ ± 30°) glass-epoxy laminates which might have been expected to exhibit matrix dominated behaviour. Strength predictions for unidirectional (0°) laminates were somewhat less successful, mainly, it is felt, because they exhibit matrix dominated failures which in cross-ply, angle-ply and multiangular laminates would amount to no more than initial failure and in most cases be delayed until a higher level of laminate loading. However, for glass-epoxies the longitudinal compression interaction with shear was much weaker than predicted, and the delamination criterion should not have been used on tubes, indeed it has been decided to eliminate it from the model. (ii) For stress v. strain predictions the model worked well for the levels of shear and transverse compressive strains for which the organisers had supplied stress v. strain data. However, no stress v. strain data was supplied in transverse tension, and as the theory relies on modelling this feature, the author was forced into attempting to read across this data from that obtained in house for other materials. Also there is no capability in the model for deriving in situ properties from those obtained from unidirectional coupons. A combination of using unidirectional data and probable errors in read across led to initial failure being underpredicted by ratios of up to 4.5 to 1. The effect of this on the strength predictions was not large, as the bulk of the laminate stiffness is provided by the fibres whose stiffness has been assumed not to be degraded. In the stress v. strain predictions this defect was inevitably more significant. Where there were fibres oriented in the loading directions, as for the 0°/90° and quasi-isotropic laminates, the underprediction led to the non-linearities observed in the experiments being predicted to occur at lower load levels, i.e. the basic nature of the nonlinearities was predicted quite well, but in practice they were observed at higher load levels than was predicted. For the ± 45° and ± 55° laminates, however, with more severe nonlinearities, the response beyond initial failure was of a different shape from that predicted. It proved impossible to rectify this situation by changing either the model or extrapolating the stress v. strain data supplied, and no purpose would appear to be served by reporting such attempts as were made. The only way forward appears to be to actually model the changes in lamina and laminate stiffnesses.

767 (iii) The features discussed in (ii) above are seen as the main shortfalls of the model, i.e. underprediction of initial failure by up to about 4.5 to 1 and inability to actually model stress v. strain behaviour of individual laminae, including embedded effects, the results of which can be seen in Figs. 7, 8 and 13. Shortcomings in modelling the shear-compression interaction can be reduced (indeed if required the interaction can be suppressed altogether) by manipulating the input data. (iv) The main lessons learnt by the author, are the need to actually model the material nonlinearities and lamina embedded effects rather than relying on given data and to take more care on the longitudinal compression interaction with shear, particularly when dealing with glass fibres with which he is less familiar than carbon. (v) The author considers his model mature enough for use in an aircraft general design case and also in research in that industry to predict the mode and loading level at failure and analyse failures. In saying this he makes assumptions about the laminate configuration, namely that if the fibre orientations are 0, 90 and ± 45, there will be at least 12.5% present of each of these orientations. Configurations with more than four fibre orientations are also accommodated provided that their stiffness and strength distributions are equivalent, e.g. 0, ± 22.5, ± 45, ± 67.5, 90 with at least 6.25% of each present. For pressure vessel design, he has confidence in the current model for final failure, but there has to be caveats with the current model for predictions of initial cracking and leakage from unlined vessels, unless reliable lamina properties are available appropriate to the actual in situ position in the laminate. (vi) Taking the quasi-isotropic AS4/3501-6 laminate (cases 9, 10 and 11), the stiffness of this laminate is so fibre dominated that the difference between a failure prediction assuming no loss of matrix stiffness and one assuming more or less complete loss of matrix properties (a pessimistic assumption in this case) is little more than 10%. For such a calculation the actual load level at initial failure is not of much significance. Initial failure is clearly of much more significance for pressure vessels in the nuclear industry, particularly unlined vessels, because of leakage problems, which can be expected when the crack density develops to a critical level. (vii) The actual values assigned to thermal stresses have had negligible effect on the final failure predictions. As pointed out in Ref. [16] (Table 9, p. 1252), they can make a significant difference to initial failure predictions. In Section 4.4 it is suggested that higher thermal stresses may be calculated by considering the lamina as heterogeneous, but on the other hand unless the specimens used in these experiments were dried before loading, the likelihood is that they contained sufficient moisture (say 0.5% by weight) to considerably reduce the residual stresses.

8. Recommendations (i) Testing is recommended to determine how much of the stiffness loss consequent on transverse matrix cracking can be recovered by unloading followed by loading in compression, and to relate this to the loading history. Work by Nensi [12] suggests that small cracks can be closed in this way and at least some of the loss of stiffness recovered. Intuitively (to the author at least) the greater the size and density of the transverse tension cracks the lower will be the recovery of property in compression. The author does however feel that the

768 assumption that any loss of stiffness under tension loading is irreversible is unduly pessimistic, nor would he believe that undamaged operation in compression following tension loading necessarily implies that no matrix cracking has taken place. (ii) The author does not recommend that unidirectional ply data, in the sense of data obtained from tests on unidirectional laminates, be the starting point of an analysis of a laminated structure. Instead he suggests that in situ unidirectional properties be calculated for the individual plies as some of the contributors to this exercise have attempted. Plies of the same orientation may have different in situ properties because of their different positions within the laminate. Surface as opposed to interior plies are obvious examples of these. The analysis would then proceed as before but with these estimated in situ properties in place of the unidirectional ones used by the author in his work. (iii) The author believes that this predictive exercise has shown that polynomial tensor failure criteria are not suitable in design models particularly under biaxial loading. He recommends that future work should be aimed at developing the discrete failure criteria approach favoured by himself and some other contributors, as the basis of laminate strength prediction. (iv) The author finds himself unable to recommend specific laminate tests to discriminate between the failure modes proposed by himself (and some other authors). There is an obvious need to carry out rigorous post-failure fractographical analyses. It might be useful if the organisers could, if at all possible, publish the failure modes observed in these experiments, so enabling those contributors whose models distinguish between different failure modes to check their predictions. (v) It is recommended that if the residual thermal stresses are to be included in the analysis, the modelling should be based on the actual heterogeneous rather the commonly assumed homogeneous laminae. It should also be remembered that moisture uptake, often an unknown quantity, works to counteracts the contraction due to cool down from cure. In general the effect of thermal stresses will be more significant for initial rather than final failure prediction.

References [1] Edge EC. Stress based Grant–Sanders method for predicting failure of composite laminates. Part A paper. Compos Sci Technol 1998;58(7):1033 42. [2] Soden PD, Hinton MJ, Kaddour AS. Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates: Failure exercise benchmark data: V62, pp. 1489–1524, 2002, see also Chapter 2.2 in this book. [3] Norris CB, McKinson PF. US Forest Products, Report no. 1328, 1956. [4] Tsai SW, Azzi VD. Strength of laminated composite materials. AIAA Journal 1966;4(2):296 301. [5] Swanson SR, Nelson M. Failure properties of carbon/epoxy laminates under tension-compression biaxial stress. In: Kawata, K., Umekawa, S., Kobayashi, A. editors. Composites’86: Recent Advances in Japan and United States. Proc. Japan-US CCM-III. Tokyo, 1986, pp. 279–286. [6] Sanders RC, Grant P. The strength of laminated plates under in-plane loading. BAe Report SOR(P)130, January 1982. [7] ESDUpac A9528. Delamination and free edge stresses in composite laminates subjected to uniform prescribed axial strain and temperature change. ESDU International plc, December 1995.

769 [8] Fleck NA. Presentation to Workshop on Compressive Failure, Cambridge University Engineering Department, July 1993. [9] Wisnom, MR. Effect of shear stress in indirect compression test of unidirectional carbon fibre/ epoxy. AIAA Journal 10, 1991;29(10). [10] Haeberle J, Matthews FL. Studies on compression failure in unidirectional CFRP using an improved test method. In: Proc. 4th. European Conf. on Composite Materials, Stuttgart, September 1990. p. 517–23. [11] Wisnom MR. On the high compressive strains achieved in bending tests on unidirectional carbonfibre/epoxy. Compos Sci Technol 1992;43:229 35. [12] Nensi T. A study of stress-induced whitening in glass-fibre reinforced epoxy laminates. University of Surrey Thesis, 1988. [13] Sanders, R.C. The influence of the matrix on the tensile behaviour of continuous fibre composite laminates. BAe Report SOR(P)157, June 1986. [14] Soden PD, Hinton MJ, Kaddour AS. Lamina properties, lay-up configurations and loading conditions for a range of fibre-reinforced composite laminates. Compos Sci Technol 1998;58(7): 1011–22. [15] Kanellopaulos VN, Wolstenholme G, Yates B. Salford University Industrial Centre Report T121 – February 1985. [16] Soden PD, Hinton MJ, Kaddour AS. A comparison of the predictive capabilities of current failure theories for composite laminates. Compos Sci Technol 1998;58(7):1225 54.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

770

CHAPTER 5.4

Comparison between theories and test data concerning the strength of various fibre-polymer composites L. J. Hart-Smith Phantom Works, The Boeing Company, Long Beach, CA, USA Abstract Predictions of laminate strengths made in three earlier papers are now compared with test data supplied by the organizers of an extensive comparison between many composite failure theories. Any discrepancies are identified and ascribed specifically to limitations of the tests or theories, as appropriate. The agreement is good, particularly in regard to fibre-dominated values. The capabilities in regard to predicting matrix failures are better than was anticipated, particularly for the Ten-Percent Rule, but they are definitely incomplete. Nevertheless, the comparison of the predictions by these theories with those from the theories of the other participants has yet to be made. The corresponding contributions from the other authors need to be examined, as well, to complete the evaluation of the many theories, and to identify what gaps might still exist in the overall capability to predict the strength of fibre polymer composite laminates. Keywords: Biaxial; B. Modelling

1. Introduction In previous papers [1–3], the author predicted the strengths of various fibre polymer composite laminates, as part of an exercise [4] organized to compare the abilities of as many as possible of the significant composite failure theories. The author covered four failure models; the widely used maximum-strain failure model as originally formulated by Waddoups [5] and as truncated by much of the US aerospace industry [6]; the author’s own generalization of the maximum-shear-stress failure model [7–9]; and the author’s empirical Ten-Percent Rule [10]. In contrast with most of the published failure models, which are highly interactive, all of these models are mechanistic (non-interactive). The author’s researches have focussed on fibre-dominated laminates, because he has worked in a world of carbon-fibre-

E-mail address: [email protected]

771 reinforced plastics. Consequently, all of his theories are weaker in regard to matrix-dominated failures than are the works of Puck [11] and his colleagues, who have worked mainly with fibre glass-reinforced plastics and correctly used the original maximum-strain failure model for glass fibres. The organizers of the contest, referred to by some as the “Olympic Games” because of the widespread origins of the participants, have now released the test data for these problems, in Ref. [12]. The objective of this new paper is to compare the predictions with the test results, to identify any limitations in the theories and, where appropriate, to suggest where something must have gone wrong with the experiments. The cited analyses have all been performed on a strength-of-materials basis, devoid of any consideration of structural effects, such as buckling or distortion of the test coupons under loads. The test data contains clear evidence of some such effects, however, so a direct comparison between test and theory is not always possible. This should not be surprising, given the very brittle nature of fibrous composites. Indeed, the author has often stated that the only thing less representative of composites than most of the failure theories is the bulk of the test data. Nevertheless, the organizers have managed to select three widely different problems that proved to be surprisingly capable of exercising the capabilities of any of these theories. The authors have not yet been given copies of the predictions of the other contestants,1 so it is not possible to make comparisons between the various theories at this stage. This paper is organized on a problem-by-problem basis, rather than a theory-by-theory sequence, because the theories employed by the author have great similarities and the small differences can thus be highlighted.

2. Test problem no. 1: biaxial (␴y ⴚ ␶xy) failure envelope for unidirectional 0° glass-epoxy lamina: type: E-Glass/LY556/HT907/DY063 Regardless of what may or may not be happening as isolated and embedded unidirectional laminae are loaded to failure under these particular components of stress, every theory represented by the author in this exercise assumes that there are no interactions between these stresses. The failure envelopes predicted for the maximum-strain and truncated maximumstrain failure models, and that for the Ten-percent Rule, are shown in Fig. 1, having been generated as Fig. 62 in Ref. [1] and Fig. 17 of Ref. [3]. These envelopes are rectangular in shape, with sides parallel to the stress axes. It is apparent that the data used to establish the reference strength for pure longitudinal compression are inferior to those generated for the same state of stress with the biaxial test specimen, and that the fit of the data would be improved by correcting for this effect. (The organizers have acknowledged in their information package [12], for Part B of the exercise that the uniaxial and biaxial test data came from different sources.)

1

The drafts of all the Part B papers were subrnitted before the publication of the Part A papers, even though their processing has extended beyond that date and the contributions by the other authors are now known. 2 Throughout this work, the complete world ‘figure’ in the text will always refer to a diagram in a referenced paper. The abbreviation Fig. will always refer to a diagram in the present paper.

772

Fig. 1. Biaxial failure stresses for 0° lamina made of GRP material (maximum-strain and truncated maximum-strain failure models, and ten-percent rule).

Figure 1 also reveals that the purely non-interactive theories are conservative for compression-dominated loads and unconservative for tension-dominated loads. There must, therefore, be some other mechanism affecting the behaviour of the composite lamina. It has been proposed in several publications that this phenomenon can be explained by internal friction, as occurs in granular materials like sandstone and the like. However, these theories failed to consider the point that it was necessary that there must have already been failures to have created the friction planes. The author believes that a far more plausible explanation lies in the omission from past theories of residual thermal stresses, in the matrix, caused by curing at elevated temperatures and operating (testing) at ambient temperatures or lower. This same sloping characteristic, with an apparent increase in shear strength associated with longitudinal compression, can also be predicted on the basis of a Mohr-circle stress analysis with the origin shifted to account for the predictable residual thermal stress in the matrix. Sufffice it to say that the level of analysis of all of this author’s analyses of this problem is incomplete. Figure 2 contains the predictions from figure 11 of Ref. [2], using an empirical formula cited there, as the author’s best fit (or educated best guess) to the data supplied. The explanation provided there for the offset of the peak shear stress involved the residual thermal stress in the matrix, circumferentially oriented around the fibres. With a correction for having been supplied with too small a reference compression strength, the fit would obviously have been far better. There is even more to this story than this, however. It has been noted by Puck and his colleagues (see, for example, Cuntze in Ref. [13]) that this problem requires the acknowledgement of the presence of three distinct failure mechanisms, each associated with a different action plane. The analysis in Fig. 2 involves only one component of residual thermal stress and, since only one equation is involved, it cannot possibly cover more than one failure mechanism. Like so many other theories for predicting the strength of composite laminae and laminates, it is simply an empirical curve fit. Its merit, if any, lies in its

773

Fig. 2. Biaxial failure stresses for 0° lamina made of GRP material (author’s empirical formula, not interacting with any other solution).

simplicity.3 As will be explained in greater detail for subsequent solutions, this particular problem refers only to unidirectional laminae in isolation; it does NOT characterize the same lamina when it is embedded within a multi-directional laminate. 3. Test problem no. 2: biaxial (␴x ⴚ ␶xy) failure envelope for unidirectional 0° carbon-epoxy lamina: type: T300/BSL914C Like the preceding problem, the non-interactive failure models predict a non-interactive rectangular failure envelope. This is shown in Fig. 3, which covers all four failure models assessed by the author. The original solutions are given in figure 8 of Ref. [1], figure 12 of Ref. [2], and figure 18 of Ref. [3]. Figure 12 of Ref. [2] offers an explanation of how the requested strength plot could be estimated more scientifically. Three independent failure mechanisms are actually involved, just as in the preceding problem. This time, however, only one is in the matrix while two (longitudinal tension and compression) are totally fibre dominated. (The premature failures in compression, of course, may be matrix influenced, but the measured strengths are way beyond those of the unreinforced resin matrix.) A rigorous accounting of the residual thermal stress in the fibres would show virtually no shift in the origin with respect to mechanically applied stresses. Although there is considerable scatter in the test data, there seems to be a trend of enhancement of the shear strength as the result of positive longitudinal stress. There should,

3

The author is indebted to Prof Alfred Puck and to Ralph Cuntze for explaining to him what is really involved in a scientific explanation of these phenomena.

774

Fig. 3. Biaxial fadlure stresses for 0° lamina made of CFRP material (maximum-strain, truncated maximum-strain, and generalized maximum-shear.

therefore, be a logical mechanistic explanation, even though it is clearly beyond the capabilities of the theories the author has championed. Given that the major Poisson’s ratio for the fibres is less than that for the matrix, the explanation is not to be found in any hypothesis that longitudinal tension would reduce the interfacial pressure4 acting between the fibres and matrix. This observation is true for residual thermal stresses associated with both the concentric cylinder model, whereby the resin tries to shrink around the fibres, and that formulated around shrinkage of the resin in the interstices between the fibres. It is also apparent that simple consideration of a three-dimensional state of stress in the matrix, in terms of Mohr circles, would not admit any increase in shear strength beyond that for pure shear, if only mechanical loads were present. This seems to leave only some phenomenon associated with the test program(s) as a possible explanation. The issue is clouded, as is also the case for the quasi-isotropic cylinders discussed later, by the combination of multiple sets of test data. As if to muddy the water further, Fig. 2 of the information package [12] supplied for Part B of this exercise indicates that the highest shear strengths were obtained in the presence of tensile longitudinal loads insufficient to cause the fibres to break – and that the shear strengths measured in their absence as part of the SAME series of tests were consistently lower than those measured as part of the axial-compression tests. Given that the intrinsic compression strength of these carbon epoxy laminae is known from other tests to be the same in both tension and compression, it is plausible to ascribe the lower pure compression results here than for tension to some form of buckling. Had the stresses parallel to the fibres been created by internal or external pressure applied to 90° wound tubes, instead, it would not be difficult to infer that the enhanced and reduced shear strengths were the result of the known stabilization of shear wrinkles by internal pressure. Only if the cylinders were unusually short could a similar kind 4

On the contrary, this is a probable mechanism for destabilizing axially compressed fibres as the matrix moves away from them because of Poisson effects, wherever the fibres may be located in a laminate.

775 of benefit be obtained from longitudinal curvature caused by circumferential Poisson contractions of 0° tubes. However, the data package contains no suggestion to this effect. In short, the author is unable to provide any definitive explanation of why there seems to be an interaction, of this form, between what he would have assumed to be non-interactive stress components, at least in the absence of gross structural effects.

4. Test problem no. 3: biaxial (␴x ⴚ ␴y) failure envelope for unidirectional 0° glass-epoxy lamina: Type: E-Glass/MY750/HY917/DY063 While, yet again, the characterization of the biaxial strength of this lamina is complicated by the usually ignored differences between the in situ and isolated strengths, the exercise is useful in revealing why such a distinction needs to be made. It also shows that any attempt to use either of the maximum-strain failure models in the traditional form, for isolated unidirectional laminae, is bound to founder, precisely because of this difference. Figure 4 shows how the test data compare with the predictions for the original and truncated maximum-strain failure models. Serious problems are apparent with the constant-transverse-strain portions of the envelope. A constant-transverse-strain cut-off passed through the measured transverse strength of an isolated unidirectional lamina would predict that it should not be possible to achieve the measured longitudinal compression strength. In this form, the theory is self-inconsistent, just like the many other (interactive) theories that the author has condemned in the past. This is obviously a deficiency of the theory, which in no small part explains why those who use these

Fig. 4. Biaxial failure stresses for 0° lamina made of GRP material (maximum-strain and truncated maximum-strain failure models).

776 theories relocate this cut-off to a less restrictive location. The solution given in figure 9 of Ref. [1] for this problem actually includes separate transverse cut-offs for this problem, to distinguish between isolated and in-situ behaviour. Setting the unidirectional transverse strains-to-failure equal to the longitudinal strains-to-failure of the fibres provides a credible failure envelope. However, it is the author’s belief that transverse-tension failures, at least, must be characterized as constant stress phenomena, as shown in Fig. 5 here, and in figure 13 of Ref. [2], for the generalized maximum-shear-stress criterion, and in figure 19 of Ref. [3] for the Ten-percent Rule. One must conclude, on the basis of the data provided by the organizers, for isolated unidirectional laminae, that transverse stress provides a better failure criterion for transverse tension loads than does a constant transverse strain limit. It is the author’s position that the same is true for in situ properties, albeit at a typically far higher stress level. In the longitudinal-tension/transverse-compression quadrant, the test data are seen to lie between the predictions by the truncated and original maximum-strain failure models. However, the author has learned during the process of solving these problems that the untruncated maximum-strain model is more appropriate for glass fibre reinforced plastics than is the model with the truncation (which is better suited to carbon-fibre-reinforced laminates). His best estimate in this regard is provided by the generalized maximum-shear-stress failure model, in Fig. 6, which is visibly better than either of the other models for failures dominated by the high longitudinal tensile loads in the fibres. Consequently, fibre failures are not the source of the explanation of why the constant-transverse-stress cut-off in Fig. 5 is overoptimistic for compressive transverse loads in combination with longitudinal tension. The real explanation must lie in a different failure mechanism for the matrix. If one were to propose that, at the

Fig. 5. Biaxial failure stresses for 0° lamina made of GRP material (generalized maximum-shear-stress failure model and ten-percent rule).

777

Fig. 6. Improved empirical transverse limits for the original and truncated maximum-strain failure models for composite laminates.

macro5 level, failure was by ductile shear, a simple Mohr-circle analysis would reveal the same kind of interaction as seen in the test data in Figs 4 and 5. Such an analysis could have been provided, but the objective of the present exercise was to characterize the predictive capability of the various theories as they are customarily used. None of these theories is meant to be used to characterize matrix failures in isolated unidirectional laminae, so their imprecision in doing so is not all that serious. The process by which the original (mental-arithmetic) form of the author’s Ten-Percent Rule was expanded in Ref. [3] into a fully-fledged failure envelope covering fibres in arbitrary directions suggests that a minor, but significant, improvement could be made to the two maximum-strain failure models. This is done in Fig. 6. These new transverse cut-offs are, in effect, non-interactive constant-stress limits set sufficiently high that the theory is incapable of predicting matrix failures that precede fibre failures in other plies. The transverse-tension cut-off would move closer to the longitudinal stress axis for laminates with such thick plies that there is only a weak interaction between the adjacent plies that could retard the formation or spread of microcracks. In the extreme case, it would end up as a constant-transverse-stress cut-off for isolated unidirectional laminae whenever transverse tension loads dominated the behaviour. This is consistent with the test data provided by the organizers.

5

At the micro level, these failures have been better characterized, by others with more knowledge of the subject, as microcracking, with minute cracks not spreading, but being arrested by the abundance of discrete fibres as crack-stoppers.

778 Insufficient data are provided to either substantiate or refute the author’s position that the best characterizations possible for glass-fibre-dominated strengths are non-interactive constant-longitudinal-strain, rather than-stress, characteristics, at least when expressed at the lamina level. The best possible characterization for transverse-compression-dominated failures in the matrix may not be so simple. Certainly, the author’s theories presume that, at the laminate level, fibre failures in some transverse plies precede these undefined matrix failures. This position remains realistic for well-designed (thoroughly interspersed) laminates reinforced by carbon fibres. While he has learned during the conduct of this contest that this cannot be relied upon for fibreglass-reinforced plastics with plies under transverse tension loads, there is no indication that such a simplification is not still reasonable for fibreglassreinforced plastics with plies under transverse compression loads. If not, the readers will be forced to rely on the capabilities of other theories with greater capability in this regard. No predictions can be made for the compression compression quadrant in the absence of either good test data or a reliable mechanistic failure model. As noted earlier, the author’s theories are obviously limited in regard to predicting real matrix failures, although not necessarily any more so than several other theories for which no such admissions of limitations have been made. His personal opinion is that a prediction which can be made, but which is invariably wrong, is not necessarily an improvement over a theory that cannot make any such predictions. He is, consequently, not perturbed by the inability to completely solve the first three of the assigned problems, for unidirectional laminae. What is needed, of course, is matrix-failure theories that can reliably predict matrix failures. The real worth, or otherwise, of the author’s fibre-failure dominated theories will be revealed by their ability to explain the observations for multi-directional structural laminates, in the remaining problems. (In the process, it is revealed that the Ten-Percent Rule [3], had surprisingly good unanticipated capability to also predict some matrix failures as well.)

5. Test problem no. 4: biaxial (␴y–␴x) failure envelopes for (90°/ ± 30°)s glass-epoxy laminate: Type: E-Glass/ LY556/HT907/DY063 The test data supplied for this problem confirm the reliability of all four of the theories assessed by the author, in both magnitude and form of the failure envelopes. Figure 7 compares the test data with predictions made by the original and truncated maximum-strain failure models (figure 14 of Ref. [1]); the generalized maximum-shear-stress failure model (figure 17 of Ref. [2]); and the expanded Ten-percent Rule (figure 15 of Ref. [3]). All theories predict totally fibre-dominated failures for all states of stress. Somewhat embarrassingly, it seems as if the predictions made by the humble Ten-Percent Rule seem to be the best of the lot, even though all of the solutions are quite acceptable as explanations of the test data.6 The failure envelopes are basically skewed parallelograms, with

6

Obviously, they are not conservative enough to use directly for design. One would need to add matrixfailure cut-offs not considered here, if appropuate, or provide physical explanations of why some test results under-estimated the true material strength, if that were the explanation for some of the discrepancies and scatter. The fibre-dominated failure theories being evaluated here cannot do either of these additional tasks. Predictions from other theories that can realistically characterize matrix failures will help differentiate between the two possible explanations.

779 the top and bottom defined by failures of the ± 30° fibres and the sloping sides established by failures in the 90° fibres. (There is a minor local truncation of the basic envelope in the form of predicted failures of the 90° fibers under transverse tension according to the Ten-Percent Rule.) No definite matrix failures are involved, except for a small region of probable failure of the matrix between the 90° fibres, at the top of the envelope, although the bottom-right and top-left corners are close to predicted matrix shear failures in the ± 30° plies. The truncated maximum-strain and generalized maximum-shear-stress failure models have slight truncations in the upper-left and lower-right quadrants. However, as noted earlier, these regions are characterized far more precisely, for fibreglass-reinforced laminates, by the generalized maximum-shear-stress failure model than by the truncated maximum-strain failure model, so the latter truncation should be disregarded here. Figure 17 in Ref. [2] shows how matrix-cracking failures under transverse tension would be predicted, if only the relevant test data were available. So does figure 15 in Ref. [3]. These predictions go a long way to supporting the practice, in the U.S. aerospace industry at least, of rejecting matrix failures predicted on the basis of transverse strengths measured on unidirectional coupons. If the maximum-strain model [1] were used blindly, it would have predicted that the laminate discussed here would have “failed” at such a low transverse strain in the 90° plies that the entire top half of the failure envelope would have been omitted. Such predictions are clearly contrary to this (and other) test data. The unthinking use of the arbitrary transverse cut-off shown at a strain level equal to that of hypothetical 0° fibres, as is

Fig. 7. Biaxial failure stresses for (90°/130°)s laminate made of GRP material (maximum-strain and truncated maximum-strain failure models, generalized maximum-shear-stress failure model, and expanded ten-percent rule failure model).

780 customarily used for laminates with fibres in the family of 0°, ± 45°, 90° directions would also been unrealistic, as is discussed below. This same issue is addressed again in the context of a request by the organizers to explain kinks noted in the stress–strain curves for the quasiisotropic carbon epoxy tubes discussed later (Test Problems 10 and 11). The unquantified physical explanation suggested is as follows, in the present context. Let us suppose that real matrix cracking does occur between the 90 fibres under the 0° component of the applied loads, in such a manner that it extends uniformly throughout the laminate. The analysis provided here established that the 90° fibres will be axially compressed across the top of the failure envelope in Fig. 7. If the cracking is sufficiently extensive, these 90° fibres will be destablized. They will no longer act as one side of a triangulated structure of fibres, leaving the ± 30° fibres more as a mechanism than as part of a structure. There would then be scissoring between these two fibre directions at essentially constant transverse stress on the 90° plies. One would anticipate further 0° strains to failure as a result of this hypothesis, but no increase in load capacity. This reduced stiffness of the laminate before failure might be detected in some of the test data or be predicted by scientific analysis of matrix failures. The author would obviously welcome such additional information. However, since there are no 0° plies in this laminate, deleting the longitudinal stiffness of the 90° plies would increase the Poisson’s ratio of the remaining effective plies tremendously, to something like 0.6, so that there would be substantial compression of the now destabilized 90° fibres, possibly enough to even buckle them. It should also be noted that figure 17 of Ref. [2] and figure 15 of Ref. [3] also include the form of a possible cut-off resulting from splitting the matrix between the ± 30° fibres that could go a long way in explaining the differences between test an theory evident in Fig. 7 on the right-hand side of the upper right quadrant. The consequent softening of the ± 30° plies would then overload the 90° fibres causing them to fail instantly with no kinks in any stress– strain curves. In the absence of a theory with which to quantify this effect, the author has simply reproduced the cut-offs illustrated in Refs. [2,3] in Fig. 6. It is obvious that the form of these cut-offs is consistent with the test data, if moved down and to the left. The quantified cut-offs of the Ten-Percent Rule need little moving at all. These data are particularly useful in justifying the transversely extended lamina-level failure envelope shown in Fig. 6. The customary use of a constant-transverse strain limit with the original (untruncated) maximum-strain failure model is appropriate for fiber patterns in the 0°, ± 45°, 90° family. But for this laminate, that would result in a truncation sloping upward from about 550 MPa on the left side of the envelope to some 650 MPa on the right. This would lie well inside many of the data. The combination of any real 90° plies with the 0° plies on which Fig. 6 is based would lead to the traditional analyses used for the standard fibre patterns, so the transverse expansion recommended here would not change any of these predictions. It can be looked upon as a generalization of the intent of the earlier empirical transverse limits to arbitrary fibre patterns. Since the lower-left corner of the envelope has been shown to be far removed from matrix failures, it would seem that the discrepancies between test and theory are probably due to the traditional difficulties in performing compression–compression tests. The author suspects buckling failures there, which are even more clearly evident in the testing of the quasiisotropic carbon-epoxy laminate in the final problem and, indeed, in virtually all such published test data. This is a structural, rather than material, failure. In Ref. [12], the organizers have acknowledged that these particular tests were observed to fail by buckling.

781 There is a possible materials explanation of the lower-than-predicted test results in the lower-right quadrant, however. The corner point shown there is almost coincident with matrix shear failures in the ± 30° plies. The compressive strength of fibres is known, by both test and theory, to be reduced if the matrix is insufficiently stiff to fully stabilize the compressed fibres. The secant modulus of the matrix is decreased greatly between the initial tangent modulus and the point of failure, per the data supplied by the organizers for Part A of this exercise. The author cannot quantify what kind of reduction in axial compressive strength this would cause; this is clearly a micromechanical issue. Worse, this issue could be resolved experimentally only by establishing a lower unidirectional compression strength with the same fibres embedded in a different resin matrix with a lower modulus even in the absence of the added in-plane shear loads. Nevertheless, the slope of the calculated matrix shear failures in figure 17 of Ref. [2], for example, is similar to a line of best fit through the test data in the lower-right quadrant in Fig. 7, so the hypothesis is plausible. In the upper-left corners of these failure envelopes, this same hypothesis would have no effect on the ± 30° fibres, since they would be under longitudinal tension there. Matrix shear failures in the 90° plies are shown in figure 19 of Ref. [2] to be remote from the x–y plane, and incapable of influencing any of the test results under consideration here. (Such deductions, incidentally, are possible only with mechanistic failure criteria for composite laminates; interactive theories could provide no clue as to what might be happening.) In summary, all of the present theories do a better than adequate job of explaining the fibredominated failures for this laminate. Plausible explanations have been given for the remaining discrepancies between test and theory, some ascribed to testing difficulties and others to limitations of the theories covered here that cannot comprehensively address matrix failures. 6. Test problem no. 5: biaxial (␴y ⴚ ␶xy) failure envelopes for (90°/ ± 30°)s glass-epoxy laminate: Type: E-Glass/LY556/HT907/DY063 The results for this other cut through the total failure envelope for this same laminate as in the previous problem are compared with the almost indistinguishable predictions of the four failure models in Fig. 9. The agreement is not as good as in the previous problem, even though the predictions are of the correct general magnitude. (An attempt is made below to explain the discrepancies in terms of the same hypotheses as offered in relation to Fig. 7.) Again, the failure envelopes are totally fibre dominated, as also is the other section cut, given in Fig. 22 of Ref. [2]. The relevant solutions are given in Fig. 15 of Ref. [1], Fig. 21 of Ref. [2] and Fig. 16 of Ref. [3]. The theories overestimate the laminate in-plane-shear strengths for longitudinal tension loads, and under-estimate them in the presence of compression. However, the estimates for uniaxial strength of the laminate are realistic in both tension and compression. Lower-thananticipated test results are always easier to explain that higher ones, because one can always blame the test data (in most cases justifiably for brittle fibre-polymer composites), so the discrepancy in the left quadrants in Fig. 9 is the more critical to explain. The theories are predicting that the in situ compression strengths of the ± 30° fibres are higher than those measured on the 0° reference test coupons. Only one hypothesis seems plausible to the author; its credibility rests on the fact that the measured uniaxial compression strengths are limited by the support given to the fibres by the matrix rather than by the intrinsic strength of the fibres

782 themselves. It is now quite apparent to those within the MILHDBK-17 organization that the in situ compression strength of unidirectional laminae is not universal. It is strongly dependent on ply thickness, being unrealistically high, for most design purposes, when deduced from measurements of the strength of cross-plied laminates containing only single 0° plies. A more realistic value is achieved by testing laminates made with pairs of 0° plies stacked together. With stacks thicker than four plies, or entirely unidirectional coupons, the fibre wash (and consequent fibre misalignment) is so great as to prevent the fibres from developing anything near their true strengths, particularly in compression. This effect might well explain why the 0° compression strengths recorded in Fig. 4, on biaxially loaded specimens, are some 18% higher than the supplied reference strength from different uniaxially loaded coupons. However, an 18% increase in compression strength for the ± 30° plies would not eradicate all of the discrepancy in the left quadrant of Fig. 9, even though it would help reduce it. Similarly, it would help eliminate the underpredictions of strength on the positive vertical axis in Figs 7 and 8. The compressive strength of a fully stabilized fibre must be at least as great as its tensile strength – the same if fibre failure is by shear, and higher if by brittle fracture. The properties supplied by the organizers for this fibre-glass-epoxy material clearly indicate that the uniaxial compression strengths measured were nowhere near as high as were those for tensile loads. If the reference data were established on unidirectional test coupons (flat or tubular), it is very likely that they are lower than the strength that those same fibres would develop in a multidirectional laminate with thin stacks of fibres stabilized by layers in other directions that would also serve to limit the fibre wash that occurs in unidirectional coupons, particularly

Fig. 8. Biaxial failure stresses for (90°/ ± 30°)s laminate made of GRP material showing probable modifications to account for matrix transversetension cracking between fibres in each ply (generalized maximum-shear-stress failure model).

783

Fig. 9. Biaxial failure stresses for (90°/ ± 30°)s laminate made of GRP material (maximum-strain and truncated maximum-strain failure models, generalized maximum-shear-stress failure model, and expanded ten-percent rule failure model).

those made by filament winding. While it is therefore likely that higher reference compression strengths could be validated by testing crossplied laminates more representative of the ply thickness in the biaxially loaded test tubes, such a modification would only raise the sloping line on the left side of Fig. 9, without adding either the plateau on the top or the steeper loss of strength at the left end of the envelope. Additional failure mechanisms must be involved. An examination of figure 16 in Ref. [3] suggests that the plateau in the test data in Fig. 9 could be associated only with some effect caused by matrix shear failures in the 90° plies, since all other potential failure mechanisms are associated with different sloping characteristics. In particular, compression failures of the 90° fibres and in-plane-shear failure of the matrix between the ± 30° fibres are both far removed from the region of small positive 0° loads where the discrepancies are greatest. Unfortunately, it is not evident that such a loss of stiffness in the matrix between the 90° fibres could affect the strengths of the ± 30° fibres that define the failure envelope in this region. The agreement between this set of test data and the theories employed by the author is credible in magnitude but in need of improvement in regard to form, since the predictions lack any plateau around the xy axis in Fig. 16. There are also some unexplained truncations needed to make the predictions in Fig. 15 match the data, whether the resolution is by improved analysis or by improved testing. 7. Test problem no. 6: biaxial (␴y ⴚ ␴x) failure envelopes for ( ± 55°)s glass-epoxy laminate: Type: E-Glass/MY750/HY917/DY063 The predictions of the analyses, figure 24 of Ref. [1], figure 17 of Ref. [2], and figure 11 of Ref. [3], are compared in Fig. 10 with the test data provided in Ref. [12]. Two highlights are noteworthy. The use of thick-walled tubes, or maybe it was the reduced ply stack thickness, has stabilized the biaxially compressed fibres so much better than for the uniaxial test coupons that the apparent fibre strength has increased to match that measured under tension, as it

784

Fig. 10. Biaxial failure stresses for ( ± 55°)s laminate made of GRP material (maximum-strain and truncated maximum-strain failure models, generalized maximum-shear-stress failure model, and expanded ten-percent rule failure model).

should if both failures were by shear in the fibres under axial loads. The failure to predict such high biaxial compression strengths is entirely a function of the uniaxial reference strengths (supplied by the organizers) used in the analysis; it is not a problem with these theories. This is apparent from the extension of the Ten-Percent Rule predictions in Fig. 10 for assumed equal tensile and compressive reference strengths. The equality of fibre strengths in tension and compression, for T-300 carbon fibres, has been confirmed experimentally in Ref. [14], where the authors had independently postulated the shear failure mechanism for axially loaded carbon fibres even before the present author had. (Much the same problem with a low supplied reference strength for longitudinal compression is evident in Fig. 2.) It is also noteworthy that these laminate test data are consistent with equal transverse tension and compression strengths, too, even though most test data generated on coupons would suggest far greater strengths in compression. The other significant issue is that there is a need to explain why the results in the upper-right (tension-tension) quadrant appear to sag consistently below the predictions. But by far the most outstanding result from the three analyses is that the expanded TenPercent Rule method of analysis was even able to predict the locations for the long matrix shear-failure sides of the skewed parallelogram when absolutely NO matrix reference strengths or stiffnesses were involved in the analysis. Yet it obviously did so astonishingly well, in comparison with both the other theories and most of the test data. Even more amazingly, it predicted that there should be a truncation imposed by transverse-tension cracking between the fibres at the top-right corner of the failure envelope. (These matrixfailure predictions are not believed to be a fluke, but such agreements should not be expected for composites other than of strong, stiff fibres and soft polymers. Although it was not

785 recognized when this theory was first proposed, the Ten-Percent Rule implies a specific inplane shear strain at failure via the obvious relationship between in-plane-shear strength and stiffness.) The short ends of the parallelograms in Fig. 10 represent fibre failures. Figure 17 of Ref. [2] and figure 11 of Ref. [3] include a possible matrix cracking cut-off, at the top of the upper right quadrant. As noted before, its position needs to be established on the basis of deduced in situ strengths. The upper limit of the present test data would tend to confirm the need for such a cut-off in this case. The validity or otherwise of this cut-off might yet be established on the basis of observations of cracking or loss of pressure. It seems very significant that, in the test data supplied, the hoop pressures withstood prior to failure were consistently higher when the tubes were fitted with liners than when they were not. It is also relevant that the matrix-cracking cut-off shown in Fig. 10 would fit well as a cap to those data noted by the organizers as weeping oil, if scaled suitably. The location of the cut-off suggested by the author for the Generalized Tresca model [2] was purely illustrative, but its relocation as an upper limit of the data for internally pressurized tubes without liners would be so good as to inspire the hope that micromechanicians participating in this contest might be able to confirm the existence of this failure mode. (The corresponding upper limit suggested by the TenPercent Rule is not adjustable, but this set of test data suggests that this cut-off should not be moved further from the origin than is necessary to avoid false predictions of failures. Sometimes the predictions might be real.) The author is unable to provide any other explanation capable of explaining, to the extent necessary, why the test data are lower than the theories in the upper right corners of the failure envelopes. A physical, rather than mathematical, assessment of how and where matrix cracking between the fibres might occur under intralaminar transverse-tension loads would identify this same region on the failure envelope as the most likely possibility. These transverse-tension cut-offs for the ( ± 55°)s laminate are shown in Fig. 11, where located for the Ten-percent Rule and fitted to the data for the Generalized Tresca model. This leaves what appears to be an overall drooping, clockwise, of the test data, to be explained. After a couple of seemingly plausible explanations were evaluated and rejected, because they were not sufficiently powerful, numerically, to explain the magnitude of the discrepancy, an extremely simple solution was recognized. The basic problem had little to do with structural distortion under load, even though the calculated hoop strains could have been low by as much as 10%, and the pressure contribution to the axial load high by an equal amount, both of which factors would have moved these plotted test data down and to the right. (The data cited by the organizers would place the average correction from this source at a much smaller percentage, still.) The discrepancy also has nothing to do with poor experimental technique or errors, rather than possible omissions, in the analyses used to establish the failure envelopes. It was simply a case that the pressurized tubes did not know when they had failed! The first point to be clarified is that the test failures outside the predicted failure envelope are not premature because they occurred at lower x stresses than the theory predicted. They are actually higher y strengths than predicted – and higher in-plane shear strains  xy. The shear strain achieved is proportional to the distance from the diagonal equalstrain line half way between the upper and lower sides of the calculated failure envelopes, as indicated in figure 16 of Ref. [1] and figure 23 of Ref. [2]. The failure envelopes plotted in Fig. 10 are established primarily by two totally unrelated mechanisms. The short ends represent fibre failures and the long sides matrix shear failures. However, in the triangle

786

Fig. 11. Biaxial failure stresses for ( ± 55°)s laminate made of GRP material, showing probable transverse-tension failures in the matrix between the fibres in each ply (generalized maximum-shearstress failure model).

formed by the positive y axis, the lower matrix shear-failure cut-off and the projection (extension) downwards of the tensile fibre-failure characteristic, the effectively infinitely long pressurized cylinder can still withstand tensile hoop stresses. The failure envelope is expanded appropriately in Fig. 12, using the maximum-strain failure model, but any of the present theories would predict much the same result. The right-hand limit, at 1 = 0.02807, is composed of two straight lines, the original (upper) portion being defined by the initial shear modulus of G12 = 5.83 MPa as in Fig. 10. The new (lower) limit is defined by a final constant shear stress limit of xy = 75 MPa, associated with a steadily reducing shear modulus for higher and higher shear strains, far beyond the supplied limit of 0.04, which is reached at the left end of the second portion of this limit. This second line crosses the y axis at 692.3 MPa, the upper right end (at y = 996.1 MPa, x = 449.3 MPa) being defined by the combination of strains 1 = 0.02807, 2 = –0.03559, and 12 = 0.175. For reference, had the matrix strains to failure not be extended beyond the data provided, the matrix would have been predicted to fail when 12 = 0.04, at a y stress of 261.0 MPa, with an axial strain in the fibres of $1 = 0.00811 and a subcritical transverse strain of $2 = –0.00645. If only this shear-strain limit had been overridden, the matrix would have then been predicted to fail under transverse compression, at $2 = –0.012, when y = 363.7 MPa, 12 = 0.06594 and $1 = 0.0120. These predicted failures would have occurred long before the test result, indicating the inadequacy of applying linear analysis to matrix-failure problems unless the matrix properties are characterized by secant moduli that would need to be changed for each and every problem. This line was extended analytically beyond (below) the axis into the axial compression zone, but these projections, based on reaching the tensile fiber strain to failure without developing tensile transverse strains, extend way beyond the test data. This suggests

787

Fig. 12. Expansion of biaxial failure envelope, for post-failure residual strength in ( ± 55°)s laminate.

that there is an additional failure mechanism involved, possibly exceeding a maximum transverse compression strength when the axial stress in the fibers is no longer the dominant effect.7 According to the lamina properties supplied, the matrix would be fully crazed throughout the triangular area of expansion shown, and should be incapable of transmitting shear from fibre to fibre – but it doesn’t need to.8 (The actual calculations involve small non-zero shear stresses associated with variable shear strains extended far beyond the limit of 0.04 supplied by the organizers for pure shear in the absence of compression. The consequent values of the secant shear moduli G12 are very small for large shear strains.) The fibres cannot break until sufficient hoop load is applied to strain them all the way to the fibre-failure cut-off at the right of the failure envelope. Outside the originally defined failure envelopes, the matrix will be

7

An otherwise similar series of similar calculations was performed in which the transverse compressive stress perpendicular to the fibers was capped at 150 MPa. Curiously, it was then predicted that the longitudinal stress x in the tube could not be reduced to zero without exceeding this limit. This confirms the original hypothesis that the matrix had really “failed” with respect to all loads other than transverse compression. 8 An interesting test would be to see whether or not this same triangular extension would be present if the hoop and longitudinal directions were effectively interchanged by altering the winding angle to + 35°. Under the right fractions of hoop and longitudinal tensile loads, the matrix could be guaranteed to fail first. What is not immediately apparent is the nature and expanse of any regime of residual strength.

788 riddled with closely-spaced cracks – primarily along the length of the cylinder for this triangular extension of the measured, rather than calculated, failure envelope. This will leave the matrix fully capable of withstanding axial compressive inter-fibre loads, which is all that the matrix is called upon to endure while the fibres, which are oriented more around the circumference of the tube than along its length, resist the internal pressure. (The resolved axial tensile forces in the fibres must be balanced by axial compression in the matrix between them to create a zero average longitudinal stress in the cylinders.) The addition of small amounts of tensile longitudinal stresses, at the macro level, would not cause the tubes to disintegrate, either, until they became so great as to fail the matrix by transverse tension. Until then, they would mainly be relieving the intralaminar transverse compression associated with the macrolevel hoop stresses. So the apparent mystery is very easily explained using mechanistic models that properly distinguish between the fibre and matrix constituents of the composite. It should be noted, however, that the triangular extension of the failure envelope is a situation restricted to the monotonic application of tensile in-plane loads to test coupons in the absence of in-plane shear. In the context of structural laminates subjected to arbitrary loads in random sequence, this laminate should be considered to have really failed where the earlier analyses here predicted it to have. It now remains only to justify why this enhancement of the calculated failure envelope does not extend to other states of stress. The diagonally opposite triangle cannot be filled in because the fibres could not withstand hoop compression unless stabilized by a matrix that was not riddled with microcracks. The analysis predicts that the transverse lamina strain throughout that region would be large and tensile. A matrix already cracked by in-plane shear loads cannot provide the inter-fibre tension strength needed to balance the resolved axial component of the longitudinal load in the axially compressed fibres caused by the macro-level compressive hoop loads. Obviously, in the presence of axial compression along the length of the tube, below the horizontal axis in Fig. 12, the matrix cannot disintegrate under biaxial compression. But neither can it stabilize axially compressed fibers, so there is no mechanism to justify filling in the triangle between the bottom left of the failure envelope and the negative vertical (y) axis. The use of reduced secant moduli, rather than the initial linear values, was needed to generate the failure envelope shown in Fig. 12. An attempt was made to use the same approach with constant-transverse-stress limits to further refine the envelope to introduce cut-offs at the top-right and lower-left corners akin to those predicted by the Ten-Percent Rule (figure 11 in Ref. [3]). The in-plane shear modulus G12 was progressively reduced for the predicted very high shear strains, while the transverse modulus E2 was maintained constant until it was dropped to zero. Unfortunately, this model failed to produce realistic cut-offs for both transverse tension and transverse compression limits. Both y and x stresses were acceptable at the remote extremities away from the origin, but the x stresses were numerically far too small closer to the origin. It seems that adjusting material properties without the guidance of a matrix-failure criterion covering the combination of the governing strains is not a fruitful exercise. The analysis is further complicated by the appreciable changes in angles between the fibres if the very high shear strains predicted for the analyses in Figs 11 and 12 really do occur. Clearly, the further refinement of the failure envelope in Fig. 12 is a task for other researchers with physics-based matrix-failure criteria to apply. It is evident from the test data provided that the axial ax tensile strength of the tubes with liners and subjected simultaneously to y stresses is consistently higher than for those tubes

789 without them. There must be a physical explanation, of course, but it seems to be necessarily unrelated to the expansion of the y failure stresses discussed above, since the predicted matrix shear failures exceed the measured tube strengths in this case. One can reasonably postulate that the final matrix shear failures would have been preceded by short tensile hackle fractures at 45 to the individual fibres, with a resulting progressive reduction in in-plane shear stiffness. The question is as to whether or not this could lead to “premature” failures at the global level. It would clearly lead to leakage of the internal pressure, which could have been detected experimentally. What both the lined and unlined tubes achieved by test showed, in this context, is that the tubes “failed” without ever straining sufficiently to break the fibres. With an excess of axial stress beyond the pure internal pressure case, the matrix between the fibres would have been under tension. In the absence of fibres in any third direction, there could be no mechanism to prevent the tube from failing once the matrix had. The only issues are why the tubes failed before predicted, and how did the liners enhance the strength. Having proved that this is entirely a matrix-dominated phenomenon, the author will defer to explanations by those with better matrix-failure theories. The one contribution he can make is the addition to Fig. 11 of matrix shear cut-offs not at ultimate failure but at the “knee” of the in-plane-shear stress–strain curve, which is when one should expect matrix cracking to initiate. This point is estimated from the data pack supplied for Part A of this exercise as Fse 50 MPa, e 0.01, and Ge 5.0 GPa. (There is appreciable nonlinearity in the curve, so no unique location can be established as a linear elastic limit. In any event, the point being made is still valid with an approximate estimate.) The shear-strain intercepts on the axes in figure 16 of Ref. [1] are relocated to $x = $y = 0.01064, and the intercepts on the axes in Fig. 11 become x = ± 84.184 MPa and y =  141.254 MPa. These lie well inside the intercepts for ultimate matrix shear failures, which are roughly twice as far from the origin, confirming that the liners should have been able to boost the maximum strength for all but fibre failures. The general shape of the theoretical failure envelopes has thus been confirmed for generalpurpose structures. The quantitative assessment has also been confirmed, except for the need to locate a matrix-cracking cut-off in the upper-right corner and to extend the failure envelope down and to the left to match the higher lamina compression strengths than were achieved by the unidirectional test coupons. A rational explanation has even been found to justify extending the failure envelope, for this set of monotonic loads only, to cover the test results between the hoop tension axis and the calculated failure envelope. Although this may not be obvious to those who have not studied composite test data extensively, the experimentalists who achieved such phenomenally high biaxial compression strengths in this series of tests are to be commended. Here, finally, is the irrefutable proof that the primary cause of discrepancies between test and theory in this compression-compression quadrant is that the earlier tests, for a variety of reasons, failed to demonstrate the intrinsic capabilities of the fibre polymer composites. They achieved only structural failures that preceded the material failures they sought.

8. Test problems no. 7 and 8: stress–strain curves for ( ± 55°)s glass-epoxy laminate: Type: E-Glass/MY750/HY917/DY063 Although the author made no attempt to solve these problems in Part A of the exercise, it is appropriate to comment on the data given by the contest organizers in Figs 8 and 9 of the data

790 package for Part B of this exercise [12], because they have a direct bearing on the explanation offered in the preceding section to reconcile test and theory. Their Fig. 8 is annotated to identify what must be matrix failures at strains of about 5%, only half of those withstood before failure under equal circumferential and longitudinal tensile strains. The circumferential stress achieved at the first failure, in the matrix, is some 400 MPa, roughly two thirds of the ultimate strength of 600 MPa. Their figure 9, for pure internal pressure in the absence of mechanically applied longitudinal loads, identifies matrix failures, by weeping of the oil from inside the tubes, at a hoop stress as low as 300 MPa, with ultimate failure not occurring until some 700 MPa had been achieved. A series of possible stress–strain curves, based on specific failure mechanisms in the matrix, is included in the Appendix for both of these requests. The curves in Figs A1 and A2 show appreciable nonlinearities as the result of assumed incomplete failures in the matrix. 9. Test problem no. 9: biaxial (␴x ⴚ ␴y) failure envelopes for quasi-isotropic (0°/ ± 45°/90°)s carbon-epoxy laminate. Type: AS4/3501-6 Some of these test data had been compared with the author’s theoretical predictions during the published evolution of his failure theories for composite laminates. Only the compression compression data are new to him. Unfortunately, this last set only confirms his earlier conclusion that the agreement between test and theory would have been perfect, had it not been for some premature experimental failures. This said, it should also be noted that so many of the tests, particularly in the first quadrant, did not fail prematurely that these data are a remarkable tribute to those who performed them. Many others have fallen far short of these results, and had their results published with no apparent awareness that they should have attained far higher strengths. The basic problem is that, in this case, the equal-biaxial-tension strength should be 50% higher than the corresponding uniaxial strength. The same is true for compression loads, There is no counterpart to this phenomenon in the world of isotropic homogeneous materials, so many experimentalists set their sights too low, even to the extent of building elaborate biaxial test fixtures for flat laminates that were inherently incapable of developing higher biaxial stresses in the middle of the test coupons than would fail those same coupons under the uniaxial loads around their periphery. (The author’s efforts at proposing a suitable form of test coupon to resolve the issue finally settled by the very high biaxial compression strengths achieved in the preceding problem are recorded in Ref. [15], where the use of a laterally loaded circular honeycomb sandwich panel was suggested.) Figure 13 compares the predictions by the three theories evaluated by the author against the test data supplied by the organizers in Ref. [12]. The original derivations are shown in figure 20 of Ref. [1], figure 26 of Ref. [2], and figure 9 of Ref. [3]. All failures are predicted to be fibre dominated, so this is the exercise that the author had expected to show the theories he had developed and endorsed in the best light. It is evident that there is very little to choose between the theories, other than in regard to the need or otherwise for the cut-off in the tension compression quadrants. Notwithstanding published statements [16] that these data contradict the need for such a cut-off, only one biaxial test result lies outside the failure envelope for the original (untruncated) failure envelope in Fig. 13, while nine lie well inside it.

791

Fig. 13. Biaxial failure stresses for (0°/ ± 45°/90°)s laminate made of CFRP material (maximum-strain and truncated maximum-strain failure models, generalized maximum-shear-stress failure model, and expanded ten-percent rule failure model).

Before discussing what differences exist between test and theory, it should be noted that the tests were actually run in multiple sets, and that the quality of the specimens was seen to vary with time. There are clear differences in the measured uniaxial tensile strength from set to set. The same is true for the measured uniaxial compression strength, with only the stronger set matching the measurements the experimenters made on flat laminates of the same material. Worse, despite the fact that the fibre pattern was quasi-isotropic, the compression strengths measured under hoop and axial-compression, respectively, in the compression-compression quadrant, differ by a factor of almost two, that for axial (rather than hoop) compression being the greater. The presence of some buckling failures in these tests is confirmed in Ref. [12], particularly for the tubes under external pressure. Under the circumstances, the use of single measured reference strengths for the analyses is questionable, particularly in regard to the use (by one of the author’s critics [16]) of weaker reference data from the first quadrant, in combination with the stronger data in the second quadrant, to “prove” that the author’s predicted truncation in the second quadrant simply did not exist. The only deviation to be explained in the tension-tension quadrant is that the theoretical failure envelopes do not take account of the strain hardening nature of these fibres. Data supplied by the organizers [4], were used in Ref. [1] to refine the estimated tensile equalbiaxial-strain point to 1098 MPa, rather than the linearly estimated 995 MPa. This is in accordance with the test data, but such a complication is beyond the scope of normal structural analysis. Other than the slight concave curvature from this effect, the test data in this quadrant would confirm those theories that predicted straight-line characteristics and reject those that

792 didn’t. The data thus confirm both the form and the magnitude of all of the projections shown in the tension-tension quadrant of Fig. 13. The issue in the tension compression quadrant is not so straightforward since some, at least, of the data were more influenced by buckling of the externally pressurized cylinders than by the inherent strength of the composite laminate under purely in-plane loads, which is all that was accounted for in the analyses. All that can be claimed with reasonable certainty is that none of these data are inconsistent with any of the theories discussed here, although the probability that there should be no cut-off is extremely remote. Regrettably, the data in the compression-compression quadrant have yet again confirmed the extreme difficulty in generating reliable test data of this kind. The primary basis for claiming that the calculations, rather than the test data are the more credible in this quadrant, for this problem, is the highest of the compression-compression tests achieved with the ( ± 55°)s laminate in the preceding exercise. They are precisely what these same three theories would predict if the fibres had been sufficiently well stabilized to not fail by some form of instability, but to develop the same longitudinal strength as the fibres had shown under tensile loads. There is a further potential complication for this kind of tests; developing the highest compressive strengths, even for uniaxial loads, requires that the parallel plies not be clustered together but be thoroughly interspersed between the fibres in other directions. Nevertheless, the author would submit that theories predicting lower compression compression strengths than shown here, but which did not take account of failure being due to structural rather than material strength, are no more confirmed by these test data than are any of the theories discussed here. Despite the experimental difficulties with instabilities for compression-dominated loads, there is nothing in these test results to cast doubt upon the validity of any of these theories. The correlation in the tension-tension quadrant is precise, if a slight correction is made for the strain hardening of the fibres. Although not absolutely conclusive, the preponderance of this, and other, evidence favours the existence of the cut-offs in the tension compression quadrants rather than their absence. There are no published data to prove their absence, particularly for carbon-fibre-reinforced plastics, for which the effect should be most apparent. The issue is unresolved by the present set of data in the compression compression quadrant, but there is every reason to suspect the data rather than these theories. Unfortunately, for precisely the same reasons, these same compression compression data are unlikely to either disprove or confirm any other failure theory, either – unless the other predictions are so great as to exceed not only the test results but also the predictions of the maximum-strain failure model by a large margin.

10. Test problems no. 10 and 11: stress–strain curves for quasi-isotropic (0°/ ± 45°/90°)s carbon-epoxy laminate: Type: AS4/3501-6 The author made no attempt to solve these tasks, because he possessed no special expertise on the subject. However, the organizers have requested, as part of their review, that he try to explain observed nonlinearities in the stress–strain curves for this laminate. While not quantified, the following physics-based explanation is provided in response. Their comment is that, “for the quasi-isotropic laminates, the test results show some form of cracking and reduction in the stiffness of the stress–strain curves at around 450 MPa. The theoretical

793 predictions of final failure stress were some 30–40% higher that the stress observed at cracking.” The first response to point out is that this level of linear response prior to detected cracking confirms the authors statements over many years that predicted first-ply failures based on measured transverse-tension strengths for isolated unidirectional laminae simply do not occur when they are predicted. The author would add, to clarify misinterpretations of some of his statements about matrix failures over the years, that he has never stated that matrix failures do not occur, only that neither his own theory nor many of the most popular interactive composite failure theories are capable of predicting when they occur. He has stated that, for carbon epoxy laminates at least, better designs are produced by assuming that such matrix failures do not occur and interspersing the plies at different orientations to minimize the likelihood that they would occur than have been achieved by believing false predictions of matrix failures and fiddling the analysis to avoid accepting these predictions. It is true that, before he was educated by Prof. Puck and his German colleagues about matrix failures in glass-fibrereinforced plastics, his life in a world of carbon-fibre aerospace structures had shielded him from exposure to the greater importance of matrix failures in non-aerospace applications. In preparing the answers to the set problems in Ref. [1], the author tried to portray the original and truncated maximum-strain failure models as they have been used in the US aerospace industry rather than to improve upon them where he could. This is why his predictions of matrix failures were confined essentially to Ref. [2] on the generalized Maximum-ShearStress model. That said, the organizers have provided real experimental evidence to explain, even if the explanation does lie beyond current aerospace practice. Having now been educated on the subject by Prof. Puck and his colleagues, and by a new colleague Jon Gosse within Boeing, the author offers the following plausible explanation. Let us accept that, at some strain level prior to that needed to fail the fibres, real matrix cracks develop between the 90° fibres in the quasi-isotropic laminate under 0° loads. The best explanations given to the author about this phenomenon are that the 90° plies will initially retain the 0° stress at which the first failures occurred, but that the density of the matrix cracks will increase as further 0° loads are applied to the laminate. Eventually, a complete loss of transverse strength would occur. According to the Ten-Percent Rule, the contribution to the overall laminate stiffness of the 90° plies, prior to matrix cracking, would be about 0.1/(1.0 + 0.2 + 0.1), or 7.7%. If the 90° plies first cracked at 70% of the strain-to-failure of the 0° plies, per the organizers’ observation above, the residual stiffness of those plies would remain constant, because the integral of even many zero-width cracks is still a negligible fraction of the total length. This loss of stiffness, at the laminate level, would be 30% of 7.7%, i.e. 2.3%, which is almost undetectable experimentally. The plot of test data supplied for Problem No. 10 (y /x = 1/0) indicates a loss in stiffness of between 2 and 3%, suggesting that the matrix between the 90° fibres had cracked but not completely failed. This is not sufficient to explain the far greater reduction in stiffness reported in the test data for Problem No. 11 (y /x = 2/1). The author’s interpretation of the data supplied for this problem is that, at the 450 MPa stress flagged by the organizers, there was an instantaneous loss in stiffness of about 6%, that was manifest as an instantaneous increase in $y strain under constant stress. There also appears to be a further reduction to about 90% of the original stiffness, just prior to failure. Gosse’s matrix-failure theory makes it far more likely that there would be matrix cracking between the 90° fibres for this biaxial load than for the uniaxial load

794 in the preceding problem. It would also predict that catastrophic matrix failure at less than ultimate biaxial strains could precede total failure under uniaxial strain, even at the higher $y ultimate strain. The magnitude of the change for biaxial loads appears to be almost enough to be explained by the 90° plies becoming totally ineffective in regard to 0° loads. No secondfailure mechanism suggests itself for the loss of stiffness between the two final measurements other than that, per Gosse’s failure model, the matrix between the ± 45° fibres would be far more likely to fail under these biaxial loads than under the uniaxial load. This might explain the further loss of stiffness, but it obviously lies beyond the capabilities of the author’s own theories. If confirmed by microscopic examination, or by someone else’s more detailed analysis, the hypotheses above would reinforce the author’s earlier statements that, for laminates subjected to different multi-axial loads in unpredictable sequences, any real matrix failures MUST be regarded as design ultimate strengths. It is regrettable that this issue has been clouded by the frequent prediction of matrix failures that do not happen. Having to over-rule so many imaginary matrix failures, or real matrix failures predicted to occur at the wrong strain level, has created an environment in which it has become difficult to focus rationally on the consequences of matrix failures that really do occur when they are predicted to.

11. Test problems no. 12–14: stress–strain curves for ( ± 45°)s and (0°/90°)s E-glass composite laminate: Type: E-Glass/MY750/HY917/DY063 The author did not solve these problems, either. However, in the process of studying them, he observed the following. For the equal biaxial tensile loads applied to the ( ± 45°)s laminate, any intermediate failure of the matrix would leave the fibres perfectly capable of carrying load, per netting theory, until they finally broke. This is a problem in which widespread evidence of a first-ply failure should be apparent before final failure. The author’s understanding of this situation is that there should be no reduction in laminate strength at the onset of matrix failure, only a reduction in laminate stiffness, by quite a small amount. The density of matrix cracks between both sets of fibres would increase steadily until the fibres finally failed. This process is progressive and not instantaneous as so many others have postulated. The evidence reported by the organizers in Ref. [12] tends to confirm the acoustic emission measurements made by others; the “event” is not instantaneous but is really quite gradual. The data supplied indicate major intervals between the observation of the first cracks, leakage of unlined specimens and final failure, which occurred at strain levels a little short of the ultimate fibre strains attained on the uniaxially loaded test coupons. The fact that the loading is by equal biaxial strain ensures that there are no matrix shear deformations. All observed matrix cracks must therefore have occurred under transverse tension between the fibres in each ply. In the case of equal-and-opposite tension and compression loads applied to the same ( ± 45°)s laminate, any deviation between the results that the organizers provided in Ref. [12] and the lamina reference data they supplied in Ref. [4] for pure-shear loads applied to a 0° lamina, should be ascribed to some structural, or testing, phenomenon rather than to a material effect. The later data show easily detectable increases in initial modulus, strain-to-failure, and peak shear strength. The author offers no explanation, only the observation that, on the basis

795 of the material reference properties supplied, no one should have been able to generate a precise match with the answers finally given. The test data for the final exercise, a uniaxial tensile load applied to a (0°/90°)s laminate, show a distinct kink in the stress–strain curve at just over 50% of the ultimate laminate strength. There is no step increase in strain at the initial failure, only a uniform reduction in the residual laminate stiffness. The final gauge readings were taken long before final failure and even before the onset of weeping in unlined tubes. It is obvious that these phenomena are clearly matrix dominated and beyond the capabilities of the author’s fibre-dominated strength predictions. The more significant observation is that the fibre-dominated ultimate strength should have been on the order of (1 + 0.1)/2 = 0.55 times the unidirectional (0°)-reference strength of 1280 MPa, if the laminate were totally fibre-dominated until failure. This would be about 700 MPa, significantly higher than the 600 MPa shown for the (0°/90°)s laminate tests. The 600 MPa measured strength is in better accord with a hypothesis of complete failure of the matrix between the 90° fibres, leaving a final strength of 640 MPa. Structurally significant matrix failures preceding fibre failures in glass-fibre-reinforced plastic laminates are events that Puck and his colleagues have been aware of for decades. Yet a corresponding analysis of the stress–strain data provided by the organizers for the quasi-isotropic (0°/ ± 45°/90°)s AS4/3501-6 carbon-epoxy laminates indicates no such effect. The Ten-Percent Rule would predict a laminate strength of (1.3/4)
1480 = 634 MPa, if the 90° plies retained their full transverse strength, or only (1.2/4)
1480 = 585 MPa if they retained only their longitudinal strength. The stress–strain curve provided shows a laminate strength just over 700 MPa, refuting any suggestion that the in situ strength of the 90° carbon-reinforced plies had been structurally degraded by early matrix failures. One key issue that the author would like to see resolved by interpreting such data with scientific matrix-failure analyses is whether or not the customarily assumed first-ply failure, in the form of matrix cracking between the transverse fibres, actually does occur to a detectable level and, if so, at what load level. The data provided by the organizers suggest that there is no universal answer for all laminates and that there is, therefore, a real need for mechanistic matrix-failure analysis methods that can – and must – be relied upon.

12. Responses to organizers’ questions Finally, the organizers have requested as part of their review of the paper that the author respond to a number of questions. (It is expected that each author received a similar request.) • The lessons learned (or confirmed) by the author from analyzing the test cases are as follows. Only mechanistic failure models are appropriate for fibre-polymer composites; interactive failure models based on artificially homogenized “composite materials” are inherently incapable of explaining the observations. While the same failure models may be used to explain the strength of isolated fibres, their effective in situ behaviours can be quite different. Specifically, the original maximum-strain failure model is excellent for glass fibres, which are very much stiffer than the matrix in the transverse direction, but the truncated maximum-strain model is a far more realistic, at the lamina level, for carbon fibres, which have far lower transverse stiffness.

796 • A recommendation as to which of these theories the author would recommend for others to use must vary with the circumstances. There is little to choose between them numerically (although this is not the case for many other published failure theories). Obviously, the author has no hesitation in recommending the Enhanced Ten-Percent Rule for novices, because there are no choices to be made as to what properties should be used. For the same reason, he would recommend the same model for anyone without sufficiently complete sets of properties to use any of the other theories. The interactive failure models that the author does not recommend continue to suffer from a lack of scientific basis, but they have also been hampered by the traditional use of linearized matrix-dominated properties applicable at best to only isolated laminae and not to multi-directional laminates. If the same set of properties were entered into either of the maximum-strain models – without the industrial modifications to over-ride artificial predictions of matrix failures – these better models would not perform as well as they have here. Given reliable data, the author would recommend the use of the original maximum-strain failure model (with modified transverse-stress cut-offs) for glass-fibre-reinforced plastics, in conjunction with someone else’s mechanistic failure model for matrix failures. The similarly modified truncated maximum-strain failure model is to be preferred for carbon-fibre reinforced laminates, again in conjunction with reliable matrix-failure models. The generalized maximum-shear stress failure model for fibre failures, including the other two fibre-failure mechanisms as employed here, serves two roles. It scientifically justifies the two preceding models that had been developed empirically initially. Because of its mechanistic background, the author would recommend the generalized maximum-shear-stress theory as the basis for researchers seeking to expand the prediction of strength to other (forms of) composite materials. It would be inappropriate for the author to recommend the exclusive use of any of these four failure models to the exclusion of the other three.9 As to the accuracy of these models perceived by the author, it should be obvious that he ascribes any major differences between truly totally fibre-dominated failure predictions and test data to problems with the test data. Conversely, without the empirical accounting (approximately) for matrix failures, the theories he developed or endorsed could not even begin to solve the problems other than for the quasi-isotropic carbon-epoxy laminate. This leaves no doubt about the need for proper analysis tools for matrix failures to complement the first three models the author employed. This is reinforced by the author’s speculations about the effects of the matrix transverse-tension cut-offs for which he could establish only the form and not the precise location. It seems likely that the upper-right quadrant of the

9

The author would now add a further answer to this question, in a different context. In 2001, he reformulated the hexagonal Tresca-based fibre failure model in Ref [2] in terms of the corresponding elliptical von Mises-based model, again needing only a single material reference strain for all states of biaxial strain. This has now been incorporated into Gosse’s computer codes for matrix failures, because it is far easier to evaluate a surface defined by an ellipsoid than by a flat-faceted polyhedron. There is little to choose between the two models numerically, but the more recent formulation has also removed earlier ambiguities about how best to characterize failures in the transverse-normal plane of the fibre that might have been governed by different mechanisms than pertained in the longitudinal-transverse and longitudinal-normal planes. More importantly, this reformulation has established that even within the tension tension quadrant, conventional aerospace carbon fibres, at least, fail by distortion rather than by brittle fracture. This new work is documented in Ref [17].

797 (90°/ ± 30°)s laminate on the stress plane should be truncated by these transverse matrix failures, even though the analyses evaluated here cannot prove that conclusively. Even so, they provide a very close upper-bound solution defined by possible transverse failures in the fibres. Subject to these caveats, the author feels his predictions of real fibre-failures are more than accurate enough for design use. The problem is that the empirical predictions for the real matrix-dominated failures, even if they prove to be better than achieved with many of the other authors’ models, do leave significant room for improvement which, unfortunately, the author cannot provide. Fortunately, he is now aware of two sources of matrix-failure predictions that can fill this need. • The organizer’s question about the shortfalls of the theories and how large the ‘errors’ might be is addressed in the preceding paragraph,. The author would add the comment that the form of additional truncations suggested by his constant-transverse-stress cut-offs does match the discrepancies between test and theory for the ( ± 55°)s and the (90°/ ± 30°)s laminates, and is not in the form of a proportional correction to the matrix-shear or fibrefailure predictions shown. The author would estimate that, for these cases, the predictions need to be reduced by about 25% to match the data. Conversely, many of the compressiondominated test failures need to be increased by 50% or more to match the theoretical predictions. • The organizers raised a very pertinent issue in questioning the suitability of each theory for others to use. As noted above, the author has no concerns about erroneous predictions associated with the Ten-Percent Rule. The model is unambiguous and only three properties are involved; the longitudinal lamina strengths in tension and compression and the longitudinal modulus. Given good reference data, one can hardly go wrong. The successful application of the other three models, on the other hand, relies on three factors that do need some skill. Significantly, both have been mastered by most of the US aerospace industry. They are not all that difficult, but the consequences of not applying the modifications properly can be as disastrous as using the wrong failure model. The first is to use the original untruncated maximum-strain model, at the lamina level, for fibreglass-reinforced polymers and the truncated maximum-strain or generalized Tresca model for laminates reinforced by carbon fibres. The second is to modify the transverse-tension properties in such a manner as to preclude the prediction of matrix failures that do not occur. Because of the independence of proper characterizations of fibre and matrix failures, this step is still necessary for predicting fibre failures even if one does have the benefit of parallel (or sequential) estimates of matrix-failures with a precise model. The third factor is that, even for fibre failures, one needs to enter matrix shear strengths and stiffnesses that match the fibre strains when they fail. Other than by iteration, no precise match is possible because, usually, not every ply becomes critical simultaneously. Given the relatively small contribution to laminate stiffness, and hence ply-by-ply load sharing, of soft matrices in comparison with stiff fibres, it is recommended that only two options be considered. One is for matrix properties to match fibre strains at failure (or at failure of the matrix if there are fibres in insufficient directions) and the other is for fibre strains at lower limits appropriate for laminates containing bolt holes. Those two options should suffice. • Further steps needed to produce better design models for such laminates as represented by the test cases have hopefully already been provided by other authors, using mechanistic failure models for matrix failures. They are clearly needed. The author is relying on his friends in Germany and a colleague (Jon Gosse) at Boeing found since the merger with

798 McDonnell Douglas to fill this need. He has no matrix failure criteria of his own and is grateful for opportunities to help those who do. • The issue of what constitute subcritical matrix failures, prior to catastrophic laminate failure when the fibres finally break, has been vexatious. The author has, for years, struggled against two widespread misconceptions. The first is that is acceptable to use nonmechanistic models that predict matrix failures when they do not occur as a pretext to change both failure model and matrix properties to achieve a more acceptable answer, without making the slightest effort to see whether or not the failures they predicted did occur when they predicted them. The second, which may have been encouraged by the first, is that it is permissible to disregard real matrix failures in a similar fashion merely because they precede final failure under monotonic loading of test coupons. Real matrix failures MUST be considered as design ultimate strengths in many situations. Failure to acknowledge these realities has masked a need to increase resin content to improve properties other than the specific unnotched unidirectional tension strength for test coupons, which has led to excessive fibre volumes, at least in the author’s opinion. He is obviously very sensitive to the organizers’ questions in this regard. It is very important to be able to predict real matrix failures that precede fibre failures and to be able to distinguish between tolerable and intolerable matrix failures. Unfortunately, as noted already, the author has no theories with which to assist this process. • It should be clear from the author’s solutions to the test cases that he believes that it is necessary that the analysis be formulated at the lamina level. Other publications have made it clear that he also regards it as necessary to decompose the properties of multi-directional layers of (woven) composites into two equivalent orthogonal unidirectional plies. This is the only level at which it is possible to characterize both fibre and matrix failures. However, the author has also stated clearly that the “composite material” MUST be modelled in terms of its constituents and NOT as some artificially homogenized anisotropic solid. (The homogenized composite material approach is fine for establishing lamina stiffnesses, however.) Effectively, one needs to characterize micromechanical effects at the macromechanical level. Help he provided recently to Jon Gosse in formulating his matrix-failure model, used to predict the strength of badly designed composite structures that really did fail in the matrix before the fibres could be loaded to near their capacity, has confirmed what the author said about the need to include intralaminar residual thermal stresses in any attempt to predict matrix failures. Gosse’s model is formulated at the lamina level but needs micromechanical finite-element models to relate the macro-level (laminate) strains to the states of stress and strain in the matrix and fibres as distinct constituents, in the same manner as the transverse strains in the fibre were related to those in the lamina for the generalized maximum-shear-stress fibre-failure model described in Ref. [2]. Puck and his German colleagues have, since 1968 at least (probably earlier), maintained the need to separate the characterization of fibre and matrix failures but have yet to include intralaminar effects. The German models, too, are entirely mechanistically based, but formulated in terms of homogeneous laminae. • Actually, the organizers may have not fully envisaged the total extent of what their question relates to. The question is not just micromechanical versus lamina-level analysis; the question should be “which lamina?” The compressive strength of fibres is so sensitive to fibre waviness that any deliberate or unintentional crimping must also be accounted for. This means different properties for plain-weave cloths of very different tow size. And

799 further properties, necessarily measured at the laminate level for satin-weave cloths and filament-wound structures, and the like, must also have equivalent unidirectional lamina properties “backed out” to establish the properties needed for analysis. There is also a strong ply-thickness effect for unidirectional laminates. The fibres are actually stronger when laid down in thinner layers, with periodic crossplies to prevent fibre wash from ply to ply. Both fibre and matrix must be characterized in the form in which they exist in their structural laminates. • Micromechanical relationships are important to the extent that they are needed to characterize the in situ properties of the constituents of the composite laminates. And there is no doubt that it is necessary to know the mechanical (elastic constant) and thermal (coefficients of thermal expansion) properties of both constituents, as well as properties measured at a higher level. However, it would be inappropriate to try to formulate strengthprediction methods based only on fibre and matrix properties measured in isolation. This is easy to see for the fibres. Suppose that there are 60 broken fibres on a tow of 3000 fibres. If tested dry, one would expect a loss of strength by 2%. But, if the same fibres were embedded in a resin matrix and none of the breaks was at the same axial position as any other, effectively only one fibre would be broken at any cross section. The major stress component in the matrix for typical high-temperature-cured carbon epoxy laminates, regardless of the external mechanical loads, is the intralaminar residual thermal stress. This affects all measured matrix strengths. Gosse’s matrix-failure model now accounts for this effect, but the actual failure criterion is best established by extracting the same property from what appear to be different tests, at the macro level. This gives far greater confidence in the results that separately characterize dilatational and distortional failures. While consideration of micromechanical effects is vital in predicting fibre-polymer laminate strengths, the need for in situ properties overpowers the traditional approach to the subject. • The above observations lead directly into the organizers’ next question about the need for different or additional properties. The characterization of fibres appears to be adequate, provided that they reflect the actual ply thicknesses and any crimping present in the actual laminate. Some “lamina” properties need to be backed out from laminate tests. Nevertheless, it is also apparent to the author that so many of the data provided for this exercise involve premature failures due to structural effects, rather than material effects, that the use of cross-ply testing [18] that improved flat-laminate tension and compression strengths so much needs to be applied to tubular coupons as well. One benefit from such tests is that load-introduction problems have been alleviated by using laminates that have proportionally far less strength and far more bending stiffness than for the same number of plies, but all unidirectional. Gosse’s work has exposed what is probably the primary reason for such low measured transverse-tension strengths on unidirectional laminae. By adding thin surfacing plies of 0° fibres, he was able to consistently deduce the same transverse strength, regardless of the total thickness, and at a far higher level than for conventional tests, which he showed to be just as sensitive to surface flaws as the 0° tension tests have been to edge effects. The author would recommend that it is worth while to try surfacing plies of adhesive film instead of 0° fibres, to see if the same benefits can be achieved while still leaving the property capable of being measured directly. What the author knows of mechanistic matrix-failure theories is that some of them rely on properties that can only be inferred, rather than be measured directly. Validating such properties requires multiple tests

800 causing the same failure mode. Any single such test may lead to no more than a curve fit, with no validation for other situations. • Thermal stresses are customarily omitted from consideration because the standard interactive failure theories taught in universities permit no consideration of intralaminar residual thermal stresses. Accounting for inter-laminar thermal effects alone has led to so many inconsistent predictions that it has become customary to exclude all thermal effects rather than to face up to admitting to inherent deficiencies in the most widely taught theories. Fortunately, thermal stresses have only a negligible effect on the strength of strong stiff fibres embedded in soft resin matrices, so their omission in the author’s fibre-failure models is understandable, and justifiable. The author believes very strongly that these effects should be included in any matrix-failure models, as Gosse now does. However, Puck’s success in predicting matrix failures without such effects cannot be denied. One observation might reconcile these two approaches; Gosse’s model used to “work” before he included intralaminar effects, but not as well as it does today, and with a number of unresolved inconsistencies that have since been eliminated. What the inclusion of these effects in Gosse’s theory accomplished was a dramatic improvement in consistency between different load cases and an appreciable reduction in scatter in interpreting the test data. There is no doubt that it is now a far better theory than it used to be. And this was achieved without any change to the matrix-failure criteria he postulated many years ago. It is the author’s belief that the German matrix-failure models would be enhanced by the inclusion of such effects, too, with no risk that they would be invalidated. Their physical basis is secure even if the author does think that they are capable of being made even better. • The issue of the importance of in situ properties that the organizers raised has been addressed above. They are necessary and the only way to generate them is to test laminates in which the fibres and matrix are disposed in the same manner as they are in the real laminate. It is then necessary to “back out” equivalent unidirectional lamina properties with which to perform the analyses. This will usually involve a mixture of tests on multidirectional and unidirectional layers. This issue is discussed in Ref. [18]. • The author’s fibre-based models alone are incapable of explaining nonlinearities observed in stress–strain characteristics, so his answer for how best to predict (or explain) them must be based on the capability, or improvements, of other author’s theories. The author addressed this issue in the body of the paper in reference to the quasi-isotropic (0°/ ± 45°/90°)s carbon-epoxy laminate. The author would suggest that this is impossible with a less than comprehensive model of matrix failures formulated at the constituent level – and a thorough understanding of the mechanisms that are and might be involved.

13. Suggested modifications to the models The organizers have also requested comments on possible improvements or modifications that ought to be made to the theories evaluated in light of this exercise. The first point to be made is that having to solve problems outside the customary (0°/ ± 45°/90°) family of fibre patterns used in the aircraft industry has already caused major improvements to be made to the TenPercent Rule, as described in Ref. [3], just to be able to solve the other problems. The ( ± 55°)s and (90°/ ± 30°)s problems likewise exposed a need to expand the traditional empirical transverse lamina strengths beyond limits set for orthogonal fibres in the original and

801 truncated maximum-strain failure models that had been formulated for laminates in the (0° + 45°, 90°) family of fibre patterns. The appropriate expansions are inherent in the generalized maximum-shear-stress failure model and were discussed at length in the context of the Ten-Percent Rule. They have the form of constant-transverse-stress loci passed through the equal-biaxial-strain point in the tension-tension quadrant. The second point is that there are very strong suggestions from the author’s analyses in Refs. [1,2] that there is a real need for scientific analyses of matrix failures to complement the present models for fibre failures. Since the author strongly advocates the exclusive use of mechanistic failure models, these would not be expected to alter the theories presented here; they would be needed only to solve the aspects of the problems that his theories and those he has endorsed can’t. A third point concerns the inclusion of both glass-fibre and carbon-fibre reinforced plastics in the exercise. This exercise has convinced the author that his life in a “black-string” environment has shielded him from effects that researchers like Alfred Puck experienced with the much-higherstrain glass fibres. He has already responded to these phenomena in recommending the use of the original maximum-strain failure model for glass fibres and of the truncated maximum strain model for carbon fibres, with corresponding modifications to the Ten-Percent Rule. The fourth point to be made concerns the validity of the test data. It is quite obvious to the author that even the best data the organizers could find, after rejecting most of what was published for the same reasons that the author had, includes some results that do not reflect pure material strengths. Indeed, some of the biaxial compression results given by the organizers prove the presence of structural effects in other such data they provided. This comment is not meant to criticize those who perform very difficult tests on brittle materials, other than to suggest the need for them to establish more realistic targets,10 but in the context of dissuading those who would distort material failure models to curve-fit effects that have nothing to do with composite material properties. It is clear to the author that the discrepancies between the test data and his fibre-failure predictions involve some need for the addition of better matrix-failure predictions, but also a need to acknowledge that other discrepancies, particularly under compressive loads, are due to the inability of certain tests to characterize pure material properties. As he has noted elsewhere, one legitimate higher measured strength is sufficient to invalidate hundreds of premature failures that had been recognized as such through higher variability than there should have been, but never acknowledged as such, owing to the anticipated cost of repeating the tests. This exercise has confirmed to the author that there is no point in applying statistics to sets of test data that clearly include variability from sources outside true material variability. The best measure of the value of test data is not the cost of having generated it, or of duplicating it; it is the savings it can generate in production if it is a reliable result. 14. Concluding remarks While the author should confess to harbouring a desire that the true colours of some of the composite failure theories he has condemned for so long would be revealed by this exercise, 10

It is conceded that the apparent acceptability of many unscientific composite failure models has contributed to this uncertainty as to what the target results should be, but comparisons with other sets of data can also provide useful reference points.

802 he can now see the possibility of an even greater accomplishment as the result of this comparison between theories and test data. The capabilities and limitations of his own theories (the generalization of the maximum-shear-stress and Ten-Percent Rule) are now apparent for all to assess. Hopefully, the same will be true of the works of all of the other participants. While it would have been more appropriate to have Max Waddoups champion the original (untruncated) maximum-strain failure model, there is no single origin for the truncation in the tension – compression (shear) quadrants. The author trusts that he has done justice to these theories as they are used in America, in industry at least. He has consciously refrained from trying to plant his own slant on them. (However, he has prepared a separate paper suggesting improved empirical transverse strength limits in the light of what he learned during this exercise, to be published elsewhere.) These two theories are far too significant to have been omitted from this exercise. The author deeply regrets that the developers of certain other failure models, for whatever their claim to significance, did not participate. He would recommend that, even at this late stage, some further theories be evaluated by their originators or by some impartial government agency. Indeed, there is no reason why anyone who has ever proposed such a theory should not complete this same exercise. There are sufficient discrepancies between test and theory, which the author would ascribe to difficulties with the experiments, that any attempt to simply “fit” the theory to the data would not result in a useful, let alone credible, analysis tool. The author, and presumably all of the other participants also, can offer assurance that these three problems will thoroughly exercise any theory. Possibly the only additional fibre patterns worthy of consideration are a few which could be selected to highlight the idiosyncrasies of scientifically unsound theories, even without test data to confirm them. What the author had not anticipated at the start of this exercise was the tremendous improvements (needing to be) made to his understanding of the subject, and of the theories he used to characterize the strength of fibre polymer composites, during the completion of the exercises. If the organizers inspired comparable improvements by every other participant as well, this exercise will have been more than worthwhile from this vantage alone. In summarizing the capabilities and limitations of the theories he has championed, it is the author’s position that, for failures of the fibres, the discrepancies are primarily due to shortcomings in the test program. While any author could claim this, the author has provided specific explanations in each case. In regard to the truncations for fibre failures in the tension compression (shear) quadrants, the improved analysis during this exercise, whereby a quantified distinction was established between the transverse strains in the fibres and the matrix, the author would recommend the following. The truncation, at 45° on the lamina strain plane, is appropriate for carbon fibres, while it should not be considered, at the lamina level, for isotropic fibres like glass, even though the cut-offs are always appropriate for the isolated fibres. The test data provided by the organizers very strongly support this position, even though some obvious doubts about some of the test data prevent a claim of absolute confirmation. There is, on the other hand, absolutely no indication of any justification to disregard the cut-offs. It has surprised the author that the matrix failures implied by the Ten-Percent Rule, as fractions of fibre-dominated properties, agreed so well with the test data at failure. Obviously, they failed to identify any non-catastrophic matrix failures on the way. But they also avoided predicting matrix failures that obviously did not occur. They could well serve as first-estimates of matrix-dominated strengths quite generally, subject only to being rendered ineffective if

803 they are followed shortly thereafter by failure of fibres perpendicular to the cracks. The relative softness of skewed fibres appears, from the test data and from common sense, to be insufficient to justify ignoring predicted failures in such cases. It was always expected that any weaknesses in the author’s theories would be in regard to real matrix failures. The same is true for both maximum-strain failure models, for which empirical cut-offs traditionally replace the blind use of measured transverse strengths that are appropriate only for isolated unidirectional laminae and not when they are embedded in a structural laminate. (The strength of these theories in this regard is that they then do not predict matrix failures when they don’t occur.) The need for some such improvements was recognized while the individual problems were being solved in Part A, while one more major effect was not anticipated before the data were supplied for Part B of the exercise. The more interesting issue concerns the author’s reliance on empirically located constant-transverse-stress cut-offs to characterize in situ transverse cracking between the fibres in each ply, whenever real failures of this type occurred. The failure envelope for the ( ± 55°)s fibreglass-epoxy laminate, under biaxial loads with a predominant 0° component, definitely seems to confirm both the existence of this failure mechanism, and that the theory is of the right form to characterize it, if only it were possible to position the cut-off. Hopefully, the micromechanicians will have been able to do so. There is also a strong indication that such effects were observed in the tension-tension quadrant for the (90°/ ± 30°)s laminate. The other technique of interest is the modelling of in-plane-shear failures in the matrix, when there are no fibers to constrain it, as an elastic-plastic phenomenon mathematically, just as for ductile adhesives (even though the physics are actually very different), with the shear strains possibly extending until the laminate finally fails when some fibres break. This phenomenon seems to have been evident for some of the combined stresses acting on the ( ± 55°)s laminate.

Acknowledgements Presumably like every other participant, the author embarked upon this exercise confident that he had already developed the worlds’ best composite failure models, conscious only of some undefined weaknesses in regard to matrix failures. What he has since learned that he did not already know is another matter all together. There is a giant world for composite structures outside aerospace, and the level of analytical capability (needed and available) is quite variable. Although he has long been apprehensive about the likelihood that any test data would ever succeed in discrediting any published composite failure theories, particularly those with disposable parameters, the author cannot refrain from commending the organizers for their excellent choice of fibre patterns for this contest. They forced him to expand his vision beyond the world of carbon-fibre-reinforced polymers with orientations confined within the (0/ ± 45/90) family with which he was familiar. Working with the other fibre patterns, without the help of any computer program, has identified simple changes needing to be made as to how transverse strengths should better be handled for the popular maximum-strain failure models (in all variants). Having to analyze a laminate for which it was not possible to rely on simplifications arising from fibres failing before any matrix failures has made him aware of just how much more difficult a challenge has been faced by his peers who had worked with fibreglass-reinforced laminates. Even the fibre pattern for which it was not obvious a priori that it was fibre dominated proved to be a real challenge, particularly in regard to the

804 transverse cross section of the (90°/ ± 30°)s laminate. He is particularly thankful to his German counterparts who have educated him so well in regard to what is needed to predict real matrix failures, rather than to use abstract mathematical models to predict matrix failures that do not occur. No comparable brain-dump of wisdom was forthcoming from America (until he was approached by Jon Gosse after the merger of their employers) but, because he worked there, the help and encouragement he did receive there (and from colleagues in England) was vital in sustaining him through Parts A and B of this exercise, particularly in the face of so much opposition from a number of sources. But his greatest debt is to Profs Keith Kedward of UCSB and Sam Kaddour of UMIST (now at DERA/Qinetiq) for challenging his belief that there was no hope of developing the Ten-Percent Rule beyond a simple empirical rule-ofmixtures formula.

References [1] Hart-Smith LJ. Predictions of the original and truncated maximum-strain failure models for certain fibrous composite laminates. McDonnell Douglas Paper MDC 97KOO10. Composites Science and Technology 1980;58:1151–78. [2] Hart-Smith LJ. Predictions of a generalized maximum-shear-stress failure criterion for certain fibrous composite laminates. McDonnell Douglas Paper MDC 97KOOII. Composites Science and Technology 1998;58:1179–208. [3] Hart-Smith LJ. Expanding the capabilities of the ten-percent rule for predicting the strength of fibre polymer composites. McDonnell Douglas Paper MDC 97K0012. Composites Science and Technology, V62, pp 1515–1544, 2002, see also Chapter 3.17 in this book. [4] Soden PD, Hinton MJ, Kaddour AS. Lamina properties, lay-up configurations and loading conditions for a wide range of fibre reinforced composite laminates. Composite Sciences and Technology 1998;58(7):1011–22. [5] Waddoups ME. Characterization and design of composite materials. In: Tsai SW, Halpin JC, Pagano NJ, editors. Composite Materials Workshop. Technomic; 1968. p. 254 308. [6] Hart-Smith LJ. The truncated maximum-strain composite failure model. Composites 1993;24(7):587–91. [7] Hart-Smith LJ. A new approach to fibrous composite laminate strength prediction. Douglas Paper 8366. Presented to MILHDBK-17 Committee Meeting, Singer Island, Palm Beach, FL, October 3–5, 1989. Represented to 8th DoD/NASA/FAA Conference on Fibrous Composites in Structural Design, Norfolk, VA, 28–30 November 1989. In: Proceedings, NASA CP-3087, part 2, pp. 663– 693. [8] Hart-Smith LJ. Predicting the strength of fibrous composites by an orthotropic generalization of the maximum-shear-stress (tresca) condition. McDonnell Douglas Paper MDC 91K0068. Presented to Second SERC-I.Mech.E. Expert Meeting, failure mechanism(s) and criteria for predicting the performance of polymer-based composite structures, St. Albans, London, UK, 20 25 September 1991. In Proc Instn Mech Engrs Part G, J. Aerospace Engng. 1994:208;9–18. [9] Hart-Smith LJ. The first fair dinkum macro-level fibrous composite failure criteria. McDonnell Douglas Paper MDC 97K0009. Presented to 11th ICCM Conference, Gold Coast, Australia, 14–18 July, 1997. In: Proceedings 1; I-52 I-87. [10] Hart-Smith LJ. The ten-percent rule for preliminary sizing of fibrous composite structures. McDonnell Douglas Paper MDC 92K0024, SAWE Paper No. 2054. Presented to 51st SAWE International Conference, Hartford, CT, 18–20 May 1992. In: SAWE Journal, Winter 1992. p. 29 45. Aerospace Mat. 5 (2):1993;10–16. [11] Puck A. Festigkeitsanalyse von Faser-Matrix-Laminaten: Modelle fur die Praxis. Munchen, Wien: Hanser; 1996.

805 [12] Soden PD, Hinton MJ, Kaddour AS. Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates: Failure exercise benchmark data: V62, pp. 1489–1514, 2002, see also Chapter 2.2 in this book. [13] Cuntze RG. Evaluation of multiaxial test data of UD laminae by so-called fracture-type strength criteria and by supporting probabilistic means. In: Proc. 11th ICCM Conference, Gold Coast, Australia, 14–18 July 1997. [14] Ewings PD, Potter RT. Some observations on the nature of fibre reinforced plastics and the implications for structural design. Phil Trans Roy Soc Lond, Series A;294:507 17. [15] Hart-Smith LJ. A biaxial-strength test for composite laminates using circular honeycomb sandwich panels. Douglas Paper 7974. Presented to 33rd International SAMPE Symposium and Exhibition, Anaheim, CA, 7–10 March, 1988. In: Proceedings. pp. 1485–1498. [16] Sun CT, et al. Comparative evaluation of failure analysis methods for composite laminates. FAA technical report DOT/FAA/AR-95/109, 1996. [17] Hart-Smith LJ, Gosse JH. Formulation of fiber-failure criteria for fiber-polymer composites in terms of strain invariants. Boeing Paper MDC 01K? J Comp Mat [in press]. [18] Hart-Smith LJ. Backing out composite lamina strengths from cross-ply testing, Boeing Paper MDC 00K0099. In: Kelly A, Zweben C, editors. Comprehensive composite materials. Elsevier; London: 2000 [chapter 7]. pp. 149–161.

Appendix A. Test problems nos. 7 and 8: stress–strain curves for ( ± 55°)s glass-epoxy laminate: Type: E-Glass/MY750/HY917/DY063 The following discussion about the seemingly abnormal behaviour of the ( ± 55°)s laminate is separated from the body of the text because it refers to an after-the-fact explanation of phenomena the author did not predict prior to receipt of the test results. As such, it is not really a discussion of the solutions he provided during Part A of this exercise. The lower limit of the triangular extension in Fig. 12 is a uniaxial stress locus, x = 0. Therefore, a belated attempt to solve the organizers’ Problem No. 7 might uncover some of the phenomena involved ± 45. It should be noted that, if the failure envelopes in Fig. 10 do represent failure of the laminate, the triangular extension in Fig. 12 should involve a “failed” matrix throughout. Nevertheless, it did not take many trial solutions in this area to confirm that a netting-theory solution, for a totally failed matrix with zero residual stiffness, cannot possibly explain the phenomena. The absence of any x stress in each lamina requires that there be compressive transverse stress to balance the tensile stress in each fibre. The compressive stress could not exist in the absence of any matrix. The issue, therefore, is one of establishing the residual strength of the matrix outside the original failure envelope. Figure A1 shows a series of stress–strain curves for uniaxial y loads, some proposed on the basis of intuitive logic, others with the intent of matching the test results provided. The goal is to be able to reject some hypotheses because of discrepancies between the test data provided and the predictions based on those hypotheses and, hopefully, to find one plausible hypothesis not contrary to the data. In each case, the end of the curve is established by the limiting fibre strain-to-failure of 2.807%, and the point at which the matrix shear strain reaches 4.0% is also identified. (The latter is easy to recognize in curves A and B, because it is where the step discontinuity occurs in curve A.) All calculations start by establishing the orthogonal unidirectional lamina strains for which the transverse laminate strain is zero and the shear strain matches one of the data points provided. The corresponding laminate strains and stresses are then deduced by standard lamination theory.

806

Fig. A1. Uniaxial stress–strain curves for ( ± 55°)s fibre glass epoxy laminates.

The lowest curve, A, is for the seemingly plausible hypothesis that there is no residual shear stiffness beyond a lamina shear strain of 4%, the limit of the data provided. There is a discontinuity predicted when the matrix shear strength and stiffness vanish. The transverse stiffness E2 for the lamina is held constant at its initial value, 16.2 GPa. A reduction to zero transverse stiffness, whether instantaneous or progressive, leads to obviously unacceptable predictions, as explained above. Since this concept is the basis of progressive failure theories starting from a prediction of first-ply failure when the matrix-dominated properties of isolated unidirectional laminae are exceeded, one is forced to question their validity. The present data has no abrupt discontinuity, in either strength or the slope of the stress–strain curve, but the predictions based on complete failure of the matrix do. Curve B presents a different hypothesis, whereby the lamina shear strength does not drop to zero, but is maintained at a constant value of 75 MPa, for all shear strains greater than 5%, while the in-plane shear stiffness G12 decreases steadily as the shear strains are increased with no limit. (They are predicted to reach almost 17.5% before the fibres break.) For what it may be worth, the justification for this hypothesis is that the combination of fibres in two intersecting directions, in conjunction with the transverse compression that must exist within each ply, could inhibit the catastrophic shear failure that would occur at a shear strain of 4.0% in a unidirectional laminate (tubular or flat). The discontinuity in curve A is thereby eliminated, and it is evident that this hypothesis comes quite close to explaining the test results, with catastrophic failure identified by the fibre axial tensile strain to failure. It is significant that, despite the severe discontinuity when the matrix shear strain reaches 4%, as indicated in curve A, curves A and B predict quite similar fibre-failure points, indicating that the major role of the matrix by then is to resist the compression developed between the fibres rather than to transmit the shear between them. This suggests that imprecision in identifying the lamina in-plane-shear stiffness and strain at “failure” will not have a profound effect on the explanation of the ultimate test results. (Nevertheless, as indicated below, failure to acknowledge that the matrix does fail and lose much of its shear stiffness would lead to gross

807 overestimates of the laminate strength.) It seems to the author that the presence of intersecting fibres suppresses the spread of the initial micro-cracking that does occur in the matrix, for otherwise the in-plane shear stiffness of the lamina would not reduce. Tests on unidirectional laminae could not exhibit this phenomenon. Curve B terminates at a hoop stress of only about 690 MPa, while the projection of the fibre-failure characteristics in Fig. 10 would suggest a strength in excess of 1000 MPa. The resolution of this apparent discrepancy is that the slope of the fibre-failure lines in Fig. 10 is quite sensitive to the value of the matrix shear stiffness G12. The value used in Fig. A1 is progressively reduced to less than a quarter of that used in Fig. 10, because the matrix was presumed to have “failed” in Fig. 10 at a strain of 4.0%, rather than the 17.5% sustained in the analysis in Fig. A1. Until the applied load would be released, the broken matrix would not have had an opportunity to disintegrate as soon as it was no longer compressed. Curve C provides an absolute upper-bound solution, in which the initial lamina stiffnesses are maintained until failure, even though the matrix in-plane-shear characteristics provided by the organizers would not substantiate this simplified linear approach. Significantly, this prediction does exceed the measured laminate strengths, but at such low laminate strains as not to be a credible solution. What this particular hypothesis does is make the point that matrix nonlinear behaviour, in shear at least, cannot be ignored if one expects to make realistic predictions of laminate failures. None of these three analyses provides a perfect explanation of the relevant test results. Nevertheless, the predictions in Curve B, based on the assumption that the matrix retains its shear strength as its stiffness continues to decline with added strain, comes quite close. As noted in the body of the paper, the lamina transverse compression strength must also be allowed to increase beyond the strength measured on uniaxially loaded test coupons, when the broken matrix cannot be displaced from between the fibres. Conversely, just below the positive y axis, where the tensile axial stresses in the fibres are lower, and their constraining capability is less, the numerical experiments suggest that the matrix between the fibres really does fail under compression since the continued projection of the new limit established in Fig. 12 would otherwise lie outside the test data. The other two hypotheses, for curves A and C, are clearly unacceptable. The matrix does crack, and its stiffness is reduced, but not instantaneously to zero. This is not totally implausible, since it is known that the actual “shearfailure” process is by tensile cracks developing in the matrix perpendicular to the resolved peak tensile stress component inherent in the overall shear loads. These cracks are arrested when they reach the fibres. The application of further load causes the initial cracks to spread along the fibres, but this secondary failure process would clearly be impeded by orthogonal fibres and, to a lesser extent, by inclined fibres as in this case. The intense transverse compression between the fibers in each layer would contribute even more resistance to subsequent spreading of the initial hackles. A possibly significant observation about these calculations is that, while the defined lamina elastic constants were well behaved, as were the laminate constants for curve C, the Poisson’s ratios vxy and vyx for the laminate were not. This was traced numerically to the influence of very low values for the lamina stiffness G12 on the laminate stiffnesses for curves A and B in Fig. A1. These three analyses might not have generated a definitive single stress–strain curve for the ( ± 55°)s tubes under pure hoop tension, even though they have clearly eliminated two common hypotheses. Yet they have once again confirmed the wisdom of the organizers who,

808 by providing test results that could not be explained by traditional analysis models, have identified the need for a far more intensive assessment of matrix failures than has been understood in the past. They also highlight the wisdom of Alfred Puck and his colleagues in having pioneered the analysis of matrix failures in terms of distinct failure mechanisms, each characterized by its own governing equation. Much the same method for modifying basically linear analyses can be applied to the analysis of the same ( ± 55°)s laminate under biaxial pure pressure loads, (y /x = 2/1), as requested by the organizers. The Excel spread-sheet analysis was prepared to record all stresses and strains at both lamina and laminate level. Again, three solutions have been provided, in Fig. A2. Curves A represent perfectly linear analyses, in the traditional manner. Both stress–strain curves are straight, but the transverse strains $22 developed between the fibers exceed the matrix capability by far. The nature of the fiber orientations for this load is that there should be extremely little shear deformation between the fibers, as was confirmed by the analysis. Indeed, there was so little in-plane shear deformation within each lamina that the shear stresses were not reduced to zero even after it was obvious that the matrix between the fibers had completely failed and that the loads were now being reacted only by the fibers, in a state akin to netting analysis. (The same simplification would not apply to very different ratio of stresses, however.) A second analysis, shown as curves B, was prepared by reducing the transverse lamina modulus E2 to zero once the transverse lamina strain had exceeded the specified value of 0.0025. (Similar results were achieved when the critical transverse strain was set at 0.010.) In contrast with the straight curves A, radiating from the origin, the stress– strain curve B in the longitudinal direction is displaced to the right. Also, whereas in curves A, the longitudinal strain $x was always less than the hoop strain ey, by about a factor of three, the longitudinal strain exceeded the hoop strain at the high-load end of curves B.

Fig. A2. Biaxial (2 : 1) stress–strain curves for ( ± 55°)s fibreglass epoxy laminates.

809 Unfortunately, there is a physically unrealistic step discontinuity at the quite low load at which the transverse lamina strain was first exceeded. It would seem that the matrix failure is not instantaneous. Therefore, a third solution was prepared, by hand iterations, to maintain the transverse stress 2 at 40 MPa. This solution, curves C, eliminated the discontinuity, as would be expected. It also introduced curvature in the stress–strain curve in the longitudinal direction. Logic suggests that the real behaviour would best be described as a transition from curves C at low load levels to curves B before final failure. Unfortunately, without a specific physics-based matrix-failure criterion, the author cannot further improve upon the final estimates shown for the stress–strain curves requested by the organizers. In view of the apparent success in selectively modifying the constituent properties to generate specific stress–strain curves, the author’s earlier acknowledgement that he was unable to complete the refinement of the failure envelope in Fig. 12 for all combinations of biaxial stresses needs further comment. It should be evident that the same technique used here could be applied to a succession of different laminate-level stress ratios. And the use of secant moduli would ensure a continuous match between stress and strain levels. The problem is that each of Figs A1 and A2 was dominated by a single matrix-failure mechanism. In the general case, both mechanisms could be expected to be involved simultaneously. No real progress is likely until complete physics-based matrix-failure criteria have been validated – and simplified for easy use without any loss of accuracy.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

810

CHAPTER 5.5

Prediction of ply crack formation and failure in laminates L. N. McCartney NPL Materials Centre, National Physical Laboratory, Teddington, Middlesex TW11 0LW, UK Abstract For a [0/90]s cross-ply laminate subject to uniaxial loading, this paper presents the results of comparisons between the model predictions of stress/strain behaviour, carried out in Part A of the Failure Exercise, and the corresponding experimental data provided after the predictions had been made. In addition, because a new model designed for general symmetric laminates has since been developed, comparisons are shown between the stress/strain predictions of the new model applied to quasi-isotropic and [ ± ]s laminates (with  = 45 and 55°), and the corresponding experimental data provided to participants in the failure exercise. Very good correlations between the model and experimental stress/ strain data have been achieved for both the cross-ply and quasi-isotropic laminates (subject to both uniaxial and biaxial loading), principally because ply cracking in a single orientation is expected to occur for a significant part of the stress/strain curves. The correlation for [ ± ]s laminates is not as good because the model is not able to account for ply crack formation in both the +  and – plies of the laminate. In addition, the model has not yet been modified to deal with mixed-mode ply crack growth, or with non-linear matrix behaviour which could lead to significant deviations between model predictions and experimental results at large strains. It is emphasised that the models used are not able by themselves to predict strength. Additional damage modes must be taken into account (e.g. fibre fracture) before predictions of strength can be made that are based on the modelling of physical micromechanisms associated with laminate failure. A rudimentary physically based approach to the prediction of laminate failure is made by considering fibre strain within the laminate in relation to the average failure strain of a single fibre. The ply cracking models that have been assessed in the paper, for cases where they are expected to be valid, have been shown to lead to good predictions of laminate stress/ strain behaviour that is consistent with data provided for the Failure Exercise. The reliability of the models is such that they have good potential for being used as the basis of new design methodology to deal with initial ply crack formation in laminates, and the effects of ply cracking on non-linear stress/ strain behaviour for parts of stress/strain curves where ply cracking occurs in a single orientation. Keywords: A. Polymer matrix composites; A. Structural materials; B. Matrix cracking; C. Anisotropy; C. Computational simulation

E-mail address: [email protected]

811 1. Introduction In a previous paper [1], that was a contribution to the Failure Exercise – Part A [2], a theoretical model was described that took full account of the thermal residual stresses that occur in the laminate as a result of thermal expansion mis-match effects between plies. The model predicted the local stress and deformation distributions within a [0/90]s laminate subject to uniaxial loading. The solutions of the model were used to predict the dependence of the effective thermo-elastic constants of a damaged laminate on ply crack density for the special case where the crack density in the 90° ply was uniform. These data were then used together with energy methods to predict the dependence of crack density on the value of the applied stress. It then followed that the dependence of the effective elastic constants on applied stress, and non-linear stress/strain behaviour associated with damage, could be predicted. The first objective of this paper is to comment on the correlation achieved between the results given in [1] predicting the stress/strain behaviour of a [0/90]s laminate, and a biaxially loaded [ ± 45]s laminate, using the theoretical model, and the experimental results provided to contributors to the Failure Exercise [2] after the theoretical predictions had been made. The second objective is to describe briefly further developments of the theoretical methodology that enables ply cracking in general symmetric laminates to be predicted provided that cracking occurs in some or all of the plies having one orientation. It is emphasised that the models used are not able by themselves to predict strength as additional damage modes must be taken into account (e.g. fibre fracture) before predictions of strength can be made. Some strength predictions are, however, made that are based on the modelling of two physical micro-mechanisms associated with laminate failure, namely ply cracking and fibre failure. 2. Discussion of correlation for a cross-ply laminate In [1] the theoretical model of ply crack formation in laminates was applied to a [0/90]s crossply laminate made of unidirectional plies comprising Silenka E-glass 1200tex fibres reinforcing a MY750/HY917/DY062 epoxy matrix. The properties of the unidirectional plies are given by: EA = 45.6 GPa ET = 16.2 GPa A = 5.83 GPa A = 8.6
10–6/°C vA = 0.278 vT = 0.4 T = 26.4
10–6/°C. The ply thickness of the GRP material is 0.26 mm. The laminate is loaded axially leading to the progressive formation of ply cracks in the 90° plies of the laminate. When applying the methodology described in [1] the difference T between the stress-free temperature of the laminate and the testing temperature was taken to be –120°C whereas the temperature difference should have been –100°C as specified in the instructions to participants. This arose because of a data input error when using software which demanded a temperature difference rather than the stress-free temperature that was provided. The effective fracture energy (2 ) at all potential crack formation sites has the same value of 165 J/m2 i.e. the standard deviation of the fracture energy distribution is zero. For this case the energetic method of predicting ply crack formation for the case of uniaxial loading led to a characteristic initial crack density of

812 0.522 cracks/mm occurring at an initial failure stress of 123 MPa. Subsequent axial loading of the laminate led to progressive instantaneous formation of additional cracks at the midpoints between those that formed initially. The progressive doubling of crack densities at well defined stresses is a necessary consequence of assuming a unique value for the effective fracture energy for ply cracking and it leads to small isolated step changes on the predicted stress/strain curves. The resulting stress/strain behaviour is shown in [1, Fig. 2] which is duplicated in Fig. 1. Also shown in Fig. 1 are the experimental data [3] corresponding to Fig. 23 of the Failure Exercise. As the stress-free temperature specified in [2] was given as 120°C, the results given in [1] would apply for a testing temperature of 0°C. As the experimental data are for a temperature of 20°C the temperature difference needed by the model has the value T = –100°C. For the

Fig. 1. Comparison of the axial stress versus axial and transverse strain curves predicted by model and experiment for a GRP [0/90]s cross-ply laminate subject to uniaxial loading, where the temperature difference T = 120°C and the fracture energy 2 = 165 J/m2.

813

Fig. 2. Comparison of the axial stress versus axial and transverse strain curves predicted by the crossply model for a GRP [0/90]s cross-ply laminate subject to uniaxial loading, where the temperature difference has the values T = –120°C and T = –100°C and the fracture energy 2 = 165 J/m2.

case of the uniaxial loading of the cross-ply laminate, predictions have been repeated using the correct value T = –100°C for the temperature difference. A comparison showing the effect of the temperature change is shown in Fig. 2 where both the axial and transverse strains are plotted as functions of axial stress for the values T = –100°C and T = –120°C. The change in behaviour is such that the stress levels at which new ply cracks progressively form are slightly increased when T changes from –120°C to –100°C. Figure 3 shows the comparison of the new predictions of stress/strain behaviour for T = –100°C with the experimental data corresponding to Fig. 23 in the Failure Exercise. The first comment that can be made about the data shown in Fig. 3 is that the onset of nonlinear stress/strain behaviour is well predicted by the theoretical model [1]. In addition, the subsequent experimental axial stress/strain behaviour shows a significant non-linearity that is also well predicted by the model. While the theoretical predictions show discrete ply cracking behaviour (arising from a unique value of the effective fracture energy) the experimental data

814

Fig. 3. Comparison of the axial stress versus axial and transverse strain curves predicted by model and experiment for a GRP [0/90]s cross-ply laminate subject to uniaxial loading, where the temperature difference T = –100°C and the fracture energy 2 = 165 J/m2.

show a much more gradual development of non-linear behaviour. Such behaviour can be predicted by the model (see next section) if the effective fracture energies allocated to the potential cracking sites in the 90° ply are regarded as being statistically distributed in order to account for material variability arising from factors such as variable fibre/matrix interface strength, and non-uniform fibre separations. In such a case each potential crack formation site will have a different effective fracture energy leading to ply cracks that form non-uniformly one at a time, rather than the occurrence of progressive doubling of uniform crack densities. It is worth noting that the gradients of the second, third and fourth parts of the predicted stress/ strain curve are higher than those measured experimentally. This is a direct consequence of using a unique value for the fracture energy. If the fracture energy were to be statistically

815 distributed then the step changes on the stress/strain curve would be less pronounced and the gradient of the predicted stress/strain curve would be expected to follow more closely the experimental results. As the experimental data extend beyond the theoretical predictions [1] given in Part A of the exercise, the predictions have been extended as shown in Fig. 3. This is permissible as the ply cracking model does not predict failure of the laminate. The interval between successive steps in the stress/strain curves for cross-ply laminates is determined by the operation of the procedure given in section 9 of reference [1]. A similar procedure is applied when using general symmetric laminates (see Ref. [4]). The model can in fact be applied for progressively increasing applied stresses without failure, leading to ply crack saturation. Saturation defined by this method arises because the successive increases in applied stress needed to generate new cracks becomes excessively large. There are, however, other reasons for ply crack saturation, as discussed in [5] where the development of compressive loading on the midplane of the laminate arises as the load and crack density increases. Reference [5] also includes a great deal of validation data for the cross-ply models that have been used in the Exercise. In practice, other damage modes occur such as ply crack formation in the 0° plies, delamination and fibre fracture. These effects are not included in the model and will lead to reductions in laminate stiffness (see sections 8 and 9 for a description of how the modelling is being extended to deal with some of these additional damage modes). It can be seen from Fig. 3 that when the applied axial strain exceeds 2.3% the experimental data fall below the theoretical prediction and exhibit a lower gradient. This could be indicating that some other form of damage is occurring, probably delamination initiated at some or all of the ply cracks, and some fibre fracture. These damage modes cannot yet be accounted for when using the theoretical model. It is observed that at low axial strains before any ply cracks are expected to have formed, the experimental data lie above the theoretical predictions but they exhibit the same initial modulus. This behaviour is unexpected and seems to arise from the fact that the experimental data do not extrapolate to the origin. This suggests that the experimental data should be modified to ensure their consistency when the stress and strain are nearing zero value. The correlation between model predictions for the axial stress versus the in-plane transverse strain and the experimental results is shown in Fig. 3. Predictions are seen to differ increasingly as the axial stress is increased beyond the value of 0.25 GPa. The model overpredicts the magnitude of the transverse strain. This small discrepancy is very likely to be reduced if the effects of ply cracking in the 0°-plies, fibre fracture and delamination between plies were to be included in the model.

3. Effect of statistically distributed fracture energies for a cross-ply laminate When carrying out the initial predictions given in [1], a unique value of 165 J/m2 was assumed for the effective fracture energy (2 ) for ply cracking in the 90° plies of the laminate. The reason for this assumption was based on the fact that no information was available for the degree of statistical scatter to be expected. It is, therefore, useful to assess here this aspect by carrying out additional predictions of behaviour where statistical scatter is introduced into the model using nominal estimates of scatter in order to illustrate the effect. It is assumed that the

816 temperature difference T = –100°C and effective fracture energies for the potential ply cracking sites are allocated at random from a normal distribution such that the mean effective fracture energy is 165 J/m2 (corresponding to the fracture energy assumed in [1]). The standard deviations for the fracture energy distribution are taken as 5%, 10% and 15% of the mean value. The method of dealing with non-uniform ply cracking, arising from introducing statistical scatter in the fracture energies for ply crack formation, is described in [7]. The results of introducing statistical scatter into the model are shown in Fig. 4a where predicted ply crack densities (up to a nominal 4 cracks/mm) are plotted as functions of the axial strain. The effect of increasing the standard deviation of the fracture energy leads to a lowering of the stress at which a ply crack first forms. Figure 4b, and the enlargement at low strains shown in Fig. 4c show the corresponding predicted stress/strain behaviour compared to the experimental data. It is clear that the introduction of statistical variability of fracture energy leads to progressive damage formation where ply cracks form one at a time rather than progressively doubling their density. While steps are clearly seen in the results of Fig. 4a, there are much smaller steps in the corresponding stress/strain predictions shown in Fig. 4b. It should be emphasised here that only a single prediction has been made for each non-zero standard deviation selected. If predictions were repeated using the same standard deviation then different results would be obtained arising from the fact that a different set of fracture energies would be allocated to the potential ply crack formation sites in the laminate. It is not possible to ascribe realistically a value for the standard deviation on the basis of observed inhomogeneity of the fibre distribution, interface defects and quality etc. A better way to deal with the problem of determining a suitable standard deviation is to apply the model to a simple cross-ply laminate (e.g. [0/902]s) for a given material, and then to choose the mean fracture energy for ply cracking and its standard deviation so that the model reproduces the observed stress/strain behaviour (and/or observed crack density-stress behaviour if that can easily be measured). The values obtained by this method can then be used to predict the behaviour of ply cracking in other types of cross-ply laminate and more general symmetric laminates which are made of the same material. The predictions shown in Figs 5–10, to be given later, are based on a zero value for the standard deviation, i.e. statistical scatter is ignored.

4. Discussion of correlation for a quasi-isotropic laminate In [1] the theoretical model of ply crack formation could not be applied to a quasi-isotropic laminate as a suitable model was not available at the time Part A of the exercise was carried out. As a model and associated software for general symmetric laminates subject to general in-plane loading [4,6–8] now exist (including the introduction of statistical scatter in the fracture energies for ply cracking), it is appropriate to apply this model to the quasi-isotropic laminate comprising unidirectional plies made of AS4 carbon fibres reinforcing 3501-6 epoxy matrix. The model for general symmetric laminates is based on exactly the same methodology that was used for cross-ply laminates [1]. The problem becomes more complex, however, in several different ways. Shear coupling effects can occur in balanced general symmetric laminates introducing additional thermo-elastic constants that are affected by the damage. Inplane shear loading must be considered in addition to biaxial in-plane loading. Because of this two different types of stress transfer function need to be considered which are governed by

817

Fig. 4. (a) Model predictions, (b) axial stress axial strain predictions and (c) enlarged view of axial stress versus axial strain model predictions for three non-zero standard deviations (S.D.) for a GRP [0/90]s cross-ply laminate subject to uniaxial loading, where T = –100°C and the mean fracture energy 2 = 165 J/m2.

two sets of coupled ordinary differential equations. One set is of fourth order and is associated with biaxial in-plane loading while the other is second order and is associated with in-plane shear loading. For the case of cross-ply laminates the two sets of differential equations do not in fact interact so that in-plane biaxial and shear loading problems can be solved separately.

818 The properties of the CFRP unidirectional plies are given [2] by: EA = 126 GPa A = 6.6 GPa vA = 0.28 T = 26
10–6/°C.

ET = 11 GPa A = –1
10–6/°C vT = 0.4

Consider now a symmetric [0°/ ± 45°/90°]s laminate made of this CFRP material where the individual ply thickness is 0.1375 mm. The laminate is first loaded uniaxially in a way that is expected to lead to the progressive formation of ply cracks in the central plies of the laminate whose fibres are perpendicular to the loading direction. When carrying out the stress analysis using the methods described in [4], accurate predictions for the through-thickness dependence of the stress and displacement distributions in each ply can be achieved by using a ply refinement technique where each ply is subdivided into five elements (sub-plies) of equal

Fig. 5. Comparison of the axial stress versus axial and transverse strain curves predicted by model and experiment for a CFRP quasi-isotropic laminate subject to uniaxial loading, where the temperature difference T = –157°C and the fracture energy 2 = 220 J/m2.

819

Fig. 6. Comparison of the axial stress versus axial and transverse strain curves predicted by model and experiment for a CFRP quasi-isotropic laminate subject to a stress ratio of 1 : 0.05, where the temperature difference T = –157°C and the fracture energy 2 = 220 J/m2.

thickness. The elements adjacent to the interfaces between the plies are then successively subdivided three times. Such a ply refinement technique enables the singular behaviour near ply crack tips to be adequately modelled. When applying the methodology described in [4,6 8] the difference T between the stress-free temperature (177°C) and the testing temperature (assumed to be 20°C) is taken as –157°C. The effective fracture energy (2 ) at all potential crack formation sites has the same value of 220 J/m2 i.e. the standard deviation of the fracture energy distribution is zero. In Section 2 of this paper, for the case of a zero standard deviation of the fracture energies for ply cracking, the approach was to determine accurately the value of the applied stress for which an array of uniformly spaced ply cracks first form, and the stress values at which the crack densities progressively doubled. An alternative approach is to set up the ply cracking simulation for the case of a non-zero standard deviation of fracture energies and to apply the simulation for the special case when the standard deviation is set to zero. The latter approach

820

Fig. 7. Comparison of the axial stress versus axial and transverse strain curves predicted by model and experiment for a CFRP quasi-isotropic laminate subject to a stress ratio of 1 : 0.5, where the temperature difference T = –157°C and the fracture energy 2 = 220 J/m2.

has been taken when applying the model [4,6–8] to quasi-isotropic and [ ± ]s laminates, and it is expected to lead to slightly different results arising from the fact that the selected laminate length for the simulation (selected to be 10 mm) and assumed density of potential ply cracking sites (100 sites/mm) may not be consistent precisely with the initial crack density demanded by the energy balance equation that governs ply crack formation. The relatively short length of laminate selected for the simulation was chosen to minimise this difference, on the assumption that the maximum number of potential cracking sites is not greater than 1000. The laminate length selected for the simulation is expected to have a slight effect on predictions for the case when the fracture energy is statistically distributed, i.e. there will be a small size effect. The length selected is not, however, regarded as being a parameter that might be dependent on the material being investigated. For the case of uniaxial loading of the laminate, Fig. 5 shows the predicted stress/strain curves compared with the experimental results [3] corresponding to Fig. 19 of the Failure

821

Fig. 8. Comparison of the axial stress versus axial and transverse strain curves predicted by model and experiment for a GRP [–45/45]s cross-ply laminate subject to biaxial loading, where the temperature difference T = –100°C and the fracture energy 2 = 165 J/m2.

Exercise. It is emphasised that the ply cracking model alone does not predict failure of the laminate. Additional considerations are required (see discussion in section 8). The predicted stress/strain curves in the figures have been terminated at a strain that is larger than the largest experimentally measured strain. It is seen from Fig. 5 that the axial stress versus axial strain behaviour predicted the model agrees very well with the experimental data obtained from tests on tubes, while it is also seen that the model slightly over-predicts the magnitude of the transverse strain and exhibits a slightly different gradient. The information [3] provided for the Failure Exercise suggests that the uniaxial experimental data correspond to a very low degree of biaxial loading on the tube such that the hoop stress is 20 times the axial stress applied to the tube. Figure 6 shows the results of the comparison between model predictions and the experimental data corresponding to Fig. 19 of the Failure Exercise for the case when the stress ratio is 1:0.05. Again it is emphasised that the model is not directly predicting failure

822

Fig. 9. Comparison of the axial stress versus axial and transverse strain curves predicted by model and experiment for a GRP 155° laminate subject to a stress ratio of 1 : 0, where the temperature difference is given by T = –100°C and the fracture energy 2 = 165 J/m2.

of the laminate and the predicted stress/strain curves in the figures have been terminated at a strain that is larger than the largest experimentally measured strains (but see discussion in section 8). It is now seen that both the axial and transverse behaviour are very well predicted by the model. From Fig. 6 the model under-predicts the transverse strain at transverse strains whose magnitudes exceed 0.15%, perhaps because of the occurrence of ply cracking in the 45° plies, delamination and/or fibre fracture phenomena that are not being modelled. Figure 7 shows the corresponding comparisons with the experimental data of Fig. 20 of the Failure Exercise for the case when the tubes are subjected to biaxial loading where the hoop stress is twice that applied along the axis of the tube. From Fig. 7 the model is seen to underpredict the axial strain when the axial strain exceeds 0.8%, again probably because of the occurrence of damage modes that are not being modelled. Figure 7 shows that the axial stress vs transverse strain response is well predicted by the model for all values of the transverse strain. It is again emphasised that the model is not predicting failure of the laminate and the

823

Fig. 10. Comparison of the axial stress versus axial and transverse strain curves predicted by model and experiment for a GRP ± 55° laminate subject to a stress ratio of 1 : 0.5, where the temperature difference is given by T = –100°C.

predicted stress/strain curves in the figures have been terminated at a strain that is larger than the largest experimentally measured strains. 5. Discussion of correlation for a [ ± 45]s laminate For the [ ± 45]s laminates under equi-biaxial loading, previous predictions used a predictive model for a cross-ply laminate that did not allow the fracture energy for ply cracking to be statistically distributed. Now that the model has been extended to deal with statistically distributed fracture energies, and with general symmetric laminates, it is now possible to analyse the [ ± 45]s using the new model. Figure 8 compares new predictions with the experimental results provided by the organisers of the Failure Exercise. When predictions

824 were made in [1], as the model was able only to take account of ply cracking in a single orientation, two cases were considered. For the first case the ply cracks were assumed to form only in the inner plies of the laminate, while for the second case the plies formed only in the outer plies (but not in a staggered fashion usually observed in experiments). For the predictions given in [1] and in Fig. 8 the axial and transverse (hoop) strains have identical values so that only two curves can be drawn in Fig. 8, whereas the corresponding experimental data show differences between the axial and hoop strains. The initial gradient is not particularly well predicted, contrasting sharply with the good predictions obtained with the identical model for the crossply laminate (Fig. 3) and the quasi-isotropic laminate considered (Figs 6 and 7). The applied strain for initial ply cracking in the inner plies is predicted by the new model (with a zero standard deviation for the fracture energy distribution) to be 0.317% corresponding to an applied stress of 115.6 MPa. The applied strain for initial ply cracking in the outer plies (non-staggered) is predicted to be 0.291% corresponding to an applied stress of 105.9 MPa. In [1] the corresponding results differ slightly from those above. This difference arises because the current model, that can include statistical scatter in the fracture energy for ply cracking, is subject to a small error when attempting to use a zero standard deviation for fracture energy, as already discussed in Section 4. As the model does not consider ply cracking in more than one orientation the loss of stiffness during loading is much less than observed in the experiments where orthogonal cracking will occur. The inclusion of orthogonal cracking in the model is expected to close some of the gap between model predictions and experiment. Such a model modification would be necessary before predictions of leakage can be made. The inclusion of delamination from ply cracks and fibre fracture may also be required. 6. Discussion of correlation for a [ ± ␪]s laminate The general symmetric laminate model [4,6–8] can be used to consider ply crack formation in [ ± ]s laminates subject to both axial and biaxial loading. However, the model allows cracking only in a single orientation and assumes that ply crack growth occurs in mode I defined by fracture mechanics. The loading cases for [ ± ]s laminates described in [3] are such that one would expect ply cracking to occur in both the +  and – plies and that such cracking would be mixed mode; situations that are not accounted for by the model. It is nevertheless useful to apply the model to a [ ± ]s laminate and to compare its predictions, when ply cracking is restricted to the inner –55° plies, with the experimental data provided as part of the Failure Exercise [3]. The ply properties given in Section 2 are assumed for the predictions, and the ply thickness is taken as 0.25 mm. Again, in order to achieve accurate predictions for the through-thickness dependence of the stress and displacement distributions in each ply, the model uses a ply refinement technique where each ply is subdivided into five elements (sub-plies) of equal thickness. The elements adjacent to the interfaces between the plies are then successively subdivided three times enabling the singular behaviour near ply crack tips to be adequately modelled. When applying the methodology described in [4,6–8] it is assumed that the difference T between the stress-free temperature and the testing temperature is –100 °C. The effective

825 fracture energy (2 ) at all potential crack formation sites has the value of 165 J/m2 i.e. the standard deviation of the fracture energy distribution is zero. The ply cracking simulation for the case of a non-zero standard deviation of fracture energies is again applied for the special case when the standard deviation is set to zero. The laminate length for the simulation was again selected to be 10 mm and the density of potential ply cracking sites was taken to be 100 sites/mm. For the case of the uniaxial loading of the laminate, Fig. 9 shows the predicted stress/strain curves compared to the experimental results [3] corresponding to Fig. 16 of the Failure Exercise. It is emphasised that the model is not predicting failure of the laminate and the predicted stress/strain curves in the figures have been terminated at a strain that is larger than the largest experimentally measured strains. It is seen from Fig. 9 that the axial stress versus axial strain behaviour predicted by the model agrees well with the experimental data obtained from tests on tubes only for very low strains before the occurrence of any damage formation or non-linear matrix behaviour. Figure 10 shows the corresponding comparisons with the experimental data of Fig. 17 of the Failure Exercise for the case when the tubes are subjected to biaxial loading where the hoop stress is twice that applied along the axis of the tube. From Fig. 10 the model is seen to slightly under-predict the axial strain. Figure 10 shows that the transverse strain is under-predicted by the model for all values of the transverse strain, the deviation becoming very marked at strains above 0.5%. The predictions shown in Figs 9 and 10 do not agree with the experimental data for at least three important reasons. As already mentioned above, the model currently allows mode I ply crack formation only in a single orientation while the laminate lay-up is such that ply cracking in both the +  and – plies can be expected and will be mixed mode in nature. The consideration of cracking in both plies would lead to better correlations between the model and the experimental data, and could lead to a method for considering failure due to leakage from pipes at locations where the paths of ply cracks in adjacent plies cross one another. The occurrence of ply cracks in adjacent plies can lead to localised delaminations offering a route for fluid escape from the interior of the pipe to the exterior surface using both open ply cracks and the local delaminations. It also should be emphasised that, in contrast to the data shown in Figs 1–8, the maximum strains encountered are much larger leading to the possibility that non-linear matrix properties need to be taken into account.

7. Prediction of effective thermo-elastic constants While the comparisons given in this paper of model predictions with experimental data are restricted within the Failure Exercise to stress/strain curves, the cross-ply and general symmetric laminate models have been used to predict all the in-plane thermo-elastic constants of a laminate as a function of both crack density and of the applied axial stress or strain. Thus the models can be used to predict laminate behaviour for any combination of in-plane biaxial and in-plane shear loading provided ply cracking is for mode I and occurs in a single orientation. For the case of non-symmetrical multiple-ply cross-ply laminates, another model, valid for plane strain conditions, has been developed [9,10] that considers the effects of ply cracks on the effective thermo-elastic constants of a laminate that is subject to combined uniaxial in-plane loading, out-of-plane bending and thermal residual stresses.

826 8. Approximate predictions of laminate failure The models of ply cracking that have been used to predict the stress/strain curves presented in Figs 1–10 do not predict when the laminates will fail. The models predict continuing formation of ply cracks which becomes progressively harder to achieve as the loading is increased. A simple approach to the prediction of failure is to assume that laminate failures occur when the fibres begin to fail. When a fibre is tested uniaxially in isolation it is possible to define both a failure stress and a failure strain; quantities that are related through the fibre modulus. This approach ignores the effects of fibre defects on fibre strength and the resulting statistical variability in fibre strength properties. When the failure of a fibre is considered within a composite, the fibre will not be in a state of uniaxial stress or strain. A choice has to be made between imposing a critical axial failure strain or a critical axial failure stress. It is thought (on the basis of considering an atomistic model for a fibre) that the better selection is to assume that a fibre in a composite fails when its axial strain reaches the axial failure strain for a fibre tested uniaxially in isolation, which is independent of the degree of multiaxial loading that the fibre experiences within the composite. When applying this fibre failure criterion within a composite, the relevant fibre strain is that measured in the fibre relative to the stress-free state of the UD ply or laminate, not that measured relative to the initial loading of a laminate during testing in the laboratory. Because of thermal residual stresses in the laminate, and within each ply at the fibre/matrix level, the axial strain in a fibre will not be zero when the axial strain applied to the laminate is zero. The axial failure strain for a laminate is defined to be that which occurs when the maximum axial strain in any fibre within the laminate exceeds the critical fibre strain. The corresponding predicted axial failure stresses for the laminates are found from the stress/strain curves predicted by the ply cracking models. When imposing the critical fibre strain criterion for a laminate in which ply cracking has occurred, the predicted effective strains for the damaged laminate are assumed to be experienced by the fibres in each ply. By resolving the effective strain field (taken to be uniform) in each fibre direction of the laminate, it is possible to determine when the fibre failure criterion is satisfied during the loading of the laminate and during the progressive ply crack formation. This approach neglects the effects of fibre axial strains distributions which are nonuniform along their lengths due to ply cracking in other plies, i.e. stress concentration effects are neglected as the average axial strain in a fibre is compared to the axial fibre failure strain. The failure strain quoted [2] for the Silenka E-glass fibres is 2.905% while that for the AS4 carbon fibres is 1.488%. The results of failure predictions based on these fibre failure strains are given in Tables 1 and 2 where both laminate failure strains and failure stresses are given. For each laminate and failure parameter, three different values are given. The first values, given under the label “Experimental”, are the results provided by the organisers of the Exercise. The value quoted is always the last value provided. For the case of the [ ± 55]s laminate subject to a stress ratio of 1:0.5 the strain value and stress value do not correspond on the stress/strain curve. The value of stress corresponding to the last strain reading is 661.3 MPa which is close to the last value 668.3 MPa given in Table 2. The second values are the predictions made by applying the fibre failure strain criterion in conjunction with the ply cracking model. For this case the predictions of strains in fibres are based on resolving the effective in-plane strain field of the cracked laminate in the fibre

Table 1 Failure predictions based on criterion where laminates fail when fibre strain in 0° plies reaches the fibre failure strain (results shown in Figs. 3, 6, 7 and 8).

Laminate failure strain (%)

Failure stress (MPa)

Cross-ply GRP uniaxial loading

Quasi-isotropic CFRP uniaxial loading

Quasi-isotropic CFRP stress ratio 1 : 0.05

Quasi-isotropic CFRP stress ratio = 1 : 0.5

Experiment 2.691 Ply cracking 3.078 Undamaged 3.043 Experiment 609.1 Ply cracking 726 Undamaged 946.3

Experiment 1.455 Ply cracking 1.533 Undamaged 1.514 Experiment 718 Ply cracking 749 Undamaged 773.1

Experiment 1.455 Ply cracking 1.546 Undamaged 1.514 Experiment 718 Ply cracking 766 Undamaged 784.6

Experiment 1.561 Ply cracking 1.555 Undamaged 1.514 Experiment 808 3 Ply cracking 882 Undamaged 905

Table 2 Failure predictions based on criterion where laminates fail when fibre strain in any ply reaches the fibre failure strain (results shown in Figs. 4, 9 and 10).

Laminate failure strain (%)

Failure stress (MPa)

[ ± 45]s GRP (outer ply cracked) biaxial loading

[ ± 55]s GRP (inner ply cracked) uniaxial loading

[ ± 55]s GRP (inner ply cracked) stress ratio = 1 : 0.5

Experiment 2.475 Ply cracking 2.658 Undamaged 3.043 Experiment 444.4 Ply cracking 778 Undamaged 1108

Experiment 2.475 Ply cracking 2.673 Undamaged 3.043 Experiment 444.4 Ply cracking 782 Undamaged 1108

Experiment 8.780 Ply cracking 6.739 Undamaged 6.317 Experiment 594.9 Ply cracking 1083 Undamaged 1482

Experiment 4.220 Ply cracking 3.499 Undamaged 4.017 Experiment 668.3 Ply cracking 995 Undamaged 1327

827

[ ± 45]s GRP (inner ply cracked) biaxial loading

828 directions. If one applies the model using a zero standard deviation for the fracture energy then the points predicted on the stress/strain curve can be widely apart and the prediction of failure is then difficult. To overcome this difficulty the stresses and strains for failure were predicted using a stress strain curve based on a 5% standard deviation for the fracture energy. As a result the predicted stresses and strains at failure are subject to statistical scatter. The results given in Tables 1 and 2 under the label “Ply cracking” are averages of five repeated predictions. The third values shown in Tables 1 and 2 are predictions made by ignoring the effects of ply cracking. The strain field for an undamaged ply is calculated and then resolved in the fibre directions in order to determine failure using the critical fibre strain criterion. Such results are given in Tables 1 and 2 under the label “Undamaged.” Table 1 provides results for those laminates that are expected to be valid for use with the predictive models of ply cracking. The laminates are such that ply cracking in a single orientation is expected for a good proportion of the stress/strain curve, and that such ply cracking will take place in mode I conditions of fracture. Also, the maximum strains in tests will be small enough for non-linear matrix effects to be negligible. The critical strains for fibre failure and the corresponding axial failure stresses are also shown in Figs 3, 6, 7 and 8. These figures show how the laminate strength predictions compare with the experimental results. It is seen that the physically based rudimentary approach to predicting failure in these laminates give very good results for the quasiisotropic carbon fibre laminates subject to uniaxial and biaxial loading, and a reasonable result for the uniaxial loading of the glass fibre-epoxy cross-ply laminate. The results for the GRP cross-ply laminate show that it is important to account for the effects of ply cracking on stress/strain curves when imposing the strain based fibre failure criterion. For the CFRP quasi-isotropic laminate the non-linearity of stress/strain behaviour is minimal indicating that ply cracking damage has very little effect on the prediction of the failure stress when using the critical fibre strain criterion for fibre failure. Table 2 provides results for those laminates that are NOT expected to be valid for use with the predictive models of ply cracking. The laminates are such that ply cracking in more than one orientation is expected for a good proportion of the stress/strain curve, and that such ply cracking will take place in mixed mode conditions of fracture. Also, the maximum strains in some cases will be large enough for non-linear matrix effects to be important. The critical strains for fibre failure and the corresponding axial failure stresses are also shown in Figs 4, 9 and 10. These figures show how the laminate strength predictions compare with the experimental results. As expected, it is seen that the physically based rudimentary approach to predicting failure in these laminates is not giving realistic predictions of laminate behaviour as the models are not yet sufficiently general to deal with all types of damage that occur during loading. The failure strains are, however, reasonably predicted. The results for these laminates show that it will be very important to account for all the effects of ply cracking and delamination when imposing the strain based fibre failure criterion and predicting laminate strengths. 9. Future developments The models used in this paper are continually being developed so that current limitations can be removed. The first limitation of current models is that ply crack formation has been

829 restricted to a fixed orientation. Work is planned that will overcome this problem by adopting a ply homogenisation technique. The models currently are able to predict all the effective thermoelastic constants of a laminate having cracks in one orientation. The approach is to determine the effective thermoelastic constants of the cracked plies so that, when used in undamaged laminate models, the predicted thermoelastic constants for the laminate correspond exactly to those predicted by the model for a damaged laminate. Having carried out this homogenisation at the ply level, the model can then be used to investigate the effects of ply crack formation in any other ply having a different orientation. By repeating this procedure ply crack formation in any number of orientations can be considered. The homogenisation procedure will enable energy methods to be used to predict the progressive formation of ply cracks in more than one orientation, although the simulation method will become complex as the location of the next ply crack to form during loading has to be determined by investigating all possible occurrences and selecting the one that can take place at the lowest load. The second current limitation of the models is that ply crack formation is assumed to occur in mode I. This assumption is valid for ply crack formation in cross-ply laminates provided that the laminate is not subjected to in-plane shear deformation. It is also valid for general symmetric laminates where cracks form in the 90° plies provided that the laminate is balanced, not subject to in-plane shear deformation, and that for every ° ply in the laminate there is a corresponding –° ply of the same thickness. For this case the shear coupling ratios of an undamaged laminate will be zero and when the laminate is damaged with cracks only in the 90° plies the shear coupling ratios will be very small. This will not usually be the case for general symmetric laminates subject to general in-plane loading as ply crack formation will occur under combined mode I and mode II conditions. While the model currently will correctly predict the energy release rate for the growth of such cracks, the difficulty of implementing the results arises because there is no information available that indicates how the fracture energy for ply cracking should depend on the degree of mixed mode loading. The third limitation concerns the prediction of the strength of multiple-ply cross-ply laminates. While the simple deterministic approach is to assume that the laminate fails when the fibres reach their mean strength, modelling is currently being developed that takes account of the statistical variability of fibre strengths in the 0° plies of the laminate by adopting a Monte Carlo approach. Failure of the laminate is regarded as occurring when the Monte Carlo simulation of the 0° plies predicts that they have failed catastrophically. Difficulties that are met concern the fact that the stress applied to the 0° plies is biaxial, and that some form of local load sharing rules need to be imposed when fibres fail. Also the loss of stiffness of the 0° plies resulting from fibre failures needs to be taken into account when carrying out simulations of ply crack formation in the 90° plies. This approach could be applied also to general symmetric laminates, where an additional difficulty is that the Monte Carlo modelling of fibre failures in a ply would need to account for the effects of shear loading if the principal loading direction did not correspond to the fibre direction of the failing ply. 10. Conclusions The following conclusions can be drawn from the results presented in this paper: (1) The predictions of ply cracking using the cross-ply laminate and general symmetric laminate models leads to very good correlations of the experimental cross-ply and quasi-

830 isotropic laminate stress/strain data given as part of the Failure Exercise, provided that ply cracking occurs only in a single orientation and that such cracking occurs in mode I. The correlations with the stress/strain data for the biaxially loaded [ ± 45]s and the [ ± ]s laminates is less good because ply cracking is expected to occur in two directions. For the [ ± ]s laminate ply cracking will be a mixed mode, and non-linear matrix behaviour is expected to be important at large strain values. (2) For the predictions of ply crack formation in cross-ply and quasi-isotropic laminates, good correlations with the experimental data are achieved when it is assumed that the fracture energy for each potential crack formation site has a single value (i.e. the standard deviation of the fracture energy distribution can be taken as zero). This assumption leads to uniform ply crack distributions that progressively double in density during loading. In practice ply cracks usually form one at a time and this can be achieved by introducing ply crack fracture energies that are statistically distributed leading to the need to specify an additional parameter, namely, the standard deviation of the fracture energy distribution. (3) The results for cross-ply laminates presented here, when taken in conjunction with other validation data obtained by experiment and using FEA [5] to check model predictions, are such that the models for the prediction of ply cracking are robust, and can be relied upon provided that they are used in situations where the damage development corresponds to that in the laminates being modelled. (4) The current major limitation of the models is that they can account reliably for ply cracking only when ply crack propagation is mode I and the ply cracks form in a single orientation. The models will, therefore, be of greatest value when predicting the first occurrence of ply cracking, and predicting stress/strain behaviour during the early stages of laminate loading in the non-linear damaged region of the stress/strain curve. (5) The models considered provide a complete description of the thermo-elastic behaviour of symmetric laminates subject to general in-plane loading and thermal residual stresses. (6) The rudimentary failure criterion where the laminate fails when the maximum average fibre strain in the laminate reaches the fibre failure strain is shown to give very good results for the quasi-isotropic carbon fibre laminates subject to uniaxial and biaxial loading, and a reasonable result for the uniaxial loading of the glass fibre-epoxy cross-ply laminate. The fibre failure criterion also leads to reasonable predictions for the failure strains of the other laminates, but the corresponding predictions of applied failure stress are far too high because all damage modes and mixed mode effects have not been taken into account. (7) The models are a very good basis for the development of new design methodology to deal with the occurrence and effects of mode I ply crack formation during loading which might be complex. Acknowledgements The author would like to acknowledge the contributions to this paper arising from thorough critical reviews of the paper by the organisers of the Failure Prediction Exercise. References [1] McCartney LN. Predicting transverse crack formation in crossply laminates Comp Sci Technol 1998;58:1069–81 [contribution to Part A of the Failure Exercise].

831 [2] Soden PD, Hinton MJ, Kaddour AS. Lamina properties, lay-up configurations and loading conditions for a range of fibre reinforced composite laminates. Comp Sci Technol 1998;58:1011–22. [3] Soden PD, Hinton MJ, Kaddour AS. Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates: Failure exercise benchmark data: V62, pp. 1489–1514, 2002, see also Chapter 2.2 in this book. [4] McCartney LN Model to predict effects of triaxial loading on ply cracking in general symmetric laminates Comp Sci Tech 2000;60:2255–79. [5] McCartney LN, Schoeppner GA, Becker W Validation of models for transverse ply cracks in composite laminates. Comp Sci Tech 2000;60:2347–59. [6] McCartney LN. Stress transfer mechanics for ply cracks in general symmetric laminates. NPL Report CMMT(A)SO, December 1996. [7] McCartney LN Framework for the prediction of ply cracking in any symmetric laminate subject to general in-plane loading. NPL Report CMMT(A)51, December 1996. [8] McCartney LN. An effective stress controlling progressive damage formation in laminates subject to triaxial loading. In Proc Deformation and Fracture Conference, 18 19 March 1999, IoM Communications, London, pp. 23–32. [9] McCartney LN. Stress transfer mechanics for multiple ply laminates subject to bending. NPL Report CMMT(A)55, February 1997. [10] McCartney LN, Pierse C Stress transfer mechanics for multiple ply laminates for axial loading and bending In: Proc. 11th Int. Conf. on Composite Materials, Gold Coast, Australia, July 1997, pp. 14–18.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

832

CHAPTER 5.6

Failure analysis of FRP laminates by means of physically based phenomenological models A. Pucka,* and H. Schürmannb,** b

a 33 Am Ahlberg, 34376 Immenhausen, Germany TUD-Darmstadt University of Technology, Institute of Lightweight Design and Construction, 30 Petersenstrasse, 64287 Darmstadt, Germany

Abstract The successive failure of different laminates subjected to a variety of loading conditions has been treated by a layer-by-layer failure analysis. Three sources of non-linearity are considered, namely material non-linearity due to microdamage, matrix cracking, and changes in fibre angle with increasing strains. In general there is good agreement between predictions and experimental results. Some disagreement is due to imperfections in certain tests, therefore, these tests should be repeated. Three categories of laminate configuration/loading condition can be distinguished: (I) laminates with 3 or more fibre directions with arbitrary loading conditions; (II) balanced angle ply laminates with stress ratios in accordance with netting analysis; (III) laminates with 2 fibre directions and loadings which are not in accordance with netting analysis. The analysis of category (I) is straight forward. Category (II) is sensitive to the stiffness degradation after the onset of matrix cracking. Category (III) fails at low stresses and large strains of the laminate due to a rapid deterioration. An intensive discussion is necessary to define a failure limit for category (III). For the detection of the different modes (A, B, C) of interfibre fracture (IFF), refined action plane related IFF-criteria developed by Puck on the basis of Mohr’s and Hashin’s considerations on brittle fracture are used. They provide much more information than has been reported from the experiments. Their unique ability to predict the inclination of the fibre parallel fracture plane is the key for assessing of the risk of delamination and local buckling due to a wedge effect which occurs when oblique fracture planes are exposed to high transverse compression. Keywords: B. Non-linear stress/strain behaviour; C. Failure criterion; Laminate failure

1. Introduction A “World-wide Failure Exercise”, Ref. [1], was launched in Britain to provide an up-to-date understanding of the prediction of failure in fibre reinforced plastic materials. In Part A of the * Corresponding author. Professor (emerit.) Dr.-Ing. Alfred Puck, 33 Am Ahlberg, 34376 Immenhausen. Tel. + Fax ( + + 49) 5673-3517. ** Corresponding author. E-mail address: helmut [email protected] (H. Schurmann).

833 Exercise, world-wide experts were invited to participate and requested to write a paper detailing their own theory and its application for the prediction of failure of 14 test cases. Puck and Schurmann [2] presented their contribution to Part A. The organisers of the exercise, Soden, Kaddour and Hinton, provided details of experimental results in [3] for the 14 test cases. They superimposed the theoretical predictions published in Part A on the experimental results and sent the graphs to the participants. The organisers requested the participants to write a short paper describing the correlation between predicted and measured data and any further refinement of the theory. All these papers will constitute Part B of the failure exercise. The present paper represents the authors contribution to Part B. It contains graphs showing the theoretical predictions for failure envelopes and stress/strain diagrams of the 14 test cases. A brief summary of the theoretical consideration of Part A paper [2] is first made in Sections 2 and 3. The original work of the present paper is based upon the investigations reported in [4–8]. Correlation between the theoretical prediction and experimental results for each of the test cases is presented in Sections 4 and 5. The authors believe that there are fundamental problems associated with the analysis of laminates subjected to “unreasonable loading conditions.” These include ± 55° laminates with stress ratios y : x ~ 2:1 and the ± 45° laminate under a stress ratio y : x = 1 : –1. These problems are discussed in Section 4.3. Finally, some conclusions are drawn in Sections 6 and 7 showing possible directions for future research and development in the field of failure analysis of FRP laminates and design assisting computer programs. The experimental results provided by the organisers contained little information about IFFevents, so a comparison of our theoretical predictions and experimental results remains incomplete in this context. Recently in Germany, measurements of IFF fracture angles and IFF failure stresses have been performed [9–11], which can be compared to theoretical predictions. These results which were made available after the launch of the Failure Exercise are very encouraging, because they agree very well with our IFF theory based on the assumption of brittle fracture. However, there is generally a good agreement between our theoretical results and the experimental results, supplied by the organisers, for the prediction of the biaxial failure envelopes and in a number of test cases an excellent prediction was obtained for stress/strain curves. One of the cases where large discrepancies were observed is that of the stress/strain curves for the ± 55°-tube tested under closed end internal pressure. Attempts are made to explain the observed discrepancy of the stress/strain curves for this laminate.

2. Highlights of the stress/strain analysis The original work on the phenomenological model of predicting the stress/strain curves behaviour and failure analysis was developed more than thirty years ago [4,5]. Since then, some remarkable improvements have been accomplished, especially regarding the analysis of what we call inter fibre fracture (IFF) [6]. Renewed thorough investigation has shown that IFFs are not always as harmless as assumed in the past. In the Anglo-American literature instead of IFF the term “matrix cracking” is widely used. In Germany the expression inter fibre fracture is preferred in order to characterise cracks running through the entire thickness

834 of a UD-layer in a laminate.1 These cracks are not necessarily caused by cohesive failure of the matrix; they could also be initiated by adhesive failure at the fibre/matrix interface. In addition to the calculation of the stresses at which IFF occurs one is now able to predict the direction of the crack, meaning the inclination fp of the fibre-parallel fracture plane measured against the thickness direction, Fig. 3. We find “straight” cracks with fp = 0° (Modes A and B) and “oblique” cracks with fp 0° (Mode C). On this basis harmless IFF (Mode A) and dangerous IFF (Mode C) can be distinguished [6]. Based on considerations of Mohr [7] and Hashin [8], it is assumed that IFF is a brittle fracture, that means a sudden separation of material without pronounced deformations in the fracturing area. Fracture is produced by a combination of normal and shear stresses. If the normal stress is tensile, fracture is caused by the tensile stress an and the shear stresses n1, nt acting on the fracture plane (Mode A). If n is a compressive stress, the fracture is caused by the shear stresses n1, nt alone. In this case, the compressive stress an impedes the shear fracture to a certain degree (Mode B or C), see Figs. 4 and 6 of Part A [2]. The application of the classical laminate theory (CLT) is well known and also its modification for non-linear stress/strain relationships is already widely used. However, there is a great uncertainty about post IFF stress/strain analysis. Therefore, this paper is focussing on this subject. 2.1. General considerations All calculations have been performed by means of the updated computer program NOLI FRAN COLAM (non-linear fracture analysis of composite laminates) which is based on the authors theory and in use at the institute for Leightweight Design and Construction at the Technical University of Darmstadt. A special feature of this program is a feedback of any small changes of the fibre orientations caused by the strains of the laminate. This is, for example, very important in the test cases dealing with a ± 45° laminate and y : y = 1 : –1. The stress/strain and failure analysis is executed layer-by-layer, because this is the only way to get a realistic image of the successive and progressive failure process of fibre polymer laminates. In nearly all cases this process starts with inter fibre fracture (IFF) in some layers and ends with a catastrophic rupture or collapse of the laminate due to fibre failure (FF) of at least one layer. In order to follow up this complex fracturing process, the analysis is carried out by loading the laminate “monotonically” by a considerable number of increments of the combined stresses x, y, xy, while the stress ratios x : y : xy are kept constant. 2.2. Pre-lFF non-linear stress/strain analysis The minor non-linearity of the 1(1) stress/strain diagrams of the UD-layers made with carbon fibres has been neglected. The properties given for the UD-layers of the different laminates [3], showed a remarkable non-linearity only for the 21( 21)-stress/strain curves and the 2(2)-stress/strain curves for 2 being a compressive stress. The main reasons for this 1

The cracks propagate very rapidly through the entire layer thickness; there is no visible crack propagation. The crack is stopped by the fibres of the neighbouring layers.

835 non-linearity are viscoelastic behaviour of the matrix and, probably above all, micro-damage2 in the matrix or at the fibre/matrix-interface long before visible “matrix cracks”, which we call IFFs, occur. For the calculation of stresses 2 and/or 21 secant-moduli E2s or G21s, respectively, have been taken from the spline functions of third order which were used to approximate the experimentally determined 2(2)- and 21( 21)-curves. In order to find the right secant shear modulus G21s for in-plane shear it does not matter whether it is taken at the shear stress 21 or at the shear strain 21 – both resulting from the (1, 2, 21) state of stress – because they are directly connected by 21 = G21s 21. According to the classical laminate theory (CLT) there is neither an influence of the longitudinal stress 1 nor of the transverse stress 2 on the shear strain 21. However, the right choice of the secant modulus E2s for transverse stress 2 is more complex. Under combined stresses 1 and 2, the transverse strain 2 = 2 /E2s  v121 /E1 is not only dependent on 2 but also on 1. Due to the Poisson’s effect activated by a longitudinal stress 2 a high transverse stress 2 can result even at 2 = 0. It is assumed that the main reason for non-linearity – that means for a considerable difference between E2s and E2 – is microdamage. This process is expected to increase with the transverse stress 2 (and not with the strain 2). Therefore under combined (1, 2)-stress, the secant modulus E2s should be taken based on 2 and not on 2 from the given 2(2)-curve for unidirectional material. In reality there is also some combined action of 2 and 21 on the 2(2)- and 21( 21)-stress/ strain diagrams, see Section 2 and Fig. 1 in [2]. This, for example, is due to the micro-damage

Fig. 1. (a) Brittle micro-damage in a UD-layer caused by a shear stress 21 leads to a marked damage hysteresis (DH) in a loading/ unloading cycle. (b) A shear stress m21 is acting in the matrix which can be transformed into the principal stresses mII = m21 and mIII = –m21. (c) The tensile principle stress mII causes numerous 45° micro-cracks long before the fibre/matrix composite fails. The microcracks are stopped by fibres which in this case prevent an early fracture of the composite. 2

Micro-damage and IFF may be mixed up: The terrn micro damage is used to characterize innumerable very small size failures in the matrix material and at the fibre/matrix interface on a microscopic scale. That means with extensions of the magnitude in the order of the fibre diameter In very transparent GFRP micro-damage sometimes becomes visible by a certain opaqueness or milkyness. IFFs in contrast are “visible” cracks which extend through the entire thickness of a UD-layer in a laminate.

836 caused by 21, which under pure shear 21 appears in the 21( 21)-diagram in the form of a considerable non-linearity and an enormous “damage hysteresis” on unloading from a high 21 stress Fig. (1). This micro-damage caused by 21 of course diminishes not only the stiffness against the stress 21 but also the stiffness against a stress 2. The same effect is found vice versa for the influence of 2 on the 21( 21)-curve. It was decided to neglect these effects in [2] because no stress/strain-curves for combined (2, 21)-stresses of the UD-materials used in the laminates were available and we wanted the results to remain comparable to those in other contributions. 2.3. Residual stresses For a realistic stress analysis, at least for carbon fibre laminates, it is mandatory to incorporate hygro-thermal effects into the calculations, because dangerous residual transverse tensile stresses result from the cooling down from curing temperature. In the Exercise it was not known to what extent the thermal stresses arising from the cooling process after hot curing were diminished by relaxation and more or less compensated by swelling of the matrix material due to moisture absorption in-between the fabrication of the test specimens and their testing. For the analysis we assumed that only 50% of the thermal stresses existing at the end of the cooling process from stress free temperature to test temperature had remained when the tests were performed. This results in the prediction of somewhat higher stresses at the onset of IFF compared to the results based on the assumption that 100% of the original residual stresses were still active when the tests were started. This is considered also in the discussion of Fig. 22. During the stepwise analytical loading process the residual stresses are diminished by micro-damage and IFFs [6]. Therefore, they have been recalculated stepwise using the appropriate secant moduli for each new load step. That means E2s, E2s and/or G21s, G21s, respectively, have been used for their recalculation ( is the degradation factor used after the onset of IFF, see Section 2.4). 2.4. Post-IFF non-linear stress/strain analysis After in one layer at a certain load level a stress exposure factor fE(IFF) = 1 has been reached, which is defined in Section 3.3, the first IFF (that means the onset of progressive matrix cracking in the affected layer) is to be expected. With increasing load more and more cracks are likely to develop which are more or less evenly distributed over the length of the layer. It is necessary to distinguish between different modes of IFF, see Section 4 in [2]. 2.4.1. Degradation procedure for cracks due to IFF of Mode A Cracks due to IFF of Mode A (Fig. 3) tend to open because the fracture angle is fp = 0° and the direct stress n = 2 perpendicular to the fracture plane is a tensile stress. That means that at the location of a crack the secant moduli E2s and G21s drop to zero. However, at a certain distance from the cracks – due to inter-laminar shear – the cracked layer takes over load from the neighbouring layers and stresses 2 and 21 are developing again. In these regions between

837 the cracks moduli E2s > 0 and G21s > 0 should be assumed. For a more “macroscopic” calculation above the crack onset limit the uneven stress distribution over the length of the layer is approximated by a constant stress which has the magnitude of the average value of the uneven stresses. This means that the effect of the cracks is “smeared” over the length dimension. In calculation practice this is achieved by using gradually reduced secant moduli E2s and G21s after IFF onset. It is expected that the Poisson’s effect is also reduced by the opening of cracks, therefore a reduced Poisson’s ratio v12 is used too. In most cases of Mode A IFFs it seems reasonable to apply equal values for the degradation factor  in all three terms. IFFs of Mode A are produced by a pure tensile stress 2 or by a pure shear stress 21 or in most cases by a combination of 2 and 21. In any case the fracture angle is fp = 0°, that means the crack appears in the action plane of 2 and/or 21. Therefore the following conclusion can be drawn: For IFFs of Mode A the degradation factor  should solely depend on the IFF crack density and not on the kind of stress or stress combination which produced the crack, and the degradation of E2s, v12 and G21s should start simultaneously. In contrast to damage mechanics solutions it was not tried to establish relationships between state of stress and crack density on the one hand and crack density and stiffness loss (E2s and G2s) on the other hand. Instead a pure phenomenological approach for the stiffness degradation is used. For this purpose a parameter controlling the degradation factor  is needed which is characteristic for the combined activity of 2 and 21 in producing subsequently more and more IFFs of Mode A. The parameter adopted for the controlling of  is the stress exposure factor fE(IFF), defined in Sect. 3.3, because fE(IFF) is the only available parameter which quantifies the risk of Mode A IFFs caused by the combined action of 2 and 21. Based on this idea the earlier degradation procedure, developed in 1969 [5], has been the following: When the stress exposure factor fE(IFF) = 1 of a layer indicated that this layer had reached its crack onset point (IFF) the secant moduli E2s and G21s were kept constant for the following calculations of the stress exposure factor fE(IFF) under higher loads. This results of course in unrealistic high stresses 1, 2, 21 which lead to stress exposure factors fE(IFF) > 1 for loads above the IFF onset load. The “surplus of stress exposure” ( fE(IFF)  1) grows with increasing load and can therefore be interpreted as a parameter representing crack density which could be used to control the stiffness degradation factor  [5]. Recently the following empirical formula for the degradation factor  [6] has been developed: =

1  r + r 1 + c( fE(IFF)  1)(

for fE(IFF) 1

(1)

The dimensionless parameters c and ( serve to fit the -curve to experimental results. If no values for c and ( resulting from experiments are available, we recommend to use c 4 and ( 2. The term r represents a small remaining stiffness ( r < 1). The curves in Fig. 2 are drawn with the recommended values of c, ( and  r . (It should be noted that the curves in Fig. 8 in [2], which have been taken from [5], were not based on any formula. They were intended only to qualitatively demonstrate the general expectations about the dependence of  on fE(IFF) the authors had in 1969.) In the model of 1969 [5], the reduced elastic properties E2s, G21s, v12 were only used to calculate the strains x, y, xy of the laminate by means of -reduced stiffnesses. It was not

838

Fig. 2. Correlations between load on the laminate, stress exposure factor fE(IFF), degradation factor  for IFF of Mode A and average stress 21(). For simplification it is assumed that 21 is the only stress responsible for IFF. The material non-linearity is neglected. Fig. 2(a) and (b). The correlations in the case of the former (1969) degradation method. The stress exposure factor fE(IFF) increases with the stress 21 resulting from the constant modulus G21. Assuming that the shear strain 21 is approximately proportional to the load on the laminate fE(IFF) is also proportional to this load. (a) The degradation factor  calculated from Eq. (1) using c = 4, ( = 2 and r = 0.1 in dependence of fE(IFF). (b) The average shear stress 21() (smeared over the length direction x2, Fig. 4 in [2]). In order to get a dimensionless magnitude 21() has been divided by 21(IFF), which is the shear stresses at the onset of IFF. (The average stresses 21() and 2() are not used in the 1969 degradation method.) Fig. (c) and (d). The different correlations in the case of the degradation method used in the Exercise, Part A (1995), [2]. Again it is assumed that 21 is proportional to the load on the laminate and that the stress/strain-diagrams are linear. After the onset of IFF the stress exposure factor is kept fE(IFF) = 1 = const. Under the assumptions mentioned above this is achieved by an average shear stress 21() which does not change its magnitude with the load and a degradation factor  which is proportional to load–1. For a certain load step  is found by an iteration process which keeps fE(IFF) = 1.

necessary to calculate the average-stresses 2(), 21(), which also could have been computed by using -reduced secant moduli, see Fig. 2b. Up to now there is neither experimental evidence that the degradation function  according to Eq. (1) is right nor that the chosen parameters c, (, r are generally valid. For the Exercise we, therefore, tried to get out of this unsatisfying situation and looked for a modified degradation procedure, which appears less arbitrary. In Part A paper [2], in contrast to the 1969-procedure, the average-stresses 2() and 21() in the cracked layer were used in order to control the degradation factor  for calculations beyond IFF. They were calculated from the strains of the laminate by using the reduced secant

839 moduli E2s, G21s and reduced Poisson’s ratio v12. These average-stresses 2(), 21() were inserted into the equation for the stress exposure factor fE(IFF), Eq. (19), in [2]. With equal values of  for E2s, G21s and v12 we then kept fE(IFF) = 1 = constant

(2)

in the whole loading range between the IFF onset in the affected layer and total failure of the laminate. This is done by choosing decreasing values for  which keep fE(IFF) = 1 = constant, in spite of increasing strains 1, 2, 12. To the authors’ best knowledge and up to now there is no experimental evidence that this procedure can provide reliable results for all kinds of laminates and arbitrary loading conditions. However, an extensive research project has been started by the Composites Department of the Institute of Plastics Processing in Aachen [12], which could provide a sound physical basis for post-IFF degradation. After IFF initiation the reduced moduli E2s and G21s are found by a superposition of “material non-linearity” and the non-linearity due to the increasing number of IFFs, expressed by the -degradation. This means that E2s and G21s are taken from the 2(2)-diagram or 21( 21)-diagram, respectively, at the average-stresses 2() and 21(). In the incremental loading process a reasonable agreement of moduli E2s and G21s on the one hand and average stresses 2() and 21() on the other hand is achieved loopwise by iteration. For rather small load increments one single iteration can sometimes be sufficient. In order to clarify the fundamental difference between the older -degradation procedure of 1969 and the new one of 1995 – the latter was used in [2] – the following considerations could be helpful. In general there are two different ways to tackle the problem: one can first make an assumption about the stiffness reduction after the onset of IFF; that means for instance an assumption about a function  = ( fE(IFF)) like the one given by Eq. (1). This leads automatically to a certain behaviour of the average stresses 2() and 21() after the onset of IFF. This procedure is explained above and in Fig. 2a and b. It was not used in the Exercise with the exception of Fig. 24 where the high flexibility of Eq. (1) was helpful. A second way starts with an assumption about the behaviour of the average stresses 2() and 21() after the onset of IFF. This approach was used in the 1995-Exercise by keeping fE(IFF) = 1 = const. after the onset of IFF. For IFFs of Mode A this assumption keeps the average stresses 2() and 21() nearly constant (not exactly because of the different non-linearities and the fact, that 2() not only depends on  · E2s but also on v12). As a consequence of this assumption the dependence of  on fE(IFF), is also fixed. In the case of Mode A there follows some kind of hyperbolic dependence of  on fE(IFF), see Fig. 2c and d. The second approach was used in 1995 in Part A, because the coeffficients of Eq. (1) were not known for the materials used in the test cases. 2.4.2. Degradation procedure for cracks due to IFF of Mode B or C Developing a reasonable degradation process for IFFs of Modes B and C is much more difficult than that for IFFs of Mode A. Based on the hypothesis that brittle IFF is influenced only by the stresses n, nt, n1 on the fracture plane, see Figs. 4 and 6 in [2], it is assumed that in Mode B (fp = 0°) a shear fracture occurs which is produced by n1 = 21. In Mode C (fp 0°)

840 the combined action of nt = –2 sin fp cos fp and n1 = 21 cos fp causes a shear fracture on an oblique fracture plane. In both modes the simultaneously acting compressive stress n = 2 (or n = 2 cos2 fp) impedes the shear fracture by increasing the resistance of the fracture plane against shear fracture. It is assumed that the additional fracture resistance caused by n increases parabolically with | n |, see Eqs. (15) and (17) in [2]. Again a degradation factor  is used to calculate average-stresses above IFF onset. These are then inserted into the equations for fE(IFF), Eq. (20) or (26) in [2]. Above the onset of IFF, similar to Mode A, the value for fE(IFF) is kept at fE(IFF) = 1 = constant (as far as possible). However, there is a fundamental difference between Mode A cracks on the one hand and Modes B and C cracks on the other hand. In contrast to Mode A cracks, Mode B and C cracks cannot open. Instead, the crack surfaces are pressed on each other by the compressive stress n = 2 cos2 fp. Therefore, it appears reasonable to apply no -degradation to the secant modulus E2s and to the Poisson’s ratio v12. This means that  = 1 is used for E2s and v12. In spite of the fact that cracks do exist, the material is likely to behave like undamaged (bulk) material as long as movements normal to the crack surfaces are prevented by the compressive stress n which is still acting on the crack surfaces. This is at least a resonable assumption for the macroscopic models used in engineering approaches. (FEA may show a more complicated picture.) During the formation of subsequent cracks above IFF onset E2s is taken – as prior to IFF onset – at the calculated stress 2 from the uniaxial 2(2)-stress/strain diagram. In contrast to this, it should, however, be reasonable to assume a certain -degradation of G21s after the onset of IFF, because some sliding movement of the crack surfaces relative to each other is to be expected. This is physically not completely correct, because before any relative sliding movement of the crack surfaces can occur a certain friction has to be overcome. For a pure 2 compressive stress, where nt = –2 sin fp cos fp is the only stress which causes the fracture, E2s and also v12 are used without -degradation in the stress calculations for loads above the IFF onset. In this case the stress exposure factor cannot be kept at fE(IFF) = 1 with increasing strains because no -reduced shear stress 21 is involved in the fracturing process. A high fracture angle of about 50° to 55° will appear, (Fig. 3) [6,9–11]. In Fig. 3 the (2, 21)-load path (a) is shown for the case of a (2, 21) combination which leads to a first IFF of Mode B with fp = 0°. When the (2, 21)-fracture envelope is reached in point b the -degradation of G21s starts. With increasing load, which also means with increasing strains, the -degradation of G21s has to keep fE(IFF) = 1. In contrast to the Mode A situation now the shear stress 21 is the only stress which is active in the fracturing process and G21s is the only modulus which is reduced by the -degradation. Because IFFs of Mode B are closed cracks the secant modulus E2s is used without -degradation. The consequence of the different treatment of G21s and E2s is that | 2 | is growing faster with increasing external load than 21(), i.e. the stress ratio | 2 : 21 | moves to higher values. Depending on the load bearing capacity of the laminate the point (2, 21) may be reached where the IFF Mode changes from B to C (point c in Fig. 3). Beyond this point the stress component nt = –2 sin fp cos fp on an oblique fracture plane helps to produce the fracture. That means that now not only 21() but also | 2 | contributes to the stress exposure factor fE(IFF), Eq. (26) in [2]. However, the -degradation of 21() alone has to keep fE(IFF) = 1. Therefore the degradation factor  has to decrease very rapidly and sometimes the calculation is coming to an early end, when  = 0 (that means pure compression 2) is reached, because  < 0 does not make sense. In that branch of the (2, 21) load-path with decreasing stresses 21 the stress ratio | 2 /21 | will change rapidly to higher values and with this to higher fracture angles fp between ± 45° and

841

Fig. 3. Fracture envolope for combined (2, 21) stresses, fracture modes A, B, C and fracture angle fp calculated by Eq. (5). D/LB = increasing risk of delamination and local buckling due to the wedge effect caused by high compressive stresses n = 2 cos2 fp acting on oblique fracture planes inclined against the thickness direction by fracture angles | fp | > 45°. Also shown is a load path (a) for combined 2 compressive stress and in-plane shear stress 21 starting with an IFF of Mode B at point b and proceeding to IFFs of Mode C.

± 55°. This means that a “wedge effect” has to be considered, Section 2.4.3, see also disscussion of Fig. 22 in Section 5.5. For the calculation of the average-stress 21() the same procedure is used as for Mode A as long as 21() is increasing. However, when the maximum of 21 on the (2, 21)-fracture envelope is exceeded, for the calculation of G21s the secant modulus G21s belonging to the maximum of 21 is used as a constant value in the product G21s. In this way the unloading behaviour of 21( 21) as shown in Fig. 1 is approximated. 2.4.3. The “wedge effect” or “explosion effect” A fracture angle fp ± 50° to ± 55° is typical for pure 2-compression [9]. With high angles fp between the fractured surface and the thickness direction the fractured pieces act like wedges, driven by a compressive force 2t (t = thickness of the layer). If the fractured layer is a relative thick one placed in the middle of the laminate the “wedge effect” may cause the laminate to “explode” causing severe delamination followed by local buckling of outer layers, Fig. 4. For more details of this “explosion effect” see also Fig. 9 and Section 8.2 in [2] and Section 6.3 in this paper.

Fig. 4. Schematic illustration of the “explosion effect” of a wedge shaped transverse compressive fracture causing delamination and local buckling (D/LB).

842 This effect can only occur on oblique shear fracture planes which are inclined against the thickness direction by an angle fp of at least ± 30° to ± 45°. It is caused by forces n3 (acting in the thickness directions (see Fig. 9d in [2]) which cannot be compensated by nt-shear stress within the UD-material, because it has lost its cohesion due to the shear fracture. A certain proportion of the shear force nt · t * – which is necessary for equilibrium can be replaced by a friction force  | n | t *, where  = coefficient of friction, n = compressive direct stress on the fractured plane and where t * is the length of the fractured plane (see Fig. 9d in [2]). The force parallel to the fracture plane, which is necessary for equilibrium of the fractured pieces (parallel to the fracture plane), is | 2 |t sin fp (Fig. 9 in [2]). It can become larger than the friction force  | 2 | t cos fp. In this case supporting forces from the neighbouring layers are necessary in horizontal (n2) and vertical (n3) directions in order to maintain equilibrium. The vertical force n3 tends to split (or “explode)” the laminate (Fig. 4). The limit above which the friction on the broken surface is no longer able to fulfil the equilibrium conditions is given by | 2 | t sin fp =  | 2 | t cos fp

tan fp = 

(3)

It is exclusively dependent on the fracture angle fp and not on the magnitude of 2. (This situation is similar to the problem of a mass sliding on an inclined plane with friction.) Some surplus of the driving force | 2 | t sin fp over the friction force acting against a sliding movement is necessary to cause a considerable explosion force n3 which may lead to delamination and local buckling (D/LB). In [2] it was assumed that a considerable risk of delamination and local buckling (D/LB) would appear, if | 2 | t · sin fp 3 ·  | 2 | t · cos fp

or

tan fp 3

(4)

For tan fp = 3 the driving force for a sliding movement is three times higher than the resisting friction force. If, for example,  = 1/3 is chosen we find that fracture angles fp 45° would be critical (Fig. 3). The figure shows, for which combinations of 21 and compressive 2 the wedge effect can be dangerous. If such a high fracture angle is computed, we stop the calculation and give a warning that the risk of “delamination and local buckling” (D/LB!) has occurred. For the determination of the fracture angle fp for Mode C Eq. (25) in [2] can be used, however, a more convenient equation has been found by solving Eq. (24) in [2] for cos fp without using the fracture condition Eq. (22) in [2], which results in:

fp = arccos

冑

1 ) 2(1 + p(⊥⊥ )

冋冉 冊 冉 冊 册 21 2

2

R A⊥⊥ S21

2

+1

(5)

For symbols see Fig. 9. ) p(⊥⊥ = slope at n = 0 of the (n, nt) fracture envelope valid for n 0; R A⊥⊥ = fracture resistance of the stress action plane against its fracture due to transverse/transverse shear stressing; S21 = shear strength transverse and parallel to the fibre direction. One should be aware of the fact that even if one starts from a Mode B situation with only a moderate 2 stress, due to the fact that only G21s is degraded, one can move to a Mode C situation and finally even end up with a pure 2 compressive fracture, see Fig. 3. This means

843 that first IFFs would occur with a fracture angle fp = 0° while later occurring IFFs would show larger fracture angles, increasing with load. It is very important to realise that the warning “D/LB!” does not mean that failure really does occur at the load level reached. It only states that from now on under increasing load there is an increasing risk of an explosive failure. That the wedge effect indeed can cause an “explosive” collapse of a highly loaded component could be learned from the “sudden death” of our early prototypes of GFRP-torsion tube springs for a passenger car, Section 8.2 in [2], and see Figs. 5–7. Further discussion of the explosion effect is found in Section 6.3. 3. Highlights of the used failure criteria 3.1. General considerations For design purposes we have to base our failure analysis of FRP laminates preliminarily on the questionable assumption that the fracture stresses of a UD-layer in a laminate are very similar to those of the isolated layer. We, therefore, start with failure criteria for the UDlayer. In 1969 it has already been disclosed that there are two extremely different failure types in UD-composites, fibre failure (FF) and inter fibre fracture (IFF) and that it is necessary to treat these two phenomena by separate failure criteria [4,5].

Fig. 5. Tubular GFRP torsional spring, made up of one thick outer + 45°-layer and one thin inner –45°-layer. These are separated by a thin Aramide crack arresting layer (CAL). As demonstrated on the outer surface of the tube a torsional load leads to f1-tension stress in the fibres of the outer layer and to f1-compressive stress in the fibres of the inner layer. The simultaneously acting transverse stresses 2 are compressive in the outer layer and tensile in the inner layer. Under pulsating torsion there are early cracks of Mode A due to the transverse tension stress in the inner layer without any detrimental effect. However, the first wedge shaped fracture of Mode C due to the transverse compressive stress in the outer layer causes a collapse of the inner layer and with this a “sudden death” of the torsional spring.

844

Fig. 6. Typical double wedge shaped transverse compression fracture as observed in fatigue tests on prototypes of GFRP-torsional springs. The high longitudinal tensile stress of the fibres in the outer layer forming a + 45°-helix result in a radial force directed to the axis of the tube. This radial force causes the thin inner layer to buckle and collapse.

The strength of a reasonably well designed laminate (a laminate with three or more fibre directions) depends to a very high degree on the load carrying capacity of its fibres, because normally most of the load is concentrated in the fibres. Therefore, the FF-criteria have the highest importance for the design of laminates. However, in most cases, before the fibres can be fully strained, IFF occurs somewhere, which means that locally the integrity of the fibre matrix composite is more or less destroyed. But in many cases IFF does not have a disastrous effect on the load carrying capacity of well designed laminates. This is not true for “laminates with unsound loading conditions” which are discussed in Section 4.3.

3.2. Failure criteria for fibre failure (FF) Since 1969 a very simple criterion for FF is widely used, which has been proposed in [4,5]: 1 =1 XT

for 1 > 0 or

1 = –1 XC

for

1 < 0

(6)

Fig. 7. Double wedge shaped fracture in an early prototype of a tubular GFRP-torsional spring which occurred under pulsatig torque after 2 · 106 cycles. (The ratio of lowest to highest torsional moment was 0.36.)

845 The counterpart of Eq. (6) written in terms of strain 1 and failure strain 1T or 1C, respectively, is also in use: 1 =1 1T

for 1 > 0 or

1 = –1 1C

for

1 < 0

(7)

Eqs. (6) and (7), which are based on experimental experience, mean that any additional stresses 2, 3, 23, 31, 21 or strains 2, 3, 23, 31, 21 do not influence FF. For the contribution to the Exercise [2], we wanted to be a little more “physically correct” and, therefore, based the FF criterion on a failure hypothesis for the fibres instead of one for the UD-composite, because the loads are concentrated in the fibres. For this reason it was started from the following hypothesis [6]: Fibre failure in a UD-composite under a (1, 2, 21)-state of stress will occur at the same fibre stress f1 as the one which is acting in the fibres when fibre failure occurs under a uniaxial 1-stress in the UD-composite. The analytical treatment of this hypothesis results in the following FF-criteria for the UD(1, 2, 21)-stresses (see Section 3 in [2]): 1 1T 1 1C

冉 冉

1 +

冊 冊

vf12 mf · 2 = 1 Ef1

for (. . .) > 0

(8)

vf12 mf · 2 = –1 Ef1

for (. . .) < 0

(9)

1 +

For plane states of stress (1, 2, 21), as assumed for the test cases given in the Exercise, the strain 1 of the composite follows from

1 =

1 v12  2 E1 E1

(10)

and the fracture strain in uniaxial tension is

1T =

XT E1

(11)

With Eqs. (10) and (11), for example, Eq. (8) can be rewritten for a critical discussion as 1 2 E1 2  v12 + vf12 mf = 1 XT XT Ef1 XT

(12)

The first term is the one used in Eq. (6). If only the first two terms of Eq. (12) are used this is just the same as using Eq. (7). Fig. 8 demonstrates that it does not matter too much whether Eq. (6), or Eq. (7) or our more sophisticated Eq. (8) or (9), respectively, is used. The results of Eq. (8) or (9) are found just in between of the solutions of Eq. (6) and Eq. (7). So, in the case of FF our more sophisticated procedure turned out to be a more or less academic exercise. However, this is not true for our new IFF-criteria which are dealt with in the next section.

846

Fig. 8. External areas of the (1, 2)-failure envelope of UD-FRP illustrating the minor differences in the FF-stresses, calculated by different fracture conditions as presented in Section 3.2.

3.3. Failure criteria for inter fibre fracture (IFF) The new IFF criteria are based on a “modified Mohr hypothesis” [13], which has been adapted to transversally isotropic material. It states that only the stresses n, nt, n1 on a fibre parallel fracture plane (see Fig. 4 in [2]) are responsible for IFF. This idea was published by Hashin in 1980 [8], but has not found any resonance in the composites community until 1992 [14]. It should be mentioned that the presented IFF-theory is completely 3-dimensional. Because in the Exercise only 2-dimensional stress situations had to be dealt with the full potential of the new IFF-criteria could not be demonstrated. From a 3-dimensional composite stress analysis one gets the stress condition of a UD-layer related to its natural axes x1, x2, x3, this means as numerical data of the stresses 1, 2, 3, 23, 31, 21 (Fig. 4 in [2]). The stresses n(), nt(), n1() on an arbitrary sectional plane with an inclination angle , which can be inserted into the IFF-criteria (see Table 1), are calculated from n() = 2 cos2  + 3 sin2  + 223 sin  cos 

(13)

nt() = (3  2) sin  cos  + 23(cos   sin )

(14)

n1() = 31 sin  + 21 cos 

(15)

2

2

In the general case of a 3-dimensional state of stress the stresses n(), nt(), n1() have now to be inserted into the IFF-criteria (16) or (17), respectively, of Table 1 depending on n 0 or n < 0. The meanings of the parameters used in Table 1 are demonstrated in Fig. 9. (The Eqs. (16) and (17) have been developed on the basis of Eqns (14) and (15) in [2] with the aim of an easier mathematical handling [6]). In order to get physically reasonable results it is necessary to respect some restrictions for ) +) the choice of the so-called inclination parameters p(⊥⊥ , p(⊥⊥ , p(⊥储 ), p(⊥储+ ), [15]. In Part A [2] ) p(⊥储 ) = 0.25; p(⊥储+ ) = 0.30 and p(⊥⊥ according to Eq. (21) were used. In general it is not known in advance which one of the infinite number of possible fibre parallel planes will become the fracture plane, when the load reaches the fracture load. It certainly is the one with the highest numerical value of the angle dependent stress exposure factor fE(IFF)(). That means that one has to search for [ fE(IFF)()]max = fE(IFF)(fp). To identify it,

847 Table 1 Puck-criterion for inter fibre fracture caused by a 3-dimensional (2, 3, 23, 31, 21) state of stress.

For n() 0: fE() =

冑冋冉

For n() < 0: fE() =

冑冉 冊 冉 冊 冋

with

1

 ( + )A

R⊥

nt() RA⊥⊥

2

+

+) +) p(⊥ p(⊥⊥ p(⊥储+ ) = A cos2  + A sin2  A R⊥ R⊥⊥ R⊥储

cos2  =

2nt  + 2n1 2 nt

R(⊥+ )A = TT;

sin2  =

RA⊥储 = S21;

冊 册冉 冊冉 冊

+) p(⊥ n() RA⊥

n1() RA⊥储

2

+

2

+

nt() RA⊥⊥

n1() RA⊥储

2

+

册

) p(⊥ n() A R⊥

2

+

+

) p(⊥ n() A R⊥

() ) () p⊥ p(⊥⊥ p⊥储 = A cos2  + A sin2  A R⊥ R⊥⊥ R⊥储

2n1  + 2n1

+) p(⊥ n() RA⊥

(16)

(17)

(18)

(19)

2 nt

RA⊥⊥ =

2

YC ) 2(1 + p(⊥⊥ )

(20)

With the aid of Eq. (16) or Eq. (17) for the angle dependent stress exposure factor fE() the global maximum of fE() = fE(fp) has to be found. The fracture condition is then fE(fp) = 1. For the choice of the parameters in Eqs. (16) and (17) see [15].

one has to compare the numerical values of the computed fE(IFF)() for a sufficiently high number of values for  between –90° and + 90°. Only for a plane state of stress (1, 2, 21) the numerical search for the fracture plane can be avoided because it was possible to develop an analytical solution for fp, see Eq. (5) of this paper and Eq. (25) in [2]. The price that had to be paid for the analytical solution was that one had to assume a coupling of four parameters introduced in Table 1 (see Fig. 9): ) ) p(⊥ p(⊥⊥ p(⊥储 ) = A = = const. R⊥ R ⊥⊥ R⊥储

(21)

There is no physical reason for this coupling, nevertheless one can be quite sure that it does not lead to any significant errors. For cases in which the fracture angle will be fp = 0° as for Mode A- and Mode B-situations (see Fig. 6 in [2]), where the action plane of 2 and 21 becomes also the fracture plane, it is quite easy to find a very simple analytical formulation of the fracture condition. One only has to insert nt = 0, n1 = 21 and n = 2 into Eq. (16) or (17), respectively, and to set fE(IFF) = 1. It does not matter whether fp has been found analytically or numerically, in either case the stresses at IFF follow from the valid equation for the stress exposure factor fE(IFF) – see for instance Eqs. (16) or (17) in Table 1 – by setting fE(IFF)(fp) = 1

(22)

In the case of plane stress (1, 2, 21) this leads to very simple IFF-conditions F (1, 2, 21, YT or YC , S21, p(⊥储 ), p(⊥储+ )) = 1, see Eqs. (19), (20), (26) and Table 1 in [2]. There it is also demonstrated how an influence of 1 on IFF can be taken into account. All the other necessary

848

Fig. 9. Illustration of the parameters describing the master fracture body in (n, nt, n1)-space resulting ) from Eqs. (16) and (17) in Table 1. R (⊥+ )A = YT , R A⊥储 = S21; R 4⊥⊥ = YC /(2 + 2p(⊥⊥ ) are “fracture resistances” of the fibre parallel action planes. Attention: The inclination parameters p have to be kept within certain limits (see [15])!

IFF conditions and corresponding relationships for the treatment of IFF in the test cases of the Exercise are also to be found in Table 1 in [2]. 4. Correlations between information from theoretical models and events in real composites It should always be kept in mind that in design practice only very imperfect models can be used for the failure analysis of FRP laminates because reality is very complex. Nowadays material scientists, design engineers and testing experts have to be highly specialised experts in their own discipline. Therefore, an extensive exchange of knowledge is very important. Section 4 should provide some guidelines for a critical comparison of analytical and experimental results. 4.1. Isolated UD-layers When performing strength tests on isolated UD-layers it might be difficult to introduce the high loads which are necessary to achieve FF and simultaneously exclude premature IFF. If this problem can be solved, very careful observations during the test and intensive inspection of the broken pieces should lead to a clear decision whether a FF or an IFF has occurred.

849 Typical for IFF are rather plane and clean fracture surfaces extending parallel to the fibres. In contrast to this very fissured regions appear where the separation of the test pieces has resulted from FF. Fibre rupture caused by 1-tensile stress makes the destroyed ends of the test pieces look like paint-brushes, while 1-compressive stress causes extensive longitudinal splitting of the test piece into numerous thin “beams”, many of which seem to have buckled and have been broken by bending. There are some photographs to be found in [6], which illustrate this. For a serious judgement of the theoretical predictions it is important to distinguish between IFF and FF in the UD-tests. Unfortunately, there is no such information available for the UDtest results belonging to Part B of the Exercise. Sophisticated experimental work that has been performed recently has proven that even the measurement of fracture angles fp of IFFs is possible [9–16]. One was able to distinguish clearly between Mode A-, B- and C-fractures. The determination of fracture angles is highly important for the confirmation of the new IFF theory based on the assumption of brittle fracture.

4.2. Laminates with reasonable loading conditions Experienced designers know that a laminate, which in service has to withstand different load combinations (x, y , xy), should be designed with at least three different fibre directions. The ( + 30°/–30°/90°) laminate and the (0°/ + 45°/–45°/90°) laminate which had to be dealt with are laminates of this robust type. Laminates having only two fibre directions may also be exposed to a “reasonable loading condition”, e.g. a “ ± 55°”-tube with closed ends and pressurised by internal pressure, which leads to a stress ratio y : x = 2 : 1. For a + 45°-tube, a torsional load resulting in a pure xystress is also a “reasonable” loading condition, [6]. The common feature of these examples is that by using netting analysis (which of course is a poor theory for a realistic stress/strain-analysis) fibre stresses of finite magnitude result. The reason is that even under the assumption that the matrix does not carry any load, equilibrium can be found between external loads and forces transferred by the different fibre layers. This also means that in reality with progressive matrix cracking the loads tend to concentrate more and more in the fibres. In laminates with three or more fibre directions the stiff fibres are decisive for the strains of the laminate. Therefore stress/strain diagrams are found which, under these conditions, are rather linear from the beginning up to laminate failure due to FF. Under monotonically increasing load laminates of this type very often show more or less similar successive and progressive failure processes: First in one or more layers IFF of Mode A occurs at rather low strains. The matrix cracks are locally bridged by neighbouring layers with their different fibre directions. With increasing number of IFFs the stresses in the laminate are redistributed and concentrate more and more in the fibres, because the fibre network alone is principally able to carry all the loads, as we learn from the netting analysis. The maximum load carrying capacity of the laminate is normally reached when FF in one or more fibre directions occurs. Looking at the stress/strain curves of laminates of this type shows that their slope becomes a little shallower not just at the point where IFF is predicted by the analysis, but at a somewhat higher load, where the cracking process has fully developed. One should be aware of the fact

850 that the theoretical prediction of IFF does only correspond with the onset of matrix cracking and not with the fully developed progressive process of cracking. One very plausible reason is the following: In tests on an isolated UD-layer the lowest possible value for YT or S21 is measured, because in the test specimen weak and strong regions are following each other statistically distributed in series over the length of the specimen. One, therefore, always measures the strength of the “weakest link.” Also the “in situ effect” and the “thin layer effect” as mentioned in Section 10 in [2] as well as the effect of combined stresses 2 and 21 on the stress/strain diagrams, which has been discussed in Section 2.2 of this paper, contribute to the fact that in tested laminates severe matrix cracking is normally observed at stresses and strains which can be up to two- to fourtimes higher than the theoretical predictions of the onset of IFF. We have no idea at the moment how to take all these effects into account by a simple but realistic phenomenological approach. By no means can it be expected that calculated IFF onset stresses, for example of an ± 55°-tube, loaded with internal fluid pressure, will coincide with the so-called weeping or leakage stress observed in a pressure test on unlined tubes. Numerous cracks must have occurred in all layers before liquid can find its long way through the labyrinth of cracks which may have or may not have a connection to a crack in the neighbouring layer and so on through the entire thickness of the multilayer tube wall. Our experience is that, depending of course on the stacking sequence and the number of layers of the laminated wall, the weeping stress may be 1.5–3 times higher than the onset of matrix cracking. Therefore we come to the conclusion that a direct comparison of IFF-stresses and weeping stresses is not possible. Perhaps measurements of acoustic emissions of the cracking process could be helpful for the desirable comparison of theoretical IFF-prediction and experiments.

4.3. Laminates with “unsound loading conditions” If we have not yet developed a certain feeling based on experience for the detection of laminates “with unsound loading condition” we can very simply check the situation by applying netting analysis to a laminate with a doubtful fibre pattern and/or loading condition. If for example we ask the netting analysis which winding angle  would be “the best” for a pressure pipe with y : x = 2 : 1 and a pipe wall made up of a ± -angle ply laminate, we would get the answer that there is only one possible winding angle which is  = arctan兹2 54.74°. Obviously a pure fibre network with  deviating from arctan 兹2 cannot provide equilibrium between the forces of the fibre layers and the loads in the pipe wall resulting from y : x = 2 : 1. The other way round, this means that if we have produced a ± arctan兹2-pipe, netting analysis tells us that the fibres alone are not able to carry loads which belong to stress ratios y : x deviating from exactly 2 : 1, even if the deviation is extremely small. If a ± arctan兹2-tube is tested with a stress ratio which considerably deviates from 2 : 1 after IFF has occurred, the forces in the laminate cannot concentrate in the fibres as it is intended, when a fibre “reinforced” component is designed. Equilibrium between external loads and internal forces of the pipe can only be found with undesirable high stresses 2 and 21 attacking the matrix and the fibre matrix interface. This leads to an unusual high non-linearity of the stress/strain curves and to heavy matrix cracking immediately after an

851 early onset of cracking. At very low stresses the laminate will start to lose more and more of its original integrity and it will not recover from these damages when it is unloaded. Because after crack onset there is no pronounced concentration of the forces in the fibres, and because the laminate has to accept large deformations in order to produce the stresses which are necessary for equilibrium with the external loads, very soon such a high degree of deterioration of the composite can be reached that the pipe later on can no longer be used even under its normal service condition y : x = 2 : 1. In this sense the laminate may have “failed” long before the maximum load which is measured in a strength test has been reached, Fig. 16. There is no clearly defined failure limit. We stopped our calculations at the latest when a shear deformation of 21 = 0.2 had been reached. This is a quite arbitrarily chosen limit, which is only set to avoid “endless” and useless iteration loops. It is not based on any physical consideration, because up to now we cannot correlate excessive shear deformation and deterioration of the laminates. The situation 21 = 20% does not appear very often in the test cases. The only example is the biaxial failure stress envelope of the ± 55° laminate, Fig. 15. More frequently an “explosion” due to the wedge effect, Section 2.4.3, was likely to occur and in this case the calculation was stopped according to Eq. (4). A similar undesirable situation is found for instance for a ± 45°-tube with a stress ratio y : x = 1 : –1 which leads to the same loading condition as found in a (0°/90°) laminate under a pure shear stress xy. The fibres cannot pick up considerable longitudinal forces, in fact they are stressed nearly by the same low shear stress 21 which is acting (with a certain uneven distribution) in the matrix. The individual layers of the laminate are exposed to the same longitudinal shear stress 21 as an isolated UD-layer under a pure 21 shear stress, at least as long as no IFFs have occurred. After IFF has occurred in such a cross ply laminate the cracks are locally bridged to a certain extent by the neighbouring layers with its fibres crossing the cracks with an angle of 90°. Therefore, the shear stress 21 at total failure of the cross ply laminate is a little higher than the longitudinal shear strength S21 of the isolated layer. However, after 21 = S21 has been exceeded the laminate will very soon become deteriorated and useless. This is not what would be expected from a fibre “reinforced” component. The conclusions to be drawn from the facts discussed in this section are: (1) In practice laminates as discussed here can only be used if nothing but very small deviations from those loading conditions occur for which the netting analysis finds equilibrium of loads and fibre forces. (2) It is extremely difficult to perform a reasonable “failure” analysis for laminates “with unsound loading conditions”, because there is no clear agreement about what a “failure” is in this case. Theoretical predictions of the behaviour above IFF depend essentially on the model chosen for the degradation after IFF. In reality the behaviour after crack onset – which probably is strongly influenced by the specific stacking sequence of the laminate – has a strong influence on “failure.”

5. Comparison of analysis and experiment In Figs. 10–23, the organisers of the Exercise superimposed the corresponding test results on the diagrams presenting the theoretical predictions in [2]. As sometimes no corresponding

852 experimental data exist, not all details in the Figs. 10–23 in [2] (same numbers as the Figs. in Part B!) – as for example the indication of the layer, in which the fracture starts or the type and the mode of the fracture were adopted by the organisers in the figures of Part B. Therefore, it would be helpful for a more complete interpretation of the experimental data to look at all the details given in Figs. 10–23 in [2]. 5.1. Fracture behaviour of an isolated UD-layer under an in-plane (1, 2, 21)-state of stress The stresses x, y, xy shown in Figs. 10 to 12 are acting in the directions of the “natural” coordinate axes x1, x2, x3 of the UD-layer and, therefore, x ⬅ 1, y ⬅ 2, xy ⬅ 21, see Fig. 4 in [2]. The curves represent fracture envelopes for stress combinations of 2 and 21 (Fig. 10), 1 and 21 (Fig. 11) and 1 and 2 (Fig. 12). That means that they show three different sections through three different (1, 2, 21)-fracture bodies, as each figure is related to another fibre/ matrix-combination. Unfortunately, for all these figures it must be stated that there is insufficient test data for a serious comparison of theory and experiment. According to the considerations of Section 2.4 and 4.1 one can be sure that the (2, 21)stress combination shown in Fig. 10 that means combination of a 2- and a 21-stress acting on the same stress action plane causes IFF. Even with the small amount of test data it is obvious that the (2, 21)-fracture envelope has approximately the same course as the one being described by the action plane related fracture criteria for Modes A, B and C after Table 1 in [2] (see also Fig. 10 in [2]). This is also verified by recent series of experiments with considerably more specimens, which gave statistically reliable results [10]. With the test results of Fig. 10 given by the organisers we are now able to present a considerably improved (2, 21)-fracture envelope by using the equations from Table 1 in [2]

Fig. 10. Biaxial failure stresses for 0° lamina made of GRP material E-Glass/LY556/HT907/DY063.

853

Fig. 11. Biaxial failure stresses for 0° lamina made of CFRP material T300/914C.

with optimised parameters YT , YC, S21, p(⊥储 ), p(⊥储+ ). These can be found by a subroutine of the program “FraCuAn” (fracture curve analysis), developed 1993/1994 at the University of Kassel, see Section 4 in [2]. The result is presented by the broken line in Fig. 10, which demonstrates an excellent adaptability of the criteria for the IFF Modes A, B, C to test data.

Fig. 12. Biaxial failure envelope of 0° GRP lamina under combined x and y stresses. Material: E-glass/MY750 epoxy.

854 For the (1, 21)-fracture curve in Fig. 11, it would have been very desirable to have information about the fracture type (FF or IFF) in order to achieve a meaningful comparison between theory and experiment (compare Fig. 11 in [2]). Usually, as known from our experience, it is possible to clearly distinguish between FF and IFF by observing the fracture process and by a close inspection of the broken area of the tested specimen, as discussed in Section 4.1. Presumably those eight (1, 21)-points, where 1 is close to XT and the other eight (1, 21)-points where 1 is close to –XC refer to FF. The remaining test points in the range of –700 MPa 1 1000 MPa probably refer to IFF. The authors are not able to explain why a moderate tensile or compressive stress 1 should increase (compared to S21) the shear stress 21 which the material is able to withstand. These results seem implausible. The only possibility one could think of is the fact that under high shear deformations 21 the fibres leave their original direction, so that the rather stiff carbon fibres might be able to pick up a certain amount of the xy-shear load by longitudinal forces in the fibres. However, this effect should have been recorded by the computer code which feeds back the changes of fibre direction due to the strains. In contrast to the presented experimental data micromechanical reflections suggest that a 1-tensile stress should not increase, but reduce the bearable 2-shear stress. It enlarges the principal tension stress within the matrix, which should be responsible for the primary fracture process in the matrix, if we assume that under combined stresses the modified Coulomb–Mohr theory of brittle fracture, [13], is valid for the matrix with all its flaws. This seems very reasonable, because obviously the brittle IFF-behaviour of the UD-composite as a whole originates from the brittle behaviour of the matrix. The micromechanical treatment of the problem is complicated by the fact that not only the magnitude but also the direction of the principal stresses with respect to the fibre direction have an influence on the fracturing process in the matrix between the fibres. If the direction of the micro cracks is not parallel to the fibres the fibres can act more or less as crack stoppers, Fig. lc. A thorough theoretical and experimental verification is recommended. In Fig. 12 only in the fourth quadrant (1 0, 2 0) of the (1, 2)-stress combinations it is possible to compare the theory with a sufficient number of test data. It would have been very interesting to get to know, if there are also FFs at other points besides 2 = 0, compare Fig. 12 in [2]. Also it would have been absolutely necessary for the judgement of the UD-failure criteria to have some results within the third quadrant (1 < 0, 2 < 0). Within this quadrant one finds the most severe deviations between the new criteria and conventional ones like the Tsai,Wu-criterion [17]. The rare test data and the basic strength values XT, XC, YT, YC, being plotted in Fig. 12, could easily be approximated by a shifted and rotated Tsai,Wu-ellipse. However, from a physical point of view, one can have serious objections, Fig. 3 in [16,18]. A fundamental experimental investigation in the third quadrant, the compression/compressionquadrant, is mandatory in order to find out if the addition of only one transverse compressive stress – in this case 2 – does really increase the sustainable 1-compressive stress markedly. In the fourth quadrant the theoretical strength over-predicted the test results. The weakening of the resistances against IFF which is caused by a 1-stress seems to be more severe than we had expected in [2]. One can easily adopt the weakening factor fw (see Section 7.1 in [2]), which describes the influence of 1 on IFF-strengths to the results shown in the fourth quadrant of Fig. 12 by using a lower exponent n, which has been set to 6 or 8 as described in Section 7.1 in [2]. But before this is recommended one would like to see more complete experimental results.

855 Referring generally to Figs. 10–12 it should be emphasised that a more complete set of the experimental results would have been most desirable, because after all the (1, 2, 21)-fracture body is the basis of any failure analysis of laminates under an in-plane (x, y , xy)-state of stress. As long as there is such a severe uncertainty about the shape and magnitude of the (1, 2, 21)-fracture body and the occurring type of failure, as it results from Figs. 10–12, much of this uncertainty is transferred to the assessment of all following figures showing the fracture behaviour of laminates made up of UD-layers, the materials of which have been tested in isolated layers. 5.2. Non quasi-isotropic laminate with 3 fibre directions, + 30°, –30°, 90° Figs. 13 and 14 (in [2] and Part B) show the failure envelopes of the ( ± 30°, 90°) laminate for (x, y)- and (x, xy)-stress combinations, respectively. The fraction of the 90° fibres is less than 1/3, therefore, the laminate is not quasi-isotropic; it is stronger in the x-direction than in the y-direction. With the help of netting analysis, it is possible to gain a rough image of the flow of forces in such a laminate. The normal stress y is mainly transferred directly by the 1-stresses of the 90°-layers. Loads in the x-direction are mainly taken over by 1-stresses of the ± 30°-layers, while simultaneously 1-stresses with opposite sign arise in the 90°-layers, because the 90°-layers restrain the natural lateral contraction of the ± 30°-layers due to the relatively high Poisson’s ratio vxy . The shear stress xy is mainly transferred by 1-stresses in the ± 30°-layers;

Fig. 13. Biaxial failure envelope for ( ± 30°/90°) laminate under combined x and y stresses. Material: E-glass/LY556 epoxy.

856

Fig. 14. Biaxial failure stresses for ( ± 30°/90°) laminate under xy and x stresses.

in one layer we find 1 > 0, in the other 1 < 0. For positive shear stress xy the 1-stresses are 1( ± 30°) > 0 and 1(30°) < 0. With the help of these network reflections the course of the fracture envelopes which are completely assigned to the failure type of FF (see Figs. 13 and 14 in [2]) is to a certain extent predictable without any calculation, see also [19]. In Fig. 13 the correlation between the theoretically predicted (x, y)-failure envelope and the experimental results is fairly good in the first and fourth quadrant and bad in the second quadrant. We would expect rather good agreement between theory and experiment in the second quadrant because 1-compressive failure is predicted here for the 90°-fibres, see Fig. 13 in [2], and this is a “straightforward” situation. In the third quadrant a risk of delamination and local buckling (D/LB) is predicted by the analysis. This may be the reason for the low failure stresses in the experiments; but this should be checked by more tests with careful observations. Fig. 14 shows the fracture envelope of a ( ± 30°/90°) laminate under (x, xy)-stress combinations. Agreement of the theoretical predictions and the test results is generally satisfying. Relatively low test results in the area with high shear stresses xy and moderate direct stress x might possibly be traced back to torsional buckling of the tubular specimen. If we look at Fig. 14 in [2] we find in the same area the prediction for a severe risk of delamination and local buckling (D/LB) of the 30°-layers. This situation could also have contributed to the pronounced dent in the experimental (x, xy)-fracture envelope on the right side of the xy-axis. 5.3. ± 55° laminate As discussed in Sections 4.2 and 4.3 a laminate having two fibre directions 1 = +  and 2 = – is – based on principles of the netting analysis – only capable of transferring one specific state of stress. For a ± 55° laminate (more strictly speaking a ± arctan兹2 laminate) y : x = 2 : 1 is such a state of stress. This is verified by Fig. 15 (in [2] and Part B), as the failure envelope for all (x, y)-states of stress can be seen as a rather narrow spindle, the central axis of which follows the line y = 2 · x. In Fig. 15 in [2] we have used a broken line for a large proportion of the failure envelope in the tension/tension-quadrant, except for the region where y 2 · x (and for a small part of

857

Fig. 15. Biaxial failure stresses for ( ± 55°) E-glass/MY750 laminates.

the envelope close to the uniaxial y tensile stress), as there is no real failure, but a continuous deterioration of the fibre/matrix-composite combined with large deformations 21 0.2 after exceeding the crack onset limit. This has been explained extensively in Section 4.3. As the calculation predicts excessive shear deformations or showed the risk of local buckling by the wedge effect [see Section 2.4.3, Eq. (4)] the calculation was stopped. This is the reason why the “failure envelope” in Fig. 15 in [2] has been partly plotted as a broken line. The stresses x and y at our more or less arbitrarily chosen deformation limit ( 21 0.2) and wedge effect limit (tan fp 3) should not be treated as points of real failure in Fig. 15 of Part B. These limits should not be seen as sharp lines, instead they should be regarded as broad ribbons. Therefore, there is not a discontinuous failure envelope as Fig. 15 might suggest. There is no universally accepted definition of failure in these areas as discussed in full detail in Sections 2.4.3 and 4.3. The “failure load” due to unacceptable deterioration of the laminate may be considerably lower than the maximum load measured in a “strength” test. For instance, in Fig. 15 the measured uniaxial tensile strength in y-direction is remarkably higher than our calculated D/LB-critical stress due to the wedge effect. This is not surprising, because when we give the D/LB-warning, this does only mean that with further increasing load the risk of delamination and local buckling will also increase. We cannot predict at what load the “explosion” will really happen. It would be rather easy to adapt the parameters of our “failure” analysis ( 21 0.2 and tan fp 3) to the experimental results shown in Fig. 15. However, it seems questionable to us whether this would make sense if we keep the discussions of Section 4.3 in mind. The measured biaxial compressive strength of the ± 55° laminate at y : x = –2 : –1 is also higher than our prediction of D/LB. It might be that not the wedge effect has been responsible

858

Fig. 16. Stress strain curves for ± 55° GRP laminate under uniaxial tension (y /x = 1/0).

for failure in this case. Instead, the longitudinal compressive strength XC might have been reached. The stress/strain-diagrams in Fig. 16 refer to the “unsound” uniaxial state of stress y : x = 1 : 0 of a ± 55° laminate. The strain x has nearly the same absolute magnitude as the strain y belonging to the same stress, but of course a negative sign. That means that the responsible Poisson’s ratio of the laminate comes very close to vyx 1. It is interesting to note that under uniaxial tension (y) theory predicts failure due to transverse compression (2 < 0) according to Mode C. Not far above this IFF of Mode C the calculation has been stopped, because the stress has reached the level which we think is critical for D/LB [see Eq. (4)]. As discussed just before, it might be that IFF of Mode C does not lead so quickly to failure of the ± 55° laminate under the unsound uniaxial loading conditions as we had assumed using Eq. (4). That means that the stress/strain curve could have been continued to somewhat higher loads. But the open question which cannot be answered from Fig. 16 is: How does the specimen look like at y = –x 6%? To what extent has it already been disintegrated? Does it look like an activated energy-absorbing crash element, or more like an intact structural component? There is a very good agreement between the lower parts of the theoretical and experimental y(x)- and y(y)-stress/strain curves. Their pronounced non-linearity demonstrates the progressive deterioration of the laminate. Of course we could have continued the y(x)- and the y(y)-curves to higher y-stresses, but this would, as we feel, only have been of some academic interest. We think that in the design of a reliable load carrying structural element one has to stay away from our arbitrary limits 21 0.2 or tan fp 3, respectively, by nearly the same “factor of safety” which is recommended for total failure due to FF. It is a little surprising that in this case the leakage stress is rather close to our predicted Mode C-IFF, because IFFs of Mode C keep the cracks mostly closed. But perhaps a few small “explosions” have already opened some passage-ways for the liquid through the tube wall. The stress/strain diagram in Fig. 17 refers to the “reasonable” state of stress y : x = 2 : 1 of the ± 55° laminate. It shows the worst case of disagreement of our theoretical prediction and

859

Fig. 17. Stress strain curves for a ± 55° laminate made of E-glass/MY750 epoxy under y /x = 2/1.

experiment. At a first glance we could not give a plausible explanation why there is not such a good correlation between the theory and the experimental results for the strains x and y, as we would have expected for this “reasonable” loading condition. Examining this problem closer one has to consider that under progressive cracking, the laminate will act more and more like a ± arctan兹2-network. The ± arctan兹2-tube with internal pressure and closed ends will show the tendency to maintain or to re-establish the angle  = arctan兹2, which for a ± -network – because of equilibrium – is related to the stress ratio y : x = 2 : 1. As long as there is no crack onset, the axial strain x is remarkably smaller than the circumferencial strain y . This is in agreement with the theory of elasticity. It leads to a minor change of the angle , i.e. the angle  increases; because only if x = y – which is the condition for expansion under the constraint of geometric similarity – the angle  remains constant. If the fibre/matrix-composite is deteriorated by severe matrix cracking, the tendency of the fibre network to re-establish the angle  = arctan兹2 could become significant, because this angle is absolutely necessary for the equilibrium between the load and the fibre-forces. Therefore x will come closer to y . We would have expected an asymptotic approach of x to y . We could not see a plausible reason for the experimental results in Fig. 17 unless the winding angle has already been too large at the production of the tube. In this case x can become clearly greater than y . Our calculations in [2] were performed with the correct network fibre angle of  = arctan兹2. It is well known that only a slight deviation from this angle will cause a significant change in deformation [20]. Additional calculations using a fibre angle of  56° show a better correlation with the test data, see the broken lines in Fig. 17. At least the intersecting of the y(y)- and the x(x)-stress/strain curves is predicted now. Therefore, we strongly believe that the fibre angle of the pipe specimen has been higher than the exact angle co = arctan兹2. That something must have gone wrong with the experiment

860 has now been confirmed by additional investigations in the course of the revision of this paper as pointed out below and in the Appendix. If it is generally predicted, based on the consideration that due to the severe deterioration of the fibre/matrix composite the strains x and y should show an asymptotic approach to the straight line for 1 = 1.5y /E1 calculated by the netting theory, we inevitably come to the question why the degradation model for Mode A did not show this tendency. Because the revised paper has been returned with the same question an intensive search for the reason for the disagreement was started. There was no hope of answering the “question of guilt” by following up the very complex computer program. One could only see a chance for clearing up the “secret” by the derivation of very clear and interpretative analytical expressions for the strains x and y as they have been presented in the PhD thesis of the author [21]. Indeed very helpful analytical relationships, the derivation of which is presented in a condensed form in the Appendix were found. The laminate is a balanced angle ply laminate with x- and y-axes as axes of symmetry. Loaded with x, y 0 and xy = 0 there is no shear deformation with regard to the axes of symmetry x and y for the entire laminate as well as for the two layers with fibre directions 1 = +  and 2 = –:

xy = 0

(23)

With the deformation constraint xy = 0 for the UD-layers and the transformation rules for strains calculating with the fixed exact angle  = arctan兹2 – two extremely helpful relationships are found for the axial and circumferential strains of the laminate using the shear deformation 21 of layer 1 with 1 = + :

x = 1  y = 1 +

1 兹2 1

2兹2

21

(24)

21

(25)

From these equations the following conclusion can be drawn: The y(x, y)-stress/strain diagrams of a ± arctan兹2-angle ply laminate with y : x = 2 : 1 can easily be drawn starting from the nearly straight line y(1) which is the fibre parallel strain 1 versus circumferential stress y . In order to find the axial strain x one has to go to the left (to lower strains) by the amount of 21 /兹2. To find the circumferential strain y one has to go half that far to the right (to higher strains). From an analytical expression for 21 it follows, see Eq. (A7) in the Appendix, that by no means 21 can change its sign. Therefore, there is no chance for the x- and y-stress/strain curve to intersect like the experimental curves in Fig. 17 do at a stress y 400 MPa. This is true even if one takes into consideration that below IFF the strain y is larger than x and that this leads to an increase of . As soon as the original exact angle  = arctan兹2 is reestablished by y = x, there is no chance for x to become larger than y , as is clearly shown by Eqs. (24) and (25). Another interesting and absolutely surprising result from the analytical treatment of the problem presented in the Appendix is the following: The unplausible predictions of x (and y)

861 in Fig. 17 in [2] and which have been transferred to Fig. 17 of Part B, possibly resulting from the degradation method for IFF of Mode A which uses equal degradation factors  for G21s and E2s. The y(x)- and y(y)-curves can easily be corrected by using a higher -degradation-rate for the modulus E2s than for the in-plane shear modulus G21s (see Fig. 24). This is discussed in more detail in the Appendix. First of all reliable strain measurements are necessary now. It will then be no problem to model thereafter the measurements with a suitable -degradation.

5.4. Quasi-isotropic laminate with four fibre directions, 0°, + 45°, –45°, 90° Due to the symmetry of the laminate with respect to the x- and y-axis the risk of laminate failure does not change, when the stresses x and y are exchanged while keeping their magnitude. This is expressed in Fig. 18 by the fact that the diamond-shaped fracture envelope for (x, y)-combinations is symmetric to the line x = y . (Note that in Fig. 18 in [2] we have indeed mixed up the symbols x and y on the axes.) Therefore, it is sufficient to examine only one half of the fracture envelope as it has obviously been done when conducting the fracture tests. The nearly straight lines of the diamond-shaped fracture envelope are representing FFlimits for the 0°- or 90°-fibres, respectively. The flat straight lines belong to FF of the 0°-fibres and the steep straight lines indicate FF of the 90°-fibres as indicated in Fig. 18 in [2]. Again,

Fig. 18. Biaxial failure stresses for (0°/ ± 45°/90°)AS4/3501-6 laminates.

862 a confirmation of these predictions by experimental observations of fracture type and failed layer would have been very desirable. The test data in the first and the fourth quadrant are coming very close to both nearly straight parts of the theoretically deduced fracture curve. In the tension/compression quadrant the theory over-predicts the real strength a little. In the compression/compression quadrant there is no IFF and no risk that an explosion effect could happen before FF has been reached. The reason for this is that the transverse failure strain 2C is higher than the longitudinal failure strain 1C; see also Section 6.1. The few experimental results in the third quadrant – the compression/compression-quadrant – are much lower than the theoretical predictions. It was reported in [3] that buckling took place and caused a premature failure of the test specimens. It is interesting to realise that no FF occurs in the ± 45°-layers except for x = y . They suffer only IFF as can be seen from the stress/strain diagrams shown in Fig. 20 in [2]. This can be explained very easily. All fibres certainly have the same failure strain, but according to the strain transformation formula under biaxial (x, y)-stress the ± 45°-fibres can never reach their failure strain before that of the 0°- or 90°-fibres. Fig. 19 shows the stress/strain curves of a (0°, + 45°, –45°, 90°)-CFRP laminate for an uniaxial load in the y-direction; therefore y > 0 and x < 0. The stiff carbon fibres of this CFRP laminate carry the main portion of the forces. That is the reason why the calculation of the strains is not at all problematic and, as expected, there is a good correlation between theory and experiment. Fig. 19 in [2] shows the Mode A crack onset point, which is exceeded in the 0°-layer when about 1/4 of the fracture load of the laminate is reached. These early cracks of the 0°-layer do not influence the strain behaviour significantly, because – as it was already the case before crack onset – the affected layer transfers only a small part of the total force.

Fig. 19. Stress strain curves for (0°/ ± 45°/90°) laminate under uniaxial tension (y /x = 1/0).

863

Fig. 20. Stress strain curves for (0°/ ± 45°/90°) laminate under biaxial stress y /x = 2/1.

Fig. 20 shows the stress/strain curves for the same quasi-isotropic laminate under a biaxial state of stress y : x = 2 : 1. There is an excellent agreement between theory and experiment. As can be seen in Fig. 20 in [2], first the 0°-layer, then the ± 45°-layers and finally the 90°-layer consecutively exceed their crack onset limit, all three in IFF of Mode A. However, this does not have a noticeable effect on y and x. An easily visible change of the slope of the experimental y-curve can only be seen at y 450 MPa, which is far above the onset of IFFs of Mode A in Fig. 20 in [2]. This is because of the effects discussed in Section 4.2.

5.5. Cross ply laminates under “reasonable” and “unsound” loading conditions Fig. 21 shows the strains x and y of a ± 45° laminate for a stress ratio of y : x = 1 : 1, which is a reasonable loading condition. The measured strain x is up to 15% higher than the calculated x. The strain y is found to be even a little higher than x. Under the assumption of complete symmetry with respect to the x- and y-axis and the exchangeability of the x- and y-direction it follows from elementary laws of logic and engineering mechanics that the strains x and y should be equal. A more sophisticated calculation which takes the deviation of the test specimen from the idealised model into account would be interesting. If this could not explain the results given in Fig. 21 it could be necessary to repeat the tests. The strains x and y are dominated by the fibres. The state of stress y = x =  in this laminate is equivalent to the case of a state of stress y = x in a (0°/90°) laminate, because

864

Fig. 21. Stress strain curves for ± 45° E-glass/MY750 laminate under y /x = 1/1.

stress transformation into a co-ordinate system which is rotated by  = ± 45° gives () = x cos2  + y sin2  =  (cos2  + sin2 ) = . The leakage stress of about y 215 MPa is considerably higher than the calculated onset of IFF at about y 60 MPa. The reason for such a difference is explained in Section 4.2. For practical reasons, we start at this point with the discussion of the last figure comparing theoretical and experimental results (Fig. 23) which also refers to a (0°/90°) laminate, but this time stressed in one fibre direction only. The state of stress is uniaxial with x only. The negative transverse strain y is of very small magnitude, since the fibres oriented in the 90°-direction strongly “oppose” the natural lateral contraction due to tensile stress of the 0°-fibres and the matrix between them. Crack onset begins in the 90°-layer and follows later in the 0°-layer (see Fig. 23 in [2]), until catastrophic fracture occurs in the form of fibre tensile failure of the 0°-fibres. As expected the agreement between theoretical and experimental stress/strain-curves is very good. From Table 1 in [23] the longitudinal tensile strength of the UD-layer can be found XT = 1280 MPa. If it is assumed that at failure only the 0°-layers are carrying the load the stress x at failure would be 0.5
1280 MPa. Our prediction of 660 MPa can be interpreted by the load sharing of the cracked 90°-layers. The observed cracking limit (of the 90°-layers) and the splitting limit (of the 0°-layers) were about twice that predicted in Fig. 23 in [2]. This difference is a little higher than expected according to Section 4.2. Fig. 22 refers to a totally different situation, i.e. the behaviour of a cross ply laminate under an “unsound loading condition.” The loading condition given for a ± 45° laminate with a stress ratio of y : x = 1 : –1 is equivalent to the case of a (0°/90°) laminate under a pure shearstress xy , as already discussed in Section 4.3. Consequently, the unidirectional layers

865

Fig. 22. Stress strain curves for ± 45° E-glass/MY750 laminate under y /x = 1/–1.

exclusively bear a 21-stress, which is “unsound” because the stress is transferred completely through the matrix. From the strain transformation rules follows that y = –x = 0.5 21. Roughly speaking the (y)-stress/strain diagram of the ± 45° laminate should look like a 21( 21)-diagram, however, “compressed” to one half in the direction of the strain axis. The shear deformation 21 at failure should be a little higher in a (0°/90°) laminate than the shear-deformation at fracture of an isolated unidirectional layer, since in the laminate the IFFcracks in one layer are bridged by fibres of the other layer, while in an isolated unidirectional layer already the first single IFF leads to total separation, see Section 4.3. We cannot explain why in the transition area from the steep to the shallow part of the stress/ strain curve the experimental results are lower than the predicted ones. It may be that we did assume too low residual stresses, which are tensile transverse stresses 2 in this case, and, therefore, lead to an earlier damage in the matrix than it happens in an isolated layer, see Fig. 1 in [2]. One would expect a “knee” in the y(y)- and y(x)-stress/strain diagram at the point of IFF and from there on a continuation of the stress/strain diagrams as straight horizontal lines, because we keep fE(IFF) = 1 If we believe that 21 is the only stress in the layers, 21 = y should also remain constant in order to keep fE(IFF) = 1 = const. In reality, however, curious things happen, which can only be discovered using the ability of the computer program to feed back the changes of the fibre directions caused by the strains of the laminate. After IFF, the strains are increasing rapidly and the resulting changes of the fibres directions activate growing 2 compressive stresses in the layers, which are increasing

866

Fig. 23. Stress strain curves for 0°/90° E-glass/MY750 laminate under uniaxial tension (y = 0).

with the strains. Since after the onset of IFF of mode B an -degradation of E2s is not allowed, because 2 and with this n is a compressive stress, the shear modulus G⊥储 has to decrease rapidly in order to keep fE(IFF) = constant. The simultaneously arising 1-tensile stresses and 2compressive stresses take over a small portion of the xy-shear load. Very soon an IFF-Mode C situation is arrived with its risk of delamination and local buckling (D/LB). The calculation is stopped according to Eq. (4). One result of this rather sophisticated treatment of the redistribution of the stresses leads to the consequence that we should have designated the onset of IFF not by IFF (Mode A) but by IFF (Mode B). At the beginning of the loading process there is a pure shear stress 21 = y = –x. But with the changes of fibre directions caused by the strains the state of stress immediately changes to a (2, 21)-combination with compressive 2, and this leads to an IFF of Mode B and later C, see Fig. 3. It was also not correct to designate the endpoint of our computed stress/strain curves as “total failure”, as it has been done in Fig. 22 in [2]. Instead we should have indicated Mode C (D/LB). The warning D/LB only gives the signal that an increasing risk of D/LB is to be expected. However, on the other hand we are quite convinced that a laminate in a structural member can not be used beyond this point.

867 6. General discussion on the comparison of theory and experiments 6.1. The three categories of laminate failure It has been learned from the Exercise that in the wide field of failure analysis of laminates under in-plane stresses three main categories of problems can be distinguished. 6.1.1. Laminates with three or more fibre directions • The failure analysis of these laminates is “the easy case” as has been demonstrated on the ( ± 30°/90°) laminate and the (0°/ ± 45°/90°) laminate. • Final failure is caused by FF in at least one fibre direction. • IFF is in most cases of Mode A. The high shear stresses 21 which are necessary to produce an IFF of Mode B do not occur because the corresponding high shear strains 21 are prevented by the fibres which take over the main portion of shear loads. A good example is a (0°/ ± 45°/90°) laminate which presents a high stiffness against shear-stresses as well as against a (y : x = 1 : –1)-loading which is the same as a shear load corresponding to the + 45°- and – 45°-axis. • IFF of Mode C is unlikely to occur in laminates of this type if these are made with carbon fibres, because the transverse compressive failure strain 2C is about twice the longitudinal compressive failure strain 1C. This is the reason why for instance compressive FF of the 0°-layers will happen before transverse compressive failure of Mode C can occur in the 90°-layers, see Fig. 18. (The opposite is true for glass fibre laminates.) • The stress/strain curves of these strong and “stable” laminates are nearly linear and the choice of a specific -degradation model is rather insignificant for the resulting stress/strain diagrams and failure predictions. Unfortunately, these laminates are not for all applications and load cases the optimum solution. It should be mentioned that (0°/90°)-cross ply laminates behave like laminates with three or more fibre directions as long as the loads are only acting exactly in one or both fibre directions. However, in real components this can normally not be guaranteed. As a consequence shear loads xy occur which have to be transferred by the matrix. 6.1.2. Angle ply laminates with two fibre directions and specific loading conditions in accordance with netting analysis • If the stress ratio is exactly that specific one which is specified by the netting analysis, these laminates can also fail by FF at the corresponding high stress level. A good example is a ± arctan兹2-pressure vessel with a stress ratio y : x = 2 : 1. • The strains of these laminates seem to be very sensitive to the degradation process. • For the theoretical degradation model it seems to be reasonable to use a faster -degradation for the transverse modulus E2s than for the in plane shear modulus G21s, see Section 5.3 and the Appendix.

868 6.1.3. Laminates with two fibre directions and loading conditions which cannot be treated by netting analysis • These laminates are characterised by the fact that the loads cannot be carried by the fibre network alone. Therefore, the laminates fail at rather low stress levels, which are not generally adequate for fibre “reinforced” materials. • Their stress/strain diagrams are non-linear below the IFF-limit if shear stresses 21 are predominant. • Straining these laminates above the onset of IFF leads to a rapid deterioration of the fibre matrix compound. It is difficult to define a limit for an allowable degree of this deterioration. • The theoretical prediction of the behaviour above IFF onset depends totally on the chosen degradation model. • It seems to be necessary to incorporate into the stress- and strain-analysis a feed back of the large deformations of these laminates which move the fibres out of their original directions. In the two examples shown, the ± 55° laminate with y : x = 1 : 0 and the ± 45° laminate with y : x = 1 : –1, the first IFFs were of Mode C and Mode A respectively. However, due to the fibre directions changing with laminate deformation and the fact that in order to keep fE(IFF) = 1 for n = 2 cos2 fp < 0 only G21s is degraded and not E2s, the stress ratio 2 : 21 proceeds very soon to a Mode C-situation. The unexpected consequence is that a risk of a wedge- or explosion-effect with the danger of delamination and local buckling (D/LB) will occur. Without a feedback of changing fibre directions one would have expected harmless IFFs of Mode A or Mode B, see Section 5.5.

6.2. Lesson learned from the ± 55° laminate An interesting lesson learned from the Exercise was how to treat the degradation of E2s, G21s and v12 for a ±  = ± arctan兹2-tube with y : x = 2 : 1 in order to get an asymptotic approach to the theoretically expected equal strains x and y at the end of the degradation process. So far only the logical conclusion was used that a completely degraded laminate cannot transfer 2-stresses and 21-stresses and that the Poisson’s effect is not present any longer. It is quite obvious that a totally deteriorated FRP laminate can only behave like a fibre network. However, we did obviously not answer the question: How does the fibre network get the required fibre angle which is necessary to provide equilibrium with the external loads? In the case of the ± arctan兹2 laminate the fibre network is compelled to restore and then fix the situation of equal strains x and y , because this is the only way to keep the fibre angle ± arctan兹2 constant which is necessary for equilibrium. In the model with fixed  one can only accomplish this by applying a higher degradation-rate for E2s than for G21s. One is able now to meet corresponding experimental stress/strain diagrams very well by a modified theoretical model, see Fig. 24. On the other hand one has to confess that it is not understood from a physical point of view how different degradation rates of E2s and G21s could be accomplished in the real laminate. We shall certainly try to find out what happens in reality. It is not believed that the different degradation rate of E2s and G21s is a general law for all kinds of laminates and loadings. At the moment the different degradation rate for E2s and G21s is regarded only as a convenient and very sensitive method, which can easily be adapted to strain

869

Fig. 24. Stress/strain curves for ± arctan兹2 laminate made of GRP material (y : x = 2 : 1) calculated with different degradation factors : a) equal  for E2s, G21s and v12; r = 0, b). equal  with r = 0 for G21s and v12; r = 0.25 for E2s, c) equal  with r = 0 for E2s and v12; r = 0.25 for G21s. In the Exercise, Part A, the version a has been used. The version c shows the expected asymptotic approach of x and y to 1. The b version makes things worse.

measurements on ±  angle-ply laminates with ±  = arctan兹y /x without the necessity of a feedback of the change of fibre direction due to the strains of the laminate. 6.3. The weak point in our failure theory We feel that the presented theoretical treatment of IFFs of Mode C is rather unique and very helpful for the interpretation of failures which looked very obscure for a long time, as for example explosive failures of some early prototypes of GFRP torsional springs. However, it is quite obvious that the recently developed model, which has been described in Section 2.4.3, is far from full maturity at this time. Therefore, we are still working on its improvement. The first step which has to be done is to incorporate the influence of a longitudinal shear stress 21, which has been neglected3 in Fig. 9 in [2] and also in Section 2.4.3 of this paper. The shear stress 21 does not contribute to the “exploding” force n3 (see Fig. 9d in [2]) but helps to overcome the frictional force (per unit width)  | n | t *. The fracture angle fp at which the driving shear force is just of the same magnitude as the restricting friction force would then follow from

tan fp =

冑

冉冊 冉冊

2 

21 2

21 1+ 2

2

2

(26)

3 In Fig. 9a and b in [2] an influence of 21 on the mutual movement of the broken pieces is demonstrated (with a negative 21 for better visability of the movement), however, the influence of 21 on the wedge effect is neglected in the theoretical treatment in Section 8.2 of [2].

870 This equation is the equivalent to Eq. (3) in Section 2.4.3, in which – for the first step – the influence of 21 has been neglected. Beyond this refinement a measure for the absolute magnitude of the explosive force n3 in Fig. 9d in [2] should be found. For this reason not only the critical fracture angle fp but also the relative thickness of the layer (or layers) with the oblique fracture plane has to be incorporated in a formula for the assessment of the risk of delamination and local buckling (D/LB). 7. Conclusions The results presented in [2] and their discussion in Part B have shown that our theoretical models work reasonably well for predicting failure for all three categories of laminate failure problems as described in Section 6.1. Naturally, one has to accept considerable uncertainty in the prediction of “failure” when too large deformations ( 21 > 0.2) due to a progressive deterioration or some “explosions” (fp 45°), followed by delamination and local buckling (D/LB), are likely to become the reason of “failure.” Probably as we learned from the experimental results of the Exercise our warnings “D/LB!” were given a little too early. The reason for this is that we want to give an early warning to the designer in order to let him try to avoid such critical situations in his components. The computer program used in the Exercise has been developed as a design tool. It is one version of a whole design assistance analysis package. It has not been conceived for the solution of fundamental problems of material science. Nevertheless, we could certainly modify the computer code to a certain degree in order to solve problems with large deformations and rapid deterioration, because – in spite of the fact that we use phenomenological approaches – the stress/strain and failure analysis has a reasonable physical basis. However, before spending more effort on this task we would like to ask for a broad international discussion on the question how relevant the problems of laminates with “unsound loading conditions” really are for design practice and for the fundamental understanding of damage and failure of laminates. In general it has to be stated for the Exercise that a complete and convincing judgement of the theoretical predictions has not been possible because for many problems the location of the failure and type of failure (FF or IFF) as well as Mode of IFF (A, B, C) have not been reported from the experiments. Some of the reported experimental results are doubtful as, for instance, the intersecting curves for x and y of the ± 55°-tube with y : x = 2 : 1 and the unequal strains x and y for the ± 45°-tube with y : x = 1 : 1. This is not at all surprising. It is known from our experience how extremely difficult tests with combined stresses are. In our broad industrial practice as designers of high strength FRP components we have very often experienced that disagreements of theoretical predictions and experimental results could finally be traced back to the fact that something has gone wrong with the experiments. In order to further increase the magnificent value of the whole Exercise an international effort should be encouraged to improve the basis on the experimental side. As already discussed in detail in Section 5.1 it seems absolutely necessary to improve the knowledge of the mechanical behaviour of the UD-layer itself – including thickness effects and the in situ effect – because this is the basis for any laminate failure analysis. In the Exercise the most sophisticated version of the computer program package NOLI FRAN COLAM (non-linear fracture analysis of composite laminates) has been used. The

871 most primitive version within this package which is called the “Quick Design Version” has been created for the design of carbon fibre laminates with three or more fibre directions. In this case the stress/strain analysis is performed with constant moduli E2 and G21 which are chosen from the 2(2)- or 21( 21)-diagram, respectively, in between the corresponding values at zero-strain and strain at failure. The value of  is found using the former method as explained in Section 2.4. The influence of the -degradation on the strains of the laminates is neglected. However, the value of  gives a realistic idea of the crack density in the layers in which IFFs have occurred. By this procedure the designer gets a very quick survey of all FFs and IFFs which are likely to occur at certain load levels and he is enabled to improve his laminate design very quickly [6]. For a while “science” should concentrate with highest priority on the development of effective design tools in order to get the cost and the time for the development of strength relevant FRP-components down to a level which is acceptable in view of the severe competition on the market. At the moment the practice of failure analysis of FRP components is far behind what is needed in industry. In this context it is a very important task to incorporate the new action plane related IFFcriteria into commercial FEM software. This is on the way now at some places. An extemal routine PUCKAN conducts a fracture analysis with the Puck-criteria on the basis of a stressanalysis with the FE-system ANSYS.4 The routine uses the ANSYS data-file as input and returns the results of the fracture analysis (FF- and IFF-stress exposure factor, fracture-mode and fracture angle) as a data-file to ANSYS. The results are visualised with ANSYSpostprocessing. The Puck-criteria exist also in the form of an internal ANSYS-routine. This version has been developed at the Institute of Plastics Processing (IKV) in Aachen and will be available soon.5 Asea Brown Boveri Corporate Research6 has developed a Puck-routine that post-processes ABAQUS *.fil data and writes a *.unv file for visualisation with IDEAS. The next logical step would certainly be to incorporate our “Quick Design Version” for post-IFF analysis into commercial stress/strain and failure analysis software. Without any doubt the results of the Failure Exercise demonstrate an urgent need for realistic failure models and reliable procedures. Thanks to the organisers of the Exercise the scientific discussion about the failure analysis of composites – which seemed to have fallen asleep for at least two decades – has been reanimated. Apparently science sometimes reacts rather slowly as has also been shown by the Exercise itself. Now – pushed by the insights which the Exercise supplies – we should really speed up. Otherwise the gap between the available instruments and what is really needed by industry could grow even further. Acknowledgements The authors would like to thank Dr. D. Huybrechts from ABB, Heidelberg, and M. Knops and Dr. J. Kopp of the Composite Department of the IKV, Aachen, for scientific discussion of the 4

Contact: CAD-FEM, Hanover, Dipl.-Ing. Martin Kracht, D-31303 Burgdorf, e-mail: [email protected] 5 Contact: IKV Composites, Aachen, Dipl.-Ing. Martin Knops, email: [email protected] 6 Contact: ABB Corporate Research, Heidelberg, Dr.-Ing. Dirk Hubrechts, e-mail: [email protected]

872 non-confidential parts of this paper. Special thanks are due to Mrs. K. Schmitt, who patiently typed several versions of this paper until it reached its final form.

Note added in proof After the authors had finished the final version of this paper they felt that the reasons for the mysterious unusual discrepancy between theoretical predictions and experimental results for the axial strain x and circumferential strain y of an ± “55°”-tube with y : x = 2 : 1 had not completely been clarified and that a more elaborate experimental and theoretical study would be necessary. In the meantime Mr. Knops from the Composite Department of the Institute of Plastics Processing (IKV), Aachen, incorporated such a study in his research program [12], and first results have just been published [24]. His experiments verify the authors suspicion that the winding angle of the specimens used by the organisers of the Excercise (their results are shown in Fig. 17) was too large ( 56°). He found an extraordinary sensitivity to small deviations from the netting theory angle arctan 兹2 = 54.74°. Unfortunately, also a mistake on the side of the calculation has obviously contributed to the extreme deviation between theory and experiment. It seems that the variations in fibre direction caused by the difference of the strains x and y had not been recognised when the theoretical predictions of Fig. 17 were computed. The reason for this error can probably not be followed back, because it is nearly impossible to trace the computing back to 1995 when the contribution [2] to Part A was written. Perhaps the ability of the computer code to feed back the deformations had not been activated, because no “large deformations” were expected in this case of a “reasonable” loading condition (compare the end of Section 8.1 in [2]). Knops demonstrated in [24] that computing with deformation feed back and using the same degradation factor  for E2s, G21s and v12 results in the expected asymptotically converging curves for x and y versus y , (similar to those shown in Fig. 24, case c, where a slower degradation of G21s – compared to the degradation of E2s was applied by choosing r = 0.25 for the degradation of G21s and r = 0 for E2s and v12).

References [1] Hinton MJ, Soden PD. Predicting failure in composite laminates: the background to the exercise. Comp Sci Technol 1998;58(7):1001–10. [2] Puck A, Schurmann H. Failure analysis of FRP laminates by means of physically based phenomenological models. Comp Sci Technol 1998;58(7):1045–67. [3] Soden PD, Hinton MJ, Kaddour AS. Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates: Failure exercise benchmark data: V62, pp. 1489–1514, 2002, see also Chapter 2.2 in this book. [4] Puck, A. and Schneider, W., On failure mechanisms and failure criteria of filament-wound glassfibre/resin composites. Plastics and Polymers 1969: 33–44. [5] Puck A. Calculating the strength of glass fibre/plastic laminates under combined load. Kunststoffe, German Plastics,7 1969;59:1819 (German text pp. 780–787). [6] Puck A. Festigkeitsanalyse von Faser-Matrix-Laminaten, Modelle fur die Praxis (Strength analysis of fibre-matrix/laminates, models for design practice). Munich: Carl Hanser Verlag, 1996 [in German].

873 [7] Mohr O. Welche Umstande bedingen die Elastizitatsgrenze und den Bruch eines Materials? (Which circumstances are causing yield limit and fracture of a material?). Civilingenieur 1900;44: 1524 30 and 1572–1577 [in German]. [8] Hashin Z. Failure criteria for unidirectional fiber composites. J Appl Mech 1980;47:329–34. [9] Huybrechts DG. Ein erster Beitrag zur Venfikation des wirkebenenbezogenen Zwischenfaserbruchkriteriums nach Puck [A first contribution in verifying the action-plane-related inter-fiber-failure criterion after Puck]. Band 44, Verlag der Augustinus Buchhandlung, Aachen, “D 82 Diss. RWTH Aachen”, 1996 [in German]. [10] Cuntze R, Deska R, Gollwitzer S, Huybrechts D, Jeltsch-Fricker R, Kopp J, et al. Neue Bruchkriterien und Festigkeitsnachweise fur unidirektionalen Faserkunststoffverbund unter mehrachsiger Beanspruchung- Modellbildung und Experimente [New failure criteria and strength analysis for unidirectional fibre/plasticcomposites under multiaxial load- modelling and experiments]. Fortschritt-Benchte VDI, Reihe 5 Nr. 506, Dusseldorf: VDI Verlag, 1997 [in German]. [11] Kopp J. Zur Spannungs- und Festigkeitsanalyse von unidirektionalen Faserverbundkunststoffen [Stress and strength analysis of unidirectional fibre/matrix composites]. Dissenation, technisch wissenschaftlicher Bencht, Verlag Mainz, Aachen, 2000 [in German]. [12] Knops M. Investigation of the successive fracture process of laminates made of UD-layers. Proceedings of the 20. Plastics Technol. Colloquium, Block 10, Aachen, Germany, 2000 [in German]. [13] Paul B. A modification of the Coulomb-Mohr theory of fracture. J Appl Mech, June 1961;2: 259 68. [14] Puck A. A failure cnteria shows the direction. Kunststoffe. German Plastic 1992;82:29–32 [German text pp. 607–610]. [15] Puck A, Kopp J, Knops M. Guidelines for the determination of the parameters in Puck’s action plane strength criterion. Comp Sci Technol 2002 Compos Sci Techno, V62, pp 371–378, 2002. [16] Michaeli W, Kopp J, Puck A. Modellbildung und Experimente zu einer physikalisch begrundeten Festigkeits hypothese fur FaserMatrix-Verbunde [Modelling and experiments to a physically based strength hypothesis for fibre/matrix-composites], Frankfud: Verbundwerkstoffe und Werkstoffverbunde, DGM Informationsgesellschaft mbH, 1997. pp 751–8 [in German]. [17] Lui K-S, Tsai SW. A progressive quadratic failure criterion for a laminate. Comp Sci Technol 1998;58(7):1023–32. [18] Puck A. Progress in composites component design through advanced failure models. 17. Conf. of the Europ. Chap. of the Soc. for the Advancement of Mat. and Proc. Engin. (SAMPE), 28–30 May 1996, Basel, pp. 83–96. [19] Hart-Smith LJ. Predictions of the original and truncated maximum-strain failure models for certain fibrous composite laminates. Comp Sci Technol 1998;58(7):1151–78. [20] Forster R, Knappe W. Experimentelle und theoretische Untersuchungen zur RiLlbildungsgrenze an zweischichtigen Wickelrohren aus Glasfaser/Kunststoff unter Innendruck [Experimental and theoretical investigations of the crack initiation limit of filamentwound glass-fibre/resin tubes with two fibre directions under internal pressure]. Kunststoffe 1971;61:583–8 [in German]. [21] Puck A. The stresses and strains of GRP-multilayer composite components, Part 1: Fundamentals of the stress and strain analysis. Kunststoffe, German Plastic 1967;57:22–4 (German text pp. 284–293). [22] Soden PD, Hinton MJ, Kaddour AS. A companson of the predictive capabilibes of current failure theory for composite laminates. Comp Sci Technol 1998;58(7):1225–54. [23] Soden PD, Hinton MJ, Kaddour AS. Lamina properties, lay up configurations and loading conditions for a range of fibre-reinforced composite laminates. Comp Sci Technol 1998;58(7): 1011–22. [24] Michaeli W, Knops M. Stress and strength analysis of structural components with inter fibre failure: experimental and theoretical work. Proc. 47th Internat. SAMPE Conf., May 2002, Long Beach, CA, USA.

874 Appendix. Analytical investigations into stresses and strains of a ± arctan兹2 angle-ply laminate with biaxial stress ␴y : ␴x = 2 : 1 Stress and strain analysis The two UD-layers of the balanced angle-ply laminate have fibre directions 1 = +  and 2 = – with  = arctan兹2. Calculated are the stresses and strains of layer 1 with 1 = + . (The angle of a UD-layer is measured from the x direction, see Fig. 8 in [23].) Since the xand the y-axis are axes of symmetry the same stress x and the same stress y are acting in the two layers, having the same magnitude as x and y , respectively, of the entire laminate. There is no external shear stress xy acting on the laminate, therefore there is also no shear deformation xy , Eq. (23), of the entire laminate and the two layers. This deformation constraint is the reason why shear stresses txy1 and xy2 of equal magnitude but with different signs do occur in the two layers. For the use of stress and strain transformation rules the values of sin  and cos  are needed. Neglecting any changes of  with the laminate strains these values can be found as sin  = 兹2/3 and cos  = 1/兹3 from tan  = 兹2. On this basis and with the deformation constraint xy = 0 the two very helpful Eqs. (24) and (25), for the axial and circumferential strains, see Section 5.3, are found, which are repeated here:

x = 1 

y = 1 +

1 兹2 1

2兹2

21

(A1)

21

(A2)

1 is the longitudinal strain and 21 the shear strain of layer 1 with 1 = + . Using the stress transformation rules very clear expressions for the stresses 1, 2, 21 of layer 1 are found, too: 1 1 = (5 + 4Da )y 6

(A3)

2 2 = (1  Da )y 3

(A4)

21 =

兹2 (1  Da )y 6

(A5)

The parameter Da is a combination of the secant moduli E2s, G21s and the major Poisson’s ratio v12. It represents the degree of anisotropy; for isotropic material Da = 0 is valid and for UDmaterial degraded to a “network” with the only elastic constant E1 left Da = 1 results. The Eqs. (A3) to (A5) show very clearly the redistribution of the stresses due to degradation (Da→1), which leads to a concentration of the loads in the fibres. The sum of 1 and 2 results in independently of the value for Da as: (1 + 2) = 1.5y .

875 Starting from xy = 0, using the law of elasticity for a UD-layer and considering already a general -degradation the following expression for Da() is found: G21s() G21s()  2[5 + v12()] E2s() E1 Da() = G21s() G21s() + 8[1 + 2v12()] 1+8 E2s() E1 1+8

(A6)

If the prediction that with increasing 11-degradation the stress/strain-curves for x and y should asymptotically approach the line for 1 = 1.5y /E1 (longitudinal strain resulting from netting analysis) is true, it follows from Eqs. (A1) and (A2) that 21 must be brought to zero at total degradation by a proper -degradation. What a proper -degradation is can be judged by an analytical expression for 21(). With the aid of Eqs. (A5), (A6) and 21() = 21()/ G21s() the following analytical expression for 21() results:

冋

21() = E1

3兹2[1 + v12()]

册

G21s() 1+8 + 8G21s()[1 + 2v12()] E2s()

· y

(A7)

This expression for 21() turns out to be extremely insensitive to an -degradation which uses the same degradation factor  for E2s, G21s and v12. With the elastic properties given in the last column of Table 1 of [23] the following results are found:

21() = 0.99y /E1

for

 = 1;

21() = 1.07y /E1

for

 = 0.5

21() = 1.09y /E1

for

 = 0.

and

This insensitivity of 21() to the value of  is the reason for the nearly straight stress/strain curves y(x) and y(y) which have been presented in Fig. 17. It has become obvious now that the shear strain 21() can only approach zero for →0 if the denominator of Eq. (A7) approaches &. This is possible only if E2() approaches zero faster than G21() as can be seen from the second term in the first bracket of the denominator. Correction of the prediction of strains ␧x and ␧y by a modified ␩-degradation In order to correct the questionable predictions of the strains x and y as shown in Fig. 17, it is most convenient to use the former degradation method described in Section 2.4.1 because the involved formula for , Eq. (1), gives the desired additional flexibility. For a corrected prediction of x and y a remaining stiffness term r for the in-plane shear modulus G21s is set to r = 0.25. However, for the transverse modulus E2s (and v12) r = 0 is used. For the factor c and the exponent ( in Eq. (1) the recommended figures c = 4 and ( = 2 are used in the calculation of  for G21s, as well as for E2s and v12. With this minor change of the degradation model, results for x and y are achieved which show the expected asymptotic approach to the nearly straight line for 1, see Fig. 24. The figure also demonstrates an extreme sensibility of the strains to small differences of the y-degradation-rate for E2s and G21s. This could explain the large divergence of the predicted strains in several contributions to the Exercise which are demonstrated by Fig. 16 of [22].

876 Comment on the experimental results for ␧x and ␧y Another important conclusion can be drawn from Eqs. (Al), (A2) and (A7). From Eq. (A7) it can be seen that for an ± arctan兹2-laminate under no circumstances 21() can change its sign. It follows then from Eqs. (A1) and (A2) that the stress/strain curves y(x) and y(y) can not intersect like the experimental stress/strain results shown in Fig. 17. Even if the real winding angle has been 55° instead of arctan 兹2 = 54.74° we do not expect intersecting y(x) and y(y) curves. Therefore, we feel that something has gone wrong with the experiment. These results should not be used for the judgement of theoretical predictions. For further discussion see Sections 5.3 and 6.2 and also Ref [24].7

7

“German Plastics” (nowadays “Plast Europe)” is the bilingual edition German + English of “Kunststoffe.”

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

877

CHAPTER 5.7

The Rotem failure criterion: Theory and practice A. Rotem Faculty of Mechanical Engineering, Technion-IIT, Haifa, Israel Abstract In part A of the exercise, the failure criterion was presented and some failure envelopes were calculated on the basis of some materials properties that were supplied by the organizers. Here, in part B, some experimental results, also supplied by the organizers, are analyzed and compared with the theoretical predictions. In some cases, there is a large deviation between theory and practice. It is shown that the huge scatter in the experimental results and the arbitrary selection of some tests and material properties might be the reason for that. It is also suggested that, in some cases, a different mode of failure that is not covered by this criterion caused the failure. Keywords: Laminate failure; C. Failure criterion

1. Introduction The Rotem failure criterion for fibrous composite laminates is based on a separation between the failure modes of the matrix and the fibers, as was explained in part A [1]. The parameters of the criterion are the static strengths in the major directions of the lamina, that is, in the fibers direction, perpendicular to the fibers and in-plane shear, as well as the stress in the matrix in the fibers direction, given by matrix and fibers moduli ratio. Predictions of laminate failure are based on the calculation of the total stress fields in each lamina of the laminate, and plug in the rotated stresses to the failure criterion and find the condition of the first failure in the first lamina. Adjacent laminae contribute to the strength of this lamina by the bridging effect and therefore the strengths are multiplied by a factor that is not known at this point and is therefore assumed on the basis of experience with some composite laminates. Afterwards, redistribution of the load and progressive change of laminae stiffness, will eventually cause first failure conditions in another lamina and so on until a condition of first failure exists in all laminae. This stage is called ‘final failure’. It is not the final rupture of the material but rather the existence of cracks in all the laminae. Sometimes the final rupture will occur at a much higher stress level because the load is transferred by the bridging effect of neighbors laminae over the cracked one. Therefore, many cracks may occur E-mail address: [email protected] (A. Rotem).

878 in each lamina before final rupture. The way to predict this load level is yet unknown. The criterion presented here is dealing only with the appearance of the first crack in all the laminae. The strengthening of the embedded lamina up to the first crack and the rate of stiffness change are two parameters that are needed for prediction of laminate behavior. These two parameters were not known and therefore were estimated. The parameters depend on the elastic moduli ratio and the angle between neighbors laminae. The predictions are based on the assumption that the failure parameters are exact quantity and the materials behavior are in the same manner each time they are tested. Of course, it might be that the strength parameters are not discrete values but rather statistical distributions, like Normal (Gaussian) [2] or Weibull [3]. However, if the distributions are sharp, i.e. concentrate around the average, with a standard deviation of few per cents, a simple calculation may be done by taking the average value as a discrete and the error in the predictions will be small. But if the distributions are spread, that is the standard deviation is not negligible, the failure prediction will also be distributed accordingly. In this exercise, given in part A [1], the materials parameters were given as discrete values, which implied that their distribution is very sharp. Unfortunately, that is not the case as was discovered in part B, here. This can be recognized easily by examining the experimental results in Figs. 1–10 of [4]. Therefore, I will explain, where it is appropriate, the discrepancies between predictions and experimental results based on these fundamental change in the way that calculations should be done.

2. Results analysis The experimental program as presented in part B of the exercise [4], is actually a gathering of few experimental programs, performed by different laboratories in different places and at different times (some tests were made over 25 years ago). The reproducibility of polymeric materials is not very good and even material made by the same manufacturer in small time interval (i.e. different batches) may exhibit large differences in the material properties [5]. Therefore, there is no wonder that in this case there was a large scatter in the experimental results for the same type of loading. In view of the wide spread of the experimental results, each test type will be analyzed, and remarks or corrections will be given to the various figures. Fig. 1: this figure established the properties of material I lamina. The experimental results for this unidirectional laminate suggest a different value for the transverse compression strength than was given by the organizers. A value of –140 MPa is more suitable as seen from the experimental results. Fig. 1(b) shows both prediction lines, the –114 MPa and the –140 MPa. However, only for the compression test there is more than one test, while there is no repeated tests of other load conditions. Therefore, there is no knowledge of the average values for these load conditions. For example, the shear strength value is also not accurate but there is only one experimental value for pure shear, which is not enough for averaging and therefore left as it was suggested. There are two points of failure with shear values higher than for pure shear. One explanation might be that compression cause friction that seems as higher strength. Fig. 2: this figure establishes the properties of a lamina made of material II. The experimental results are quite scattered. The values obtained for shear strength by MBB are

879

Fig. 1. Biaxial failure stresses for 0° lamina made of GRP material E-Glass/LY556/HT907/DY063.

higher by 50% than those obtained by DFLR. Obviously this fact put a question on the validity of the shear strength value. Another unexplained result of DFLR is the strengthening by 100% of the shear strength when axial tension is introduced. There is no logical explanation for this behavior, except for test conditions. Apart from these four test results, the others agree quite well with the prediction of separation between the fiber strength and the matrix strength. Fig. 3: this figure establishes the properties of a lamina made of material III. However, the experimental results do not include any data for pure shear, axial compression or perpendicular tension. It was not known where and how these values were measured. The graphs of shear-stress shear-strain that were given do not specify any information (like number of tests and method of measuring). Later, that is after part A of the exercise was

Fig. 2. Biaxial failure stresses for 0° 1amina made of CFRP material T300/914C.

880

Fig. 3. Biaxial failure envelope of 0° GRP lamina under combined x and y stresses. Material: E-glass/ MY750 epoxy.

completed, it was reported [4] that the tests were performed on ± 85° tubes, and the strengths were calculated in the axial and the hoop directions. In this case, the perpendicular tension strength (in the axial direction) was different from the 90° because of the bridging effect. It is not clear whether a correction was made and the strength that was given as basic property of the lamina is for 90° or ± 85°. A new graph is presented [Fig. 3(b)], which was calculated for a laminate of ± 5° where the input is the strength of unidirectional laminate (it seems that the axes X and Y were exchanged by the organizers). It can be seen in the graph that the perpendicular strength in this case is higher. However, it is not clear whether the data point for pure transverse loads is for 90° or ± 85°. From reading the test results [4], it seems that the experimental points were made on the ± 85°, but were reported as 90°. Therefore, the original graph would be more adduceable to use. For material IV there were not any experimental data on unidirectional lamina for statistical verification. Fig. 4: this figure shows some experimental results made about 25–30 years ago. There is no information on initial rupture and therefore the results for bi-axial tension are for final rupture. Of course, by definition, these experimental results show higher values than predicted here as final failure (i.e. first crack in all laminae). On the other hand, for bi-axial compression, failure was probably by buckling of the tube skin, which is not covered by the failure criterion and therefore the results are lower than predicted. It is important to note that the failure criterion, for which the organizers had provided experimental parameters, is only for unidirectional lamina. The parameters that were needed for the laminate strength were assumed by the author (i.e. bridging effect and stiffness decay rate). Moreover, the criterion presented here is for the first crack in all laminae, as explained before. Therefore, it is no wonder that experimental results in tension were much higher than predicted.

881

Fig. 4. Biaxial failure envelope for (90/ ± 30°) laminate under combined x and y stresses. Material: E-glass/LY556 epcxy.

Fig. 5: here is the same situation as with Fig. 4, only the results are for shear too. Again the experimental results are higher than predicted because it records final rupture. As mentioned before, the prediction is for the beginning of the final failure, that is the point where cracks were formed in all laminae but the laminate still held. Unfortunately, these values were not recorded. Fig. 6: this figure demonstrates the wide distribution of the experimental results. For the same load situation there are many failure points differ by multiple values. Again, as explained

Fig. 5. Biaxial failure stresses for (90/ ± 30°) laminate under xy and x stresses.

882

Fig. 6. Biaxial failure stresses for ( ± 55°) E-glass/MY750 laminates.

for Fig. 4, the final failure was assumed by hypothetical parameters. The first failure is the first crack in a lamina without bridging effect and therefore there is a difference between first and final failure even for ± 55° (i.e. loading in the axis directions).

Fig. 7. Stress-strain curves for ± 55° GRP laminate under uniaxial tension (y /x = 1/0).

883

Fig. 8. tress-strain curves for a ± 55° laminate made of E-glass/MY750 epoxy under y /x = 2/1.

Fig. 7: this figure gives the stress-strain curve of the same material as in Fig. 6, for one load situation: tension in the Y direction. Comparing with Fig. 6, failure occurs for loads between 362 and 638 MPa. The graph is for failure at 600 MPa, because test facilities were not able to read beyond this point [4]. For test without liner, which is more appropriate to the theory presented here, the prediction for the lower end is good. As mentioned before, the prediction for final failure means the first crack in the last intact lamina, and not total rupture. Therefore, it is possible that the laminate survived to higher loads until catastrophic failure occur, especially specimens with liner. Again, the lack of not having the stiffness decay rate parameter did not allow good pacing for the stress–strain curve. Fig. 8: this is another load situation for the same laminate of Fig. 6. The bi-axial load ratio was y/x = 2/1. Comparing again with Fig. 6, we see that failure at load level above 700 MPa is extraordinary. For the same load situation failure at a level of 125 MPa also exist. All the comments for Fig. 7 are valid here too. It is possible that the high strength observed by the experiment is due to fiber fracture after the matrix material was completely cracked. Of course, this situation is not covered by the present theory. Fig. 9: here again the spreading of the results is very wide. For example, failure with compression loading in the X direction only, is between –305 and –733 MPa. It was mentioned that the failure mode was buckling and thus does not enter to the prediction scheme. In tension in the Y direction, if we take into account the observed initial failure, it agrees fairly well with the prediction. The input for this material is unknown to us and

884

Fig. 9. Biaxial failure stresses for (0/ ± 45/90°)AS4/3501-6 laminates.

therefore hard to explain. As this laminate is quasi-isotropic, loading in the positive-positive quadrant is merely rotating the load direction and so is the failure prediction. However, the test results show significant strengthening with the rotation, by a factor of 200%. This is probably again because final rupture is considered and fiber strength is responsible for the high values after the matrix failed. Fig. 10: the predicted behavior is fairly well until the first failure. Since the theory does not deal with interlaminar behavior and assumed total failure of each lamina in the laminate after reaching the failure condition, there is the sudden change in strain that does not occur in practice. Stiffness decay was not available for calculating the behavior after the first crack. This is also the reason why the slope becomes moderate. Here again the high strength is due to the fibers after matrix cracking. Fig. 11: this is also a stress-strain curve for the same material under a different load situation. The predicted failure is approximately where possible matrix cracking begins. Again, the behavior after initial cracking is dependent on the assumed parameters and interlaminar behavior as explained before and fiber strength contribution. Fig. 12: this is a ± 45° angle-ply laminate that is known to stretch by cracks opening. Failure is predicted where the first crack is observed. The theory does not continue over this point (initial failure that was predicted is only for failure of one or more laminae within a laminate and the intact laminae continues to carry the load). Fig. 13: the same situation as in Fig. 12. The long elongation is not of the material but rather cracks opening.

885

Fig. 10. Stress-strain curves for (0/ ± 45/90°) laminate under uniaxial tension (y /x = 1/0).

Fig. 11. Stress-strain curves for (0/ ± 45/90°) laminate under biaxial stress y /x = 2/1.

886

Fig. 12. Stress–strain curves for ± 45° E-glass/MY750 laminate under y /x = 1/1.

Fig. 14: this is the same material as in Fig. 13, but it is loaded in the fiber direction (cross ply laminate). In this case it was assumed that the transverse plies continue to contract the axial layers after they have been cracked. The experimental result shows that it does not happen and the laminate does not contract transversally after the failure of the 90° laminae. It must be emphasized that the failure criterion is valid only as long as the laminae remains intact, because it is based on continuum behavior. Therefore, most of the discrepancy is due to the presence of cracks. On the other hand, the laminate elongation agrees quite well with the prediction. 3. Discussion The Rotem failure criterion was postulated for unidirectional lamina. Two parameters were added to account for in-laminate behavior. All the predictions were for initiation of the first crack in a lamina. The experimental program and the predictions were for multi-layered laminates. The definition of failure for these laminates is not unique. In part A of the exercise, a definition has been used that is suitable for this kind of criterion, that is, initial failure is the first crack in the first lamina. Obviously this lamina continues to carry load but its contribution depends on the strength distribution (which we do not have) and 3-D analysis. An arbitrary decay function has been used which governs the support of the failed lamina. This function depends on the extension of the laminate. It is easy to adjust this function to the actual behavior, but at least one test is needed and I did not have it. Therefore, an arbitrary value was used. Final failure was defined as the first crack in the last remaining lamina. This is also a conservative assumption as the lamina may hold with the use of other “just failed” laminae. The experimental result shows that many specimens held to higher strength values than predicted. This is mainly because specimens held to higher stress and strain levels after the

887

Fig. 13. (a) Stress–strain curves for ± 45° E-glass/MY750 laminate under y/x = 1/–1; (b) modified stress–strain curves for ± 45° E-glass/MY750.

888

Fig. 14. Stress–strain curves for 0/90° E-glass/MY750 laminate under uniaxial tension.

first crack in laminae by bridging with neighboring laminae. As an example consider the stress–strain curve of the ± 45° angle ply laminate (Fig. 13). By changing the decay parameter Rd(j) of Mat3 from 600 to 1200 we will get the curve shown in Fig. 13b. This is obviously a much better prediction. The failure criterion is based on the stress field which is developed in the laminae. However, residual and thermal stresses may be present due to other conditions like manufacturing and not loading. It is important to consider all the conditions which cause stress in the material when calculating the stress state. It is also important to make sure that the input strengths are accurate and were measured without any side effects like manufacturing residual stresses.

4. Conclusions The Rotem failure criterion was formulated for describing the failure condition of a unidirectional laminate. In order to use it for multi-directional laminate more information on the laminate behavior is needed. This information should be on the behavior of the laminate after the first crack in a lamina has occurred. It is in a form of two parameters, the in-laminate strengthening and the stiffness decay rate of the lamina. These parameters could be

889 determined from a simple bi-axial tensile test on an angle ply laminate in the form of a tube. Test results in the form of stress–strain curves, for various configurations would enable to extract these parameters for the best fit. However, it should be remembered, that the criterion does not hold beyond the point where the first crack started in the last intact lamina. Spreading of cracks in a laminate, even in a simple one like angle ply, is a very complex problem known for many years [6]. Yet, to my knowledge, it is still unsolved. Only recently, a very special case, i.e. cross ply, was partially solved [7].

References [1] Rotem A, editor. Prediction of laminate failure with the Rotem failure criterion [special issue]. Comp Sci Technol 1998;58(7). [2] Dow NF, Rosen BW, Hashin Zvi. Studies of mechanics of filamentary composites. Tech Report NASw-1144, GEC, Philadelphia, PA, 1965. [3] Weibull W. A statistical distribution function of wide applicability. J Appl Mech 1951;18:293–7. [4] Soden PD, Hinton MJ, Kaddour AS. Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates: Failure exercise benchmark data: V62, pp 1489–1514, 2002, see also Chapter 2.2 in this book. [5] Personal experience. [6] Rotem A, Hashin Z. Failure modes of angle ply laminates. J Comp Mater 1975;9(April):191–206. [7] Hashin Z. Finite thermoelastic fracture criterion with application to laminate cracking analysis. J Mech Phys Solids 1996;44(7):1129–45.

890

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

CHAPTER 5.8

The prediction of failure envelopes and stress/strain behavior of composite laminates: Comparison with experimental results C. T. Suna,*, J. Tao and A. S. Kaddourb a

School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907-1282, USA b DERA Farnborough, Farnborough, Hampshire GU14 0LX, UK

Abstract Two theoretical models were used by Sun and Tao [Comp. Sci. Technol. 58 (1998) 1125] in Part A of the exercise to predict the strength and stress/strain curves for a number of test problems. The strength model was based on linear elasticity in conjunction with a ply-discount method using parallel spring stiffness reduction, and the stress/strain curves were predicted by the linear model and a nonlinear model incorporating simplified shear-lag analysis and an ABAQUS finite-element package. This paper provides a description of the correlation between experimental results provided by the organizers of the ‘failure exercise’ and the theoretical predictions, published in Part A for (a) biaxial failure envelopes of [0°] unidirectional and [0°/ ± 45°/90°]s, [ ± 30°/90°]s and [ ± 55°]s multi-layered composite laminates, and (b) stress/strain curves for [0°/ ± 45°/90°]s, [ ± 55°]s, [0°/90°]s and [ ± 45°]s under uniaxial and biaxial loadings. Keywords: C. Failure criterion; C. Laminates; B. Strength; B. Stress/strain curves; Combined loading

1. Introduction In Part A of the failure exercise, Sun and Tao [1] presented theoretical predictions of biaxial failure envelopes of unidirectional and multi-directional composite laminates and also for seven stress/strain curves for a number of laminates under uniaxial and biaxial stresses. Details of laminates analysed, the unidirectional ply material properties and the types of loading applied are given in Ref. [2]. The strength was predicted by using the linear laminate theory in conjunction with ply-by-ply discount method using a parallel spring stiffness reduction model. The stress/strain curves were predicted by using the linear model and a nonlinear model based on simplified shear-lag analysis and finite-element analysis with

* Corresponding author. Tel.: + 1-765-494-5130; fax: + 1-765-4940307. E-mail address: [email protected] (C. T. Sun).

891 ABAQUS. The linear analysis was used to generate the biaxial failure envelopes and also the stress/strain curves. The nonlinear analysis, which was used only to predict the stress/strain curves, took into account both material nonlinearity and progressive matrix cracking. Material nonlinearity was considered by using a one-parameter plasticity model for fiber-reinforced composites. Progressive matrix cracking and the corresponding stiffness reduction were predicted by use of a simplified shear-lag analysis and finite-element analysis. The theoretical results based on the models used by Sun and Tao [1] are compared in Ref. [3] with those of other models used by ten different groups who participated in the failure analysis exercise. All these groups used the same input data to analyze 14 test problems. Soden et al. [4] provided details of test results for all the cases analyzed in Ref. [1]. Almost all the tests were carried out on tubular specimens. In some of the test results, initial failure in the form of matrix cracks or leakage was described and, in other test cases, only final failure stresses were provided. The theoretical and test results were superimposed and the figures were supplied to the authors. These figures are described in the following sections.

2. Comparison between theory and experiments 2.1. Biaxial failure envelopes Figures 1–3 show the failure envelopes of unidirectional laminae under various types of loading. Fig. 1 presents a comparison between test results and theoretical predictions for an E-glass/LY556 epoxy lamina under combined transverse and shear loading (y  xy). The correlation is good for tensile y stress. However, for compressive y stress, the correlation is rather poor as the theory predicts a quadratic envelope with no enhancement in the shear stress

Fig. 1. Biaxial failure stresses for 0° lamina made of GRP material E-glass/LYS56/HT907/DY063.

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Fig. 2. Biaxial failure stresses for 0° lamina made of CFRP material T300/914C.

due to application of compressive stress while the test results showed a degree of enhancement, especially for moderate values of y stress. It should be noted that the experimental transverse compressive strength [4] is greater than that provided for use in the analysis of Ref. [1]. If a greater transverse compressive strength were used, the predicted strength in the second quadrant would have agreed better with the experimental data.

Fig. 3. Biaxial failure envelope of 0° GRP lamina under combined x and y stresses. Material: E-glass/ MY750 epoxy.

893 Nevertheless, the experimental data indicates that shear strength is enhanced by the presence of a compressive transverse stress. Figure 2 presents the theoretical and experimental results for T300/914C carbon/epoxy lamina under combined longitudinal and shear loading (x  xy). A wide scatter is observed in the test results, especially in the values of uniaxial shear strength. In the test results, the data suggest (a) an enhancement in the shear stress above the uniaxial shear strength when the longitudinal tensile stress is around 900–1000 MPa and (b) a reduction in the shear strength when a compressive stress is applied. The interaction between the shear and tensile longitudinal stresses as displayed in the experimental data cannot be easily explained. Figure 3 shows a comparison between test results and theoretical predictions for E-glass/ MY750 epoxy lamina under combined longitudinal and transverse loading (x  y). The theoretical curve showed no interaction between the two stresses in the four quadrants while the test results in the tension-compression quadrant suggest a degree of interaction between the stresses. Therefore, as far as this quadrant is concerned, the theory over-predicted the test results and thus the correlation is rather poor. It is, however, difficult to make a coherent comparison between the theory and experiment as no sufficient test results are available in the other quadrants.

2.2. Biaxial failure envelopes for multidirectional composites Figures 4 and 5 show the experimental and theoretical failure envelopes of the [90/ ± 30°]s laminate made of E-glass/LY556/HT907/DY063 epoxy subjected to combined direct stress (x  y) and combined axial stress and shear loading (x  xy), respectively. The test results were obtained from tests on tubes and the test data available described only the final failure stresses, i.e. no initial failure stresses were supplied for the laminates. Hence, it is not possible to compare the theoretical prediction of initial failure with test data. The theoretical failure stresses under x  y loading (Fig. 4), show a trend similar to the test results. There are, however, differences between the theoretical and experimental data. The theory slightly overpredicted the biaxial failure stresses in the tension-tension and compression-compression quadrants. In the latter case buckling of the tubes may be considered as one possible cause for such differences. When the laminate is subjected to combined axial stress and shear loading (x  xy) (Fig. 5) the theory gave a good prediction of the uniaxial shear strength but it slightly underestimated the axial tensile strength. The theory also under-predicted the laminate strength when compressive x stress was combined with xy shear stress. Figure 6 shows the experimental and theoretical results of the biaxial failure envelopes of [ ± 55]s laminates made of E-glass/MY750 epoxy under combined x  y biaxial loading. Good correlation between theoretical final failure prediction and test results can be observed in a number of areas. However, the theory under-predicted the experimental strength in the compression-compression quadrant for stress ratio SR = y : x = –2 : –1. This may be partly attributed to the high fibre volume fraction of the test specimens used in this quadrant [4]. The fibre volume fraction was around 67% while the theoretical prediction was for a fibre volume fraction of 60%. It is noted that in the tension-tension quadrant, the theoretical predictions were lower than the experimental results. For instance, at SR = 10 : 1, the hoop strength y was almost three times higher than that predicted. On the other hand, the theoretical final strength

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Fig. 4. Biaxial failure envelope for (90°/ ± 30°) laminate under combined x and y stresses. Material: E-glass/LY556 epoxy.

overpredicted the experimental strength at stress ratios between SR = 0 : 1 and 2 : 1. The correlation between the initial failure prediction and the strength of unlined tubes was also poor. The difference between them was as high as five fold in some areas. Biaxial failure envelopes for [0/ ± 45/90]s laminate under x  y biaxial loading are plotted in Fig. 7. The correlation between the theory and test results is good as both indicated (a) an increase in the stress y due to biaxial loads in the tension tension quadrant and (b) a decrease in stress y in the tension compression quadrant. An apparent poor correlation between test

Fig. 5. Biaxial failure stresses for (90°/ ± 30°) laminate under xy and x stresses.

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Fig. 6. Biaxial failure stresses for ( ± 55°) E-glass/MY750 laminates.

Fig. 7. Biaxial failure stresses for (0°/ ± 45°/90°)AS4/3501-6 laminates.

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Fig. 8. Stress/strain curves for 0°/90° E-glass/MY750 laminate under uniaxial tension (y = 0).

results and theoretical prediction can be seen in the compression compression and, to a lesser extent, in the tension compression quadrants. This could be due to problems (mainly buckling) associated with the experiments. 2.3. Stress/strain curves The stress/strain curves were predicted using (a) linear analysis and (b) nonlinear analysis. In the linear analysis, a ply-by-ply discount method was used [1]. As a result, a kink was seen when matrix failure in a lamina occured. In the latter case, material non-linearity, arising from matrix nonlinear behaviour, and effects of matrix cracking were taken into account by using a progressive matrix cracking model [1]. Since the effect of matrix cracking was modeled in a progressive manner, no jumps were seen in the resulting stress/strain curves. It should be noted that, in the present exercise, the nonlinear analysis was used only to produce stress/strain curves but not for strength predictions. To use the result of the nonlinear analysis for strength predictions, it is necessary that new appropriate failure criteria be developed. Thus, the ending point of each stress/strain curve shown in Figs 8–14 does not necessarily correspond to the predicted ultimate strength of the laminate. Figure 8 shows the theoretical and experimental stress strain curves for [0/90]s laminate under uniaxial loading (y /x = 0/1). The test results were obtained from testing coupons

897 under uniaxial tension [4]. The thickness of the coupons was almost twice that of the laminate analyzed, and the fiber volume fraction, 62%, was slightly higher than that used in the analysis, 60%. In the linear analysis, the theoretical curves are linear up to around 80 MPa stress where matrix cracking took place in the layer whose fiber direction was perpendicular to the loading direction. A reduction in the modulus took place at this point. Thereafter, the slope changed and the modulus decreased slightly but remained almost parallel to that observed in the test. In the nonlinear analysis, the initial modulus, up to 80 MPa stress, was identical to that obtained from the linear analysis. When compared with the test results, the theoretical modulus, after initial failure, was slightly lower than that observed in the test. In the linear analysis, the correlation between the predicted and measured Poisson’s strains was good up to around 268 MPa stress at which matrix cracking occurred in the 0° lamina. Thereafter, the predicted strain was lower than the measured one. The Poisson’s strain in the nonlinear analysis was larger than that observed in the tests. This is in contrast to results of the linear analysis where the predicted strain was smaller than that measured in the tests. It must be mentioned here that the lamina modulus, E2, in the nonlinear analysis was assumed to decrease exponentially with increasing the crack density whereas in the linear analysis the same modulus was dropped to zero suddenly after the occurrence of transverse cracks. Again, it is noted that the ending point in the stress strain curve obtained according to the nonlinear analysis does not indicate failure of the laminate. Figure 9 shows the theoretical and experimental results for the [ ± 45]s laminate under equal-biaxial tensile loading (y /x = 1/1). The experimental results were obtained from tubes tested with and without the use of plastic liner. The main source of nonlinearity in the predicted curves is matrix cracking rather than material nonlinearity. The predicted initial failure stress was 68 MPa which is similar to that observed in the test, 70 MPa. The post initial

Fig. 9. Stress/strain curves for ± 45° E-glass/MY750 laminate under y/x = 1/1.

898 failure behaviour in the linear analysis was similar to that of the nonlinear analysis with the secant modulus in the nonlinear analysis slightly greater than that of the linear analysis, as was the case of the [0/90]s laminate. Both models used in the theory predicted the strains x and y to be identical. However, the test results showed the hoop strain y to be larger than the axial strain x. Furthermore, at any given stress the measured strains were larger than those predicted. This could be due to the short length of the test specimens, which could cause significant deviation from the boundary condition assumed in the theoretical predictions. Neither the linear analysis nor the nonlinear analysis was able to predict leakage in this laminate. Figure 10 shows the predicted and measured stress/strain curves for [ ± 45]s laminate subjected to equal biaxial tension-compression loading with y /x = 1/1. This is the only case among all the problems solved in this exercise, in which the nonlinearity is predicted to come primarily from material nonlinearity as the behaviour is dominated almost fully by the unidirectional shear stress. The linear analysis produced straight lines indicating that the first failure was the ultimate failure. In contrast, the nonlinear analysis gave nonlinear curves. But the experimental final failure strains of about 10% obtained from testing tubes under combined internal pressure and axial compression were not predicted by the linear analysis. Figure 11 shows the predicted and measured stress/strain curves for [ ± 55]s laminate subjected to uniaxial tensile loading with y /x = 1/0. The measured curves were obtained from internal pressure tests on thin tubes tested with and without the use of plastic liner. The correlation between the measured and predicted failure stresses (and thus the failure strains) with the linear analysis is poor. However, the stress/strain curves according to the nonlinear analysis seem to agree with the experimental curves quite well.

Fig. 10. Stress/strain curves for ± 45° E-glass/MY750 laminate under y/x = 1/–1.

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Fig. 11. Stress/strain curves for ± 55° GRP laminate under uniaxial tension (y/x = 1/0).

It is known that angle ply composite laminates such as the ones considered in Figs 10 and 11 tend to yield large flow strains when subjected to uniaxial loads. Large strains can significantly alter the fiber orientation with the consequence of load-shifting to fibers and, thus, yielding higher laminate strengths. The experimental results seem to confirm this conjecture. The behaviour of [ ± 55]s laminate subjected to biaxial tensile loading with y /x = 2/1 is depicted in Fig. 12. The measured curves were obtained from internal pressure tests on thin

Fig. 12. Stress strain curves for a ± 55° laminate made of E-glass/MY750 epoxy under y/x 2/1.

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Fig. 13. Stress/strain curves for (0°/ ± 45°/90°) laminate under uniaxial tension (y/x = 1/0).

tubes tested with and without the use of plastic liner. It is evident that the theoretical (linear and nonlinear) results agree with the experimental results in the early stages of deformation, i.e. up to around 120 MPa. In both the linear analysis and the measured results, the x strain curve crossed the y curve as the stress increased, see also comments on corrected results in Section 2.4. In contrast, in the nonlinear analysis, the x strain (always smaller than the measured curve) did not cross the measured y curve. Rather, it exhibited a stiffening behaviour and reached an almost constant value beyond 500 MPa stress. Since the nonlinear analysis was not used for strength predictions, the strain at the end point of the curve should not be taken as the failure strain of the laminate. Figures 13 and 14 show the theoretical and experimental stress/strain curves for [0/ ± 45/90]s AS4/3501-6 laminate under uniaxial (y /x = 1/0) and biaxial stresses (y /x = 2/ 1). It was suggested in the Part A paper by Sun and Tao [1], that material nonlinearity and matrix cracking contribute nearly the same amount to the nonlinearity of the stress-strain curve under y /x = 1/0). Also, when the laminate is subjected to y /x = 2/1 biaxial loading, matrix cracking has a greater effect on the laminate stress/strain nonlinearity. That is because the contribution to failure by the shear stress component gets smaller as the loading moves from uniaxial tension to equal biaxial tension. The correlations between experimental and the theoretical results are in general good. The worst case was that of the predicted x strain under y /x = 2/1 loading where the linear analysis resulted in a strain of 0.51%, almost 50% higher than that measured in the tests. 2.4. Modified results Revised stress/strain curves were made after the publication of Part A paper [1]. The revised figures, shown in Figs 8, 9, 12, 13 and 14, were corrected for minor numerical and plotting

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Fig. 14. Stress/strain curves for (0°/ ± 45°/90°) laminate under biaxial stress y/x = 2/1.

errors discovered after the publication of Part A. The changes are apparent as steps in the stress/strain curves predicted by the linear analysis. The most noticeable changes are in Figs 8 and 12. These changes did not affect the results of failure predictions.

3. Conclusions Employing the linear elastic laminate theory in conjunction with a ply discount method based on parallel spring stiffness reduction, together with Hashin–Rotem failure criterion gave the following features in failure prediction of composite laminates. (a) Good correlation with test results for final failure strengths of [0°/ ± 45°/90°]s AS4/ 3501-6 carbon/epoxy and [ ± 30°/90°]s glass/epoxy laminates under biaxial stresses. (b) An acceptable correlation with test results in a number of segments of the biaxial failure envelope for [ ± 55°]s glass/epoxy laminate. However, the model underpredicted the final strength in some other portion of the envelope by a factor up to 3. (c) Good agreement with the final strength of laminates whose failure is largely dominated by the fiber tensile strength, such as and [ ± 45°]s under equal biaxial tension, [0°/90°]s glass/ epoxy laminate under uniaxial tension, [ ± 55°]s glass/epoxy laminate under SR = 2/1 and [0°/ ± 45°/90°]s carbon/epoxy laminates under uniaxial and biaxial tensile stresses. (d) Very poor agreement with the experimental failure strains which are dominated by transverse and shear matrix cracking. These include both branches of the failure strains of [ ± 45°]s under SR = 1/–1, both branches of [ ± 55°]s glass/epoxy laminate under SR = 1/0, where the predicted curves are terminated prematurely, the Poisson’s strain in [0°/90°]s glass/ epoxy laminate under uniaxial tension and the Poisson’s strain of [ ± 55°]s glass/epoxy laminate under SR = 2/1.

902 (e) The assumption that the in situ lamina tensile transverse and shear strengths are one and half times those measured from individual laminae seems to work reasonably well in the prediction of laminate strengths. The laminate stress/strain curves predicted by both the linear analysis, which accounted for effects of matrix cracking by a ply discount method, and by the nonlinear analysis, which took into account nonlinearities due to material and progressive matrix cracking, generally agree well with the measured curves for fiber-dominated laminates. For matrix-dominated laminates, except for the [ ± 55°] glass/epoxy laminate under SR = 2/1, the nonlinear analysis gave reasonably accurate stress/strain curves up to moderate strain ranges. One should be cautious when drawing conclusions from the comparison between theoretical and experimental results for laminate strength. Discrepancies should not be attributed automatically to deficiencies of the theory. There are a number of factors that may result in such discrepancies. (a) The lamina strength properties in the laminate analysis were obtained from unidirectional composite specimens. These strength properties could be quite different from the in situ lamina properties due to the absence of the constraining effects from adjacent laminae in the laminate. (b) Most tests for biaxial loading were performed using short, thin-walled tubular specimens. The gripping condition could induce undesirable boundary effects, especially in tests that involve torsion. Additional failure mechanisms that were not accounted for by the theory could be responsible for failure in the test. The most notable example is buckling of thin-walled tubes under compression or torsion. (c) The effect of curvature of the tubular specimen on the composite strength is unknown. Moreover, since tubular specimens are manufactured by different methods, they may not be of the same quality as flat laminates.

References [1] Sun CT, Tao J. Prediction of failure envelopes and stress-strain behaviors of composite laminates: comparison with experimental results. Composites Science and Technology 1998;58:1125–36. [2] Soden PD, Hinton MJ, Kaddour AS. Lamina properties and lay-up configurations and loading conditions of a range fibre reinforced composite laminates. Composites Science and Technology 1998;58:7 1011–22. [3] Soden PD, Hinton MJ, Kaddour AS. Comparison of the predictive capabilities of current failure theories for composite laminates, published in. Composites Science and Technology 1998, 58(7): 1225–54. [4] Soden PD, Hinton MJ, Kaddour AS. Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates: Failure exercise benchmark data: V62, pp 1489–1514, 2002, see also Chapter 2.2 in this book.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

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CHAPTER 5.9

A progressive quadratic failure criterion, part B Akira Kuraishi, Stephen W. Tsai* and Kevin K. S. Liu Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305-4035, USA Abstract The predictions for UD and laminated composites from Part A of this exercise [Compos. Sci. Technol. (1998) 1023] are compared with test data [Compos. Sci. Technol. (2002) 1725, see also Chapter 6.1 in this book]. Questions concerning the accuracy and flexibility of our progressive criterion are answered herein. In general, the intact and degraded models provide good agreement. So do several stress/strain curves. The selective damage scheme was found to be the source of some problems at certain loading conditions. By eliminating this selective degradation, agreement between the prediction and the test data were improved. Empirical factors are easily adjusted to close the gap between theory and data. Those factors are within the bounds set by the theory in Part A. Discrepancies do happen and some can be attributed to bad data while some have no simple explanation. All in all, the exercise has been worthwhile and may have impact on the growth of composites in the years to come. An easy to learn Excel-based program for plotting failure envelopes can be downloaded free of charge for interested parties.

1. Introduction In our progressive failure theory [1], we have two states for each ply. First is an intact continuum that will be described by a continuous quadratic function. In this state, fibers and matrix are locked in together. Their individual identity is there but their combined functions cannot be easily traced and separated. Our failure criterion is modeled by a quadratic function controlled by six independent strength parameters. They can be derived from five simple tests such as uniaxial tensile and compressive tests for determining X, X , Y and Y ; and pure shear test for S. The only floating constant is the normalized interaction term Fxy* which has values between 0 and –1/2 for most materials. Figs 1–3 in the next section show this interactive failure criterion and test data provided for this study [2]. Agreement between theory and data is as good as the data scatter of the UD and combined stress tests. The other state is the degraded state, initially with dispersed micro cracks, fully saturated with finite distance between cracks. The fibers continue to carry the load, but the matrix carries less load, which is modeled by reducing the transverse stiffness Ey and the shear * Corresponding author. E-mail address: [email protected] (S.W. Tsai).

904 stiffness Es by a matrix degradation factor Em*. The longitudinal tension strength is the same as that of the intact state. The longitudinal compressive strength will be affected by the magnitude of the degraded shear modulus. We use the exponent n of a power law of the normalized shear moduli to describe the reduction in this compressive strength. All other strengths are left constant. The failure strains will be significantly increased since the corresponding strengths remain the same while the stiffness is decreased. After the second failure of the same ply, the stiffness Ex, Ey, and Es are further reduced by a fiber degradation factor Ef *. The longitudinal compression strength is reduced in the same way as before. The ultimate failure of the degraded plies can also be modeled by a quadratic relation. In Part A, the two degraded conditions, the “matrix failure” and the “fiber failure”, were chosen according to a selective degradation scheme shown in Fig. 4. This scheme works well in most cases, but has some problem in certain loading conditions. Further study of this selective degradation showed that eliminating the selection is the best choice judging from the test data provided in this exercise. The new progressive failure modeling scheme is shown in Fig. 5. The first damage of a ply is deemed “matrix failure” and Em* is applied. Next failure of the same ply is deemed “fiber failure” and Ef * is applied. The theoretical predictions of the initial- or first-ply-failure FPF and the final- or last-plyfailure LPF are compared in Figs 6–16. The agreement varies depending on the materials, laminates and particular stress curves. In most cases, agreements can be improved if empirical factors Em* and n assume different values. Their sensitivity will be illustrated, case by case, in the next section. These attempts to improve predictions follow the method described in the Part A of this paper, except for the elimination of the selective degradation. Unless otherwise stated, layup and lamina properties are identical to those described in Part A. The thermal stresses calculations are also identical to those in Part A, where a moisture content of 0.5% and a temperature difference of –100°C were assumed. In this case, both effects cancel out allowing us to use the intact material properties.

2. Comparison between theory and data In this section, each figure will be discussed individually. Attempts are made to show the sensitivity of all the relevant empirical factors Fxy*, Em*, Ef *, and exponent n on the theoretical predictions. Figure 1 is the biaxial failure envelope of E-glass/LY556 [0°] lamina. Our original prediction agrees well with the test results. Noting that the supplied value of the transverse compressive strength Y  does not match the biaxial test results, we can improve our prediction by using Y  of 137 MPa calculated from the test results. Figure 2 is the biaxial failure envelope of T300/914 [0°] lamina. Our original prediction does not agree well with the test results, although we can argue that the prediction fits within the scatter of the test results. Looking carefully at the test results, the wide scatter of the data points on the xy-axis cast a doubt on their accuracy. Considering the complexity of these biaxial tests, the lower shear strength may have been due to premature failures such as shear buckling. Ignoring these data points, and using higher shear strength S = 100 MPa as our anchor point, we can improve our prediction as shown in the figure. Figure 3 is the biaxial failure envelope of E-glass/MY750 [0°] lamina. Our original prediction agrees well with the test results. For a lamina, the only free empirical factor we

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Fig. 1. Biaxial failure envelopes for E-glass/LY556 [0°] lamina.

have is the x  y interaction factor Fxy*. To show the effect of this factor, failure envelopes were drawn for the nominal case of Fxy* = –0.5 and two extreme cases of Fxy* = –0.1 and –0.9. The results show that the nominal value of Fxy* = –0.5 lead to good agreement. The following Figs 6–16 are the failure envelopes and stress/strain curves using the modeling scheme shown in Fig. 5 (no selective degradation). The envelopes in bold are the new baseline envelopes, which use the same empirical factors as those in Part A, namely Fxy* = –0.5, Em* = 0.15, Ef * = 0.01, and n = 0.1. Compared with those in Part A, the new envelopes are smoother and have better agreement with the test results.

Fig. 2. Biaxial failure envelopes for T300/914 [0°] lamina.

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Fig. 3. Biaxial failure envelopes for E-glass/MY750 [0°] lamina.

Fig. 4. Flow chart of the progressive failure modeling used in Part A.

Fig. 5. Flow chart of the progressive failure modeling used in Part B.

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Fig. 6. Biaxial failure envelopes and the effect of the factors for E-glass/LY556 [90°/30°/–30°]s laminate under combined y  x direct stresses.

Figures 6 and 7 are the biaxial failure envelopes of E-glass/LY556 [90°/30°/–30°]s laminate. In Fig. 6, the comments in the test results indicated that the data points in the 2nd and 3rd quadrants were from the specimens that may have failed in buckling under external pressure. Ignoring these points marked in X, we find that our prediction agrees well with the test results. Among the four empirical factors that controls our prediction, Ef * has very little effect on the failure envelopes. The effect of the other three factors Fxy*, Em* and n are shown in the figures. Values used for Fxy* are –0.1, –0.5 and –0.9, where –0.5 being the nominal. Values

908 used for Em* are 0.01, 0.08, with 0.15 being the nominal. Values used for n are 0, 0.1 and 0.5, with 0.1 being the nominal and n = 0 corresponding to no degradation of longitudinal compressive strength X . As we can see from the results, each factor changes certain regions of the envelope. The choice of Em* = 0.08 leads to what we believe as the best fit, marked as “Modified” in the lower right figure. By choosing a good combination of these empirical factors, we can achieve better agreement, but may also yield an unrealistic result unless we consider the effects of these factors on the stress curves. One such example is when we used secant modulus of Es rather than the initial modulus used for Figs 8–10. The failure envelopes matched well, but the stress/ strain curve were off. This example shows the danger of relying only on the failure envelopes to determine these empirical factors. In Figure 7, the new envelopes have better agreement with the test results, compared with the original envelope in Part A. Unfortunately, none of the empirical factors could completely close the gap between the prediction and the test data. Em* = 0.08 was chosen for the modified envelope for internal consistency. Figure 8 is the biaxial failure envelope of E-glass/MY750 [55°/–55°] laminate. This is the case where we observe significant improvement from Part A, with smoother envelope and a bulge in the 3rd quadrant. The new prediction agrees well with the Test Data 2 and 3, both tested until the final failure. Test Data I shows the biaxial loads where the pressurized tubes started to leak, which our theory can not predict. First ply failure can be used for a conservative design, although the test results indicate that this may grossly underestimate the actual strength. The effect of the three empirical factors Fxy*, Em*, and n are shown in the figure. The results favor higher value of Em* for the 3rd quadrant, and lower value for the 1st quadrant. Em* = 0.10 was chosen as optimum considering other data of the same material also. Figure 9 is the stress/strain curves under the load of y /x = 1/0. The curve differs from that of Part A, which predicts the initial failure to be the final failure. In the new prediction, final failure occurs at higher loads and strains. Although the initial slope was predicted quite well, the final failure and the ultimate strains were not well predicted. The empirical factors could not completely close the gap in this case. The test result mentions the friction of the test setup, which may create some axial loading. The second figure shows the modified case with the load of y /x = 20/1, which is the same as the load used later for a similar test configuration. The value of Em* = 0.10 was also applied for internal consistency. The change in the loading condition increased the predicted failure load, while the reduced value of Em* reduced the slope of the curve. Figure 10 is for the case with the load of y /x = 2/1. Our prediction agrees very well with the test results except for the second kink in the x and its final value, which could not be reproduced regardless of the empirical factors we used. Reducing Em* to 0.10 will increase the final failure stress up to the observed failure stress. Mentioned in the test results are the local bulging of the center section of the tube under pressure loading, and the scatter of the observed ultimate strains. These suggest the possibility of localized deformation near the strain gage, which may be causing the kink in the data. Our prediction assumes homogeneous state, and ignores the effect of local damage or deformation. Figure 11 is the biaxial failure envelopes of AS4/3501-6 [0°/45°/–45°/90°] laminate. Compared to the envelope in Part A, the new envelopes are smoother. Swanson, who produced

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Fig. 7. Biaxial failure envelopes and the effect of the factors for E-glass/LY556 [90°/30°/–30°]s laminate under combined axial loading and shear.

these test results, suspected that some of the test specimens have failed in premature buckling failure, which are marked in triangle and X. Ignoring these data points, our prediction agrees well with the test results. Effects of the empirical factors are shown in the figures. We can observe that the nominal values lead to good agreement between the prediction and the tests. Figures 12 and 13 are the stress/strain curves for the same material under stress ratio of y /x = 20/1 and y /x = 2/1, respectively. The predicted curves show good agreement with the test results. From these figures and the previous figure, we conclude that the nominal values of the empirical factors are sufficient for this material.

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Fig. 8. Biaxial failure envelopes and the effect of the factors for E-glass/MY750 [55°/–55°] laminate under combined y  x direct stresses.

Figures 14 and 15 are the stress/strain curves of the E-glass/MY750 [45°/–45°] laminates under stress ratio of y /x = 1/1 and y /x = 1/–1, respectively. On Fig. 13, our prediction agrees well with the test results except for the final failure stress. Predicted failure stress can be increased up to the measured value by reducing Em* to 0.10, which is the same as those for Figs 8–10 with the same material.

911

Fig. 9. Stress/strain curves under loading ratio of y /x = 1/0 for E-glass/MY750 [55°/–55°] laminate.

Figure 15 is more complex. As the organizers have mentioned in their comments, this loading case corresponds to pure shear where non-linearity of the Es plays an important role. Our prediction is based on linear theory and can not deal with the non-linear behavior. Linear theory enables simple and quick calculation of the failure envelope, which we have shown to be in good agreement with the test results. Horizontal sections of the stress/strain curves appear when the laminate carries the same load after all the fibers failed. The curve is identical to that in Part A, except that the horizontal sections were not included in Part A considering it unrealistic. The predicted ultimate strains, not shown in the figure, are ± 63%. The reason for these high strains is because we reduce the stiffness to 1% after fiber failure while keeping the strengths almost the same. The 1st, 2nd and 3rd ply failure loads are almost identical. Therefore, small perturbations in the load or the ply orientation will change the relationship between these loads, consequently changing where the final failure is defined.

912

Fig. 10. Stress/strain curves under loading ratio of y /x = 2/1 for E-glass/MY750 [55°/–55°] laminate.

The second figure shows the case where we apply small perturbation in the load, namely y /x = –0.99/1.00. Em* is reduced to 0.10 for internal consistency. In this case we have more realistic ultimate strains around 6%, but still the curves are far from the measured curve. Figure 16 shows the stress/strain curves of the E-glass/MY750 [0°/90°] laminates under uniaxial tension. Our prediction agrees very well with the test data, including the onset of longitudinal splitting at around 400 MPa. Reducing the value of Em* to 0.10 leads to an almost identical stress/strain curve, though the longitudinal splitting is predicted at around 600 MPa. We chose not to use the value of Em* = 0.10 in this case. As a conclusion for this section, we have shown that most of our predictions agree well with the test results. We have also shown the sensitivity of the empirical factors, and how they can be used to improve the prediction. Table 1 is the summary of the empirical factors that have been used to improve the prediction. We believe that our initial choices of the factors (Fxy* = –0.5, Em* = 0.15, Ef * = 0.01, n = 0.1) are sufficient for preliminary application, given the good agreement with the test results. Modifying the factors should be done with caution since they

913

Fig. 11. Biaxial failure envelopes and the effect of various factors for AS4/3501-6 [0°/45°/–45°/90°] laminate under combined y  x direct stresses.

may be material specific. Our recommendation is to use the stress curves rather than the failure envelopes to determine these factors. Along with the four empirical factors, another factor that affects the stress curve is the perturbation of the load. No experiment is perfect, and assuming some perturbations in the load or ply orientation sometimes yields more realistic predictions that agree with the measurements. For some tests, we had to question the accuracy of the test results. For the lamina failure envelopes, some of the given anchor points did not agree with the test results to begin with.

914

Fig. 12. Stress/strain curves under loading ratio of y /x = 20/1 for AS4/3501-6 [0°/45°/Y45°/90°] laminate.

Fig. 13. Stress–strain curves under loading ratio of y /x = 2/1 for AS4/3501-6 [0°/45°/–45°/90°] laminate.

For some biaxial tests, the test results showed very low failure stresses, raising doubt that they may have failed in buckling. Our prediction based on linear theory revealed its limit for the pure in-plane shear loading case of Fig. 15, but the benefit of linear theory overplays this shortfall. The prediction is simple and fast, and is sufficiently accurate, as we have shown in this section. 3. Modification of the model We have adhered to our model as described in Part A of this exercise except the elimination of selective degradation shown in Fig. 4. Only the permissible and reasonable range of each empirical factor was applied in the previous section. If a gap still exist between theory and data, we simply accept it. One obvious conclusion is that the theory cannot reach the data, the

915

Fig. 14. Stress/strain curves under Loading ratio of y /x = 1/1 for E-glass/MY750 [45°/–45°] laminate.

data may not be accurate or both. We wish to raise the following points in support of our position. Where is the mechanism? Our failure criterion is not mechanism based. When a unidirectional ply is in its intact state, it is a continuum, which has a specific internal structure. It has parallel fibers randomly spaced in the plane normal to the fiber axis. Our approach is phenomenological and global, geared for the bridge between materials and processing engineers, and designer. It is usually accepted that when this ply is subjected to simple uniaxial tensile or pure shear stress, the behavior is either fiber or matrix controlled. The role of interface is often viewed as a part of the matrix than that of the fiber. When the load is compressive, either in the longitudinal or transverse direction, it is no longer obvious which constituent controls the strength. The often used separation of the controlling constituent is over simplified. Even the uniaxial tensile strength along the fibers is not simply fiber controlled. The matrix and interface have

916

Fig. 15. Stress/strain curves under loading ratio of y /x = 1/–1 for E-glass/MY750 [ ± 45°] laminate.

critical roles. In the case of glass/epoxy composites, the failure mechanism is one of explosion. Graphite/epoxy composites, on the other hand, fail with many splits parallel to fibers. Thus, matrix does make a difference even in these simple tests. In a composite, the matrix must be there to hold the fibers in place, to transfer internal stresses, and controls crack initiation and propagation. Its role is more than determining the fiber volume fraction. When combined stresses are present, it would seem safe to say that the interactions between the constituents and the interface will become more complex than the case of simple, uniaxial stress. It is difficult to defend the separation scheme, and to insist that failure modes do not interact. In many criteria, the mechanism of failure is over simplified. When a laminate has failed, many mechanisms have occurred simultaneously or, more likely, sequentially. There is no simple criterion that can even come close to capturing the physical process. Let us at least recognize that composites can be viewed on many levels. Our preference is to view it as a

917

Fig. 16. Stress/strain curves under loading ratio of y /x = 0/1 for E-glass/MY750 [0°/90°] laminate.

homogeneous, continuous, anisotropic material. In this continuum, we do not see fibers or matrix as different phases. Defects and cracks exist in a much smaller scale. They grow larger and eventually lead to failure of the plies and laminates. Failure criteria are limited to the macro rather than the micro level. It is a phenomenological and operational theory, not a mechanistic model. Two Continua? In our approach, we see plies in two states. One is the initial, intact, continuous state. The other is a degraded state, first saturated with micro cracks, and then complete matrix and fiber

Table 1 Summary of the empirical factors. Figure

Material

Layup

F*xy

E*f

E*m

n

Baseline 1 2 3 6 7 8 9 10 11 12 13 14 15 16

0.15

0.1

[0] [0] [0] [90/30/–30] [90/30/–30] [55/–55] [55/–55] [55/–55] [0/45/–45/90] [0/45/–45/90] [0/45/–45/90] [45/–45] [45/–45] [0/90]

–0.5 –0.5 –0.5 –0.5 –0.5 –0.5 –0.5 –0.5 –0.5 –0.5 –0.5 –0.5 –0.5 –0.5 –0.5

0.01

E-glass/LY556 T300/914 E-glass/MY750 E-glass/LY556 E-glass/LY556 E-glass/MY750 E-glass/MY750 E-glass/MY750 AS4/3501-6 AS4/3501-6 AS4/3501-6 E-glass/MY750 E-glass/MY750 E-glass/MY750

0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

0.08 0.08 0.10 0.10 0.10 0.15 0.15 0.15 0.10 0.10 0.15

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

918 failure. In both the intact and degraded states, our criterion for each is quadratic, and has many important features. (1) Six strength parameters completely define the failure envelope. Change in any one or more of the parameters will alter the envelope. A change in compressive strength, for example, will affect not only the compressive but also tensile domain. Some people would ask why so. We would say why not. It is a question whether or not one believes in interactions among strength parameters in combined stress state. (2) Quadratic failure criterion is invariant. It follows all the laws of coordinate transformation. If there is something that we believe in is right, it is automatically right in all coordinate systems. For many failure criteria, it can only be applied in one coordinate system, the material axes. If the coordinate axes change, the failure criteria cannot be applied. There is something suspicious about such criteria. For sure, they are not invariant. The closest thing that we can think of is the often used term of balanced laminate. It is suspicious because this description only applies in one coordinate system. If we say a laminate is thick or thin, heavy or light, orthotropic or anisotropic, failed or not failed, it is true whichever way we look at it. It is true in all coordinate systems; i.e. invariant. Balanced laminate, however, is a description that is coordinate dependent, not invariant. The use of invariants for failure criteria is not only correct mathematically but has physical meaning as well. The first invariant of strain, for example, measures dilatation; the second invariant, total energy; a combination of the two, distortional energy. The first invariant of stress is hydrostatic pressure; the second, total energy, and so forth. Many Russian workers have used combinations of these invariants for failure criteria. Our criterion followed the path of many of these workers. Among the US workers Feng [3] and Christensen [4–6] have proposed similar approaches. There is a lot to be learned from their works. (3) Quadratic criterion is continuous, single valued, and easy to operate mathematically (to differentiate and integrate.) As seen in the last section, quadratic criterion is simple to use and has flexibility to fit test data while maintaining internal consistency. Non-interactive criteria, on the other hand, often invoke new mechanisms to truncate regions where test data cannot support. It would seem reasonable to commit before hand whether or not there is failure mode interaction. Extension of quadratic from 2- to 3-dimensions is straightforward. Only one additional strength parameter is needed for a transversely isotropic material. The extension to 3-dimensions can be an insurmountable challenge for some failure criteria. All the required intersecting lines and planes can make faceted diamond look simple. It can be a graphical nightmare. (4) With degraded plies, a new set of strength parameters must be established. The behavior along and transverse to the fibers are de-coupled by the presence of micro cracks. Poisson’s ratio and the interaction term Fxy* in the criterion for this state are reduced by the same factors Em* and Ef *, for convenience. The longitudinal compressive strength X  is also reduced through the exponent n in the shear modulus ratio. Thus, three factors to model a degraded ply are this exponent n, the matrix degradation factor Em*, and the fiber degradation factor Ef *. In our model, micro cracks are smeared and replaced by a continuum with lower transverse and shear moduli but not zero. The quadratic criterion is applied by the same method as that to the intact plies. As just stated above and shown in the last section, the longitudinal compressive strength X  will assume a new value for the degraded ply, the failure envelope

919 will then be different in the entire stress space, not necessarily limited to the compressive quadrant.

4. Discussion The only unknown factor is the failure interaction term Fxy* for an intact ply. This term, however, has upper and lower bounds. The numerical values are for typical materials are shown in Part A of this exercise. There are no adjustable constants left. For degraded plies, there are three adjustable factors: matrix and fiber degradation factors Em* and Ef *, and longitudinal compressive strength reduction exponent n. While these possible values were cited in the last section, they were based on the materials given for the present study. For a new material, there is no guarantee that the values are correct. Like all empirical factors, they cannot be proven correct for unknown materials. In general, our criterion can be made to fit the data with remarkable ease and flexibility. The plotting routine, which was based on our Mic-Mac/Lite, can be downloaded free from www.thinkcomp.com. It shows the ease of use and flexibility of our approach. For most computers, the calculation time is in seconds. The MicMac/Lite program has five failure criteria: maximum stress, maximum strain, Hashin and Tsai-Wu with interaction factor Fxy* equal to zero and –1/2. The additional criteria can be easily added. A comparison of these failure criteria and the test data used in this paper [2] was made by Tsai and Kuraishi [7].

5. Conclusion Our model has worked in general. It works better for UD than laminates. A further reduction in the gap between the theory and the data is achieved when selective degradation is dropped. For initial stiffness of laminates, classical theory works exceptionally well. The first-plyfailure prediction has only one empirical factor for fine-tuning. The last-ply-failure has three factors, Em*, Ef *, and n. The principal shortfall lies in the difference between macro and micro modeling. What can be modeled as a continuum and what cannot. There is a major concern over the smearing of finite number of micro cracks as a degraded ply. Failure itself is also complicated by time, temperature, residual stresses, moisture effects and the list can go on and on. Still a simple quadratic criterion has done well enough for preliminary design and material screening. The experimental data may suffer as much uncertainties as theoretical models. Effects of materials, geometry, processing and boundary conditions are difficult to control. Testing conditions and instrumentation can also be sources of errors. We were surprised by the number of the people who decided not to participate in this exercise. Industry seems to stand on the sideline. Many people have clung to the same approach for a long time. The present study is not likely to change many minds. For us, we learned a great deal and improved our model and program so we can respond to any inquiry instantaneously in the future. Such capability will benefit both the stress analyst and the experimentalist. Our model is mature enough for preliminary design and material screening. It has been used by many designers over that last 40 years and in many commercial codes. The Mic-Mac/Lite,

920 cited earlier, displays the difference between the five most frequently used criteria. The predicted first-ply-failures fall within band not wider than the data scatter of this study. Our view is that we have reached a point of diminishing return. All of us have exhausted the simple macro models. New micro models can be the next generation approach. Initial or first-ply failures are theoretically sound. Post FPF leading toward the final or lastply failures are less rigorous. Our approach is based on smearing the micro cracks. It implies that within the region where failure has occurred, the area must have many parallel cracks. For homogenous stress, this condition is satisfied. For non-homogeneous situations, such as the regions of stress concentrations, the smearing is not very accurate. Another limitation is that our model is plane stress and does not take into account delamination. 6. Recommendations Internal consistency among test data obtained by different investigators and different specimens will continue to be a challenge. Although this is not easily solved, continued effort must be emphasized. The importance of more test data can not be overemphasized. More test data for different ply orientations and/or different loading conditions will be valuable in evaluating and improving the theories. Stress/strain curves are valuable in this process, because they serve as more severe tests for the theories. Some stress/strain curves in the compression-compression region would have been very useful in this exercise. Non-linear models are always useful but difficult to use and verify experimentally. The most straightforward method would be to combine a local strength analysis engine like MicMac/Lite and a global finite element analysis. Simple test specimens deserve to be modeled as carefully as any structural component. Effort to promote composite materials and structures is a duty that all of us must take seriously. We have to participate in both research and technology transition to users in industry. One airplane company would start strength analysis of composite structures with notched specimens. That is similar to many Russian workers who would start with laminate to determine the principal strength parameters. We think that both approaches are wrong. UD plies should be the basic building blocks. It is heartening that laminated plate theory and initial- or first-ply-failure (FPF) work well. While final and last-ply-failure is less certain, it is only important in certain applications. For more general applications, the FPF is good enough. Acknowledgements We wish to thank the organizers (M J Hinton, A S Kaddour and P D Soden) that have stayed with this exercise for such a long time. All of us owe the individuals and organizations a vote of gratitude. References [1] Liu KS, Tsai SW. A progressive quadratic failure criterion for a laminate. Composite Science and Technology 1998;58(7):102–332.

921 [2] Soden PD, Hinton MJ, Kaddour AS. Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates: Failure exercise benchmark data: V62, pp 1489–1514, 2002, see also Chapter 2.2 in this book. [3] Feng WW. A failure criterion for composite materials. J Comp Mater 1991;25:88–100. [4] Christensen RM. Tensor transformations and failure criteria for the analysis of fiber composite materials. J Comp Mater 1988;22: 874–97. [5] Christensen RM. Stress based yield/failure criteria for fiber composites. Int J Solid Structures 1997;34:529–43. [6] Christensen RM. The number of elastic properties and failure factors for fiber composites. J Eng Mater Technol 1998;120:110–3. [7] Tsai SW, Kuraishi A. Comparison of various failure criteria of orthotropic materials. In: Aicher S, editor. Proc Int Conf on Wood and Wood Fiber Compos. University of Stuttgart; 2000. pp. 573–84.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

922

CHAPTER 5.10

A strain-energy-based non-linear failure criterion: Comparison of numerical predictions and experimental observations for symmetric composite laminates Tarunjit S. Butalia and William E. Wolfe* Department of Civil Engineering and Environmental Engineering and Geodetic Science, The Ohio State University, Columbus, OH 43210, USA Abstract A strain-energy-based model to predict non-linear stress/strain response, failure onset and progression in composite laminates was presented by Wolfe and Butalia (Compos. Sci. Technol. 1998, 58, 1107, see Chapter 3.11). This paper focuses on some of the critical issues relating to the development of that model. Several of the assumptions made in the model are relaxed or modified. A variety of unidirectional and symmetric laminates are studied. A comparison of numerical results obtained from the model and published experimental data (Supplied by Soden, Kaddour and Hinton (see Chapter 2.2) is presented for four different material systems. The strain energy based failure theory predictions agree very well with many of the experimental observations for the material systems investigated in this study.

1. Introduction The longitudinal, transverse and shear strain energies for a material orthotropic along its material axes have been shown by Sandhu [1] to be independent parameters. A failure criterion based on the sum of suitable exponents of the longitudinal, transverse and shear strain energies along lamina axes was presented by Wolfe and Butalia [2]. This model used the non-linear response of laminae under uniaxial longitudinal or x (tension and compression), transverse or y (tension and compression) and shear (xy) loadings to predict the stress/strain response, failure initiation and progression in composite laminates. A piecewise cubic spline interpolation scheme was used to represent the non-linear stress/strain behavior under simple loading conditions. A laminated plate theory was developed which

* Corresponding author. Tel.: + 1-614-688-3048; fax: + 1-614-6883408. E-mail address: [email protected] (W. E. Wolfe).

923 combines an incremental iterative constitutive law with the strain energy based failure criterion. In that study, numerical predictions were made for a variety of laminate configurations under biaxial loading. The objective of this paper is to verify the accuracy of the strain energy based failure model presented by Wolfe and Butalia [2] by comparing the numerical predictions obtained from that model with published experimental data, supplied by Soden et al. [3]. Critical issues relating to the strain energy based model are first discussed. A comparison of the numerical predictions and experimental observations is then presented for four types of composite material systems. For all the material systems, non-linear lamina stress/strain curves were employed, but the effect of thermal stresses was neglected.

2. Strain energy based model: some critical issues A complete description of the strain energy based model was presented by Wolfe and Butalia [2]. In this section we will focus on some issues relating to the model development which can have a significant effect on the failure predictions.

2.1. Strain energy shape factors At the end of each load increment, the stress/strain state of each lamina is checked against the following generalized strain energy based criterion:

冉

冊 冉

兰ˆ 1 1 d1 兰u1 1 d1

m1

+

冊 冉

兰ˆ 2 2 d2 兰u2 2 d2

m2

+

冊

兰ˆ 6 6 d6 兰u6 6 d6

m6

=1

(1)

where ˆ refers to the current strain state while u refers to the ultimate failure strain state. The exponents m1, m2 and m6 are referred to as strain energy shape factors because they govern the shape of the failure surface in the strain energy space. For the case of all the three shape factors being equal to unity, the failure surface in a non-dimensional strain energy space would be octahedral, whereas all shape factors being equal to two would result in a spherical failure surface. Shape factors m1, and m2 can be different in tension and compression [1]. In the model presented, the shape factors are taken to be material parameters with their numerical values depending on the particular material system under consideration. Once these values have been evaluated, they remain the same for all laminate configurations made from that particular material. In Part A of the composite failure exercise [2], in the absence of any experimental data, it was assumed that m1 = m2 = m6 = 1. This octahedral failure surface condition is now allowed to be relaxed and suitable numerical values for the shape factors are evaluated for some of the material systems.

2.2. Failure modes Equation (1) explicitly defines the failure criterion for a lamina. Having determined that a particular ply has failed, the next logical step would be to evaluate the mode of failure in that

924 ply. The model incorporates two types of failure modes – fiber failure and matrix failure. Having determined that failure has occurred in a ply [as per Eq. (1)], the longitudinal strain energy ratio (LSER) is calculated as follows

冉冕 冒冕 冊 冘冉 冊 1 d1

LSER =

ˆ 1

i = 1,2,6

1 d1

m1

u1

兰ˆ i i di 兰ui 1 di

mi

(2)

The calculated LSER value is compared with an assumed critical LSER value. If the calculated LSER value is greater than the critical LSER value, the failure is determined to be fiber dominated, otherwise matrix failure is inferred. Clearly, the choice of the critical LSER value would determine the failure mode. Based on experimental observations for failure of a limited number of materials [1], Wolfe and Butalia [2] for Part A of this failure exercise used a critical LSER value of 10%. The critical LSER value is assumed to be a material property and the evaluation of more appropriate values of this parameter are expected to lead to better initial and ultimate failure strength predictions. 2.3. Ply unloading mechanisms Under a constant loading rate, a fiber mode failure results in the affected ply unloading by transferring the load being carried by it to the unaffected plies while laminate loads are maintained. For the matrix failure mode, the transverse and shear loads being carried by the ply are transferred to the unaffected plies while maintaining laminate loads. However, the process by which these unloading mechanisms occur needs further investigation. Let us consider the stress/strain curve shown in Fig. 1. Point A represents the first ply failure position

Fig. 1. Effect of ply unloading mechanisms on laminate stress/strain path.

925 on the stress/strain curve of a laminate. Some of the possible paths that can be traversed in the stress/strain space of the laminate for unloading of the ply are as follows: 1. Sudden unloading – the failed ply unloads completely and suddenly, i.e. under a constant loading rate condition, the laminate strain increases instantaneously from point A up to point B1. Further loading of the laminate results in the path B1A1. 2. Gradual unloading – the affected ply unloads but not instantaneously. It unloads gradually in small steps as indicated by path AA2. 3. Elastic-perfectly plastic behavior – the failed lamina continues to carry the failure load but does not take on any additional load. This is represented in Fig. 1 by path AA3. Wolfe and Butalia [2] had employed the sudden unloading mechanism for the results presented in Part A of the failure exercise but as will be shown later, each of the above ply unloading mechanisms gives rise to different stress/strain laminate paths beyond first ply failure and ultimate strength predictions.

2.4. Maximum allowable ultimate failure strain for a lamina At the end of each application of loading increment, in addition to the strain energy failure criterion check, the calculated strains were checked against ultimate strain values under simple loadings conditions. Wolfe and Butalia [2] suggested that unless there is experimental evidence to the contrary, it should be presumed that no calculated lamina strains can exceed the ultimate strains measured in the laboratory under uniaxial longitudinal (fiber failure), or transverse or shear (matrix failure) testing. In particular, for all the material systems used in part one of the failure exercise (refer to Table 1), the transverse tensile failure strain is the lowest among all the longitudinal, transverse and shear failure strains. Referring to Fig. 11 in Wolfe and Butalia [2], it was shown that for a unidirectional E-Glass/MY750 lamina under longitudinal compression, the predicted lamina failure load was half that of its ultimate longitudinal compressive strength because the longitudinal loading caused a transverse tensile strain to be developed (due to Poisson’s effect) and this transverse tensile strain value exceeded the transverse tensile failure strain of the material (0.246%). For the laminate to fail at the laboratory measured longitudinal compressive strength of 800 MPa, either the condition that failure strains can not be greater than ultimate measured strains under simple loading condition needs to be relaxed or the variation of Poisson’s ratio with strain needs to be experimentally measured and included in the model. In general, the numerical results presented in this paper do not allow the laminate failure strains to be greater than ultimate strains measured under simplified loading conditions.

2.5. Variation of Poisson’s ratio with strain level The strain energy based model presented by the authors used piecewise cubic spline interpolation functions for modeling the non-linear stress/strain response of a lamina under uniaxial longitudinal (tension and compression), transverse (tension and compression) and

926

Table 1 Properties of unidirectional composite material systems.

Longitudinal tensile strength (MPa) Longitudinal compressive strength (MPa) Transverse tensile strength (MPa) Transverse compressive strength (MPa) In-plane shear strength (MPa) Longitudinal tensile failure strain (%) Longitudinal compressive failure strain (%) Transverse tensile failure strain (%) Transverse compressive failure strain (%) In-plane shear failure strain (%) Major Poisson’s ratio Through the thickness Poisson’s ratio a

Refers to nonlinear stress/strain behavior of the lamina [4].

AS4/3501-6

T300/BSL914C

E-Glass/LY556

E-Glass/MY750

1950a 1480 48 200a 79a 1.38a 1.175 0.436 2.0a 2.0a 0.28 0.4

1500 900 27 200 80a 1.087 0.652 0.245 1.818 4.0a 0.28 0.4

1140 570 35 114 72a 2.132 1.065 0.197 0.644 3.7a 0.278 0.4

1280 800 40 145a 73a 2.807 1.754 0.246 1.2a 4.0a 0.278 0.4

927 shear loading conditions. The model is also capable of accounting for a spline interpolation of Poisson’s ratio as a function of strain of a lamina under uniaxial longitudinal and transverse loading. For the failure exercise, the organizers used a constant Poisson’s ratio. The possibility that the incorporation of Poisson’s ratio-strain data for uniaxial longitudinal and transverse loadings into non-linear models may lead to a more accurate prediction of laminate stress/ strain response and ultimate failure strengths is the subject of an ongoing investigation by the authors.

3. Numerical results and experimental observations The strain energy based failure criterion coupled with an iterative incremental non-linear laminate theory can be used to predict the stress/strain response, failure onset, progression and ultimate failure strength of laminates. For the numerical studies, four types of material systems commonly used in composites were investigated. These material systems were AS4/3501-6 Epoxy, T300/BSL914C Epoxy, E-Glass/LY556/HT907/DY063 Epoxy and EGlass/MY750/HY917/DY063 Epoxy. The material properties for these four systems were provided in Part A of the failure exercise [4] and are summarized in Table 1. The unidirectional and symmetric laminate configurations investigated in this study were selected by the organizers after a comprehensive review of experimental results published in the literature on the failure of composite laminates under biaxial loading. All the strain energy based numerical predictions presented by the authors in Part A of the failure exercise [2] used the following assumptions: Assumption 1. The shape energy factors (m1, m2, m6) are all unity. Assumption 2. The critical longitudinal strain energy ratio has a numerical value of 10%. Assumption 3. Failed plies unload as per the sudden unloading mechanism. Assumption 4. The calculated strains in any lamina can not exceed the ultimate strains observed in the laboratory under simple loading conditions of longitudinal (tension and compression), transverse (tension and compression) and shear. Assumption 5. Poisson’s ratio does not vary with strain level. In this section, we present the predictions of the numerical model using the above assumptions as well as improved predictions obtained by modifying one or more of the above assumptions. A detailed discussion on the results obtained by making the above assumptions has been presented by Wolfe and Butalia [2] and will not be repeated here. Instead, we will focus on comparing the numerical model predictions with experimental data made available by the organizers in Part B of the failure exercise and investigate possible relaxations of the above assumptions to achieve more accurate predictions for stress/strain behavior and ultimate laminate strength. Unless stated otherwise, all the experimental results included in this paper were obtained by tests conducted on tubular specimens under axial tension or compression, internal or external pressure and torsion loading combinations. While comparing the numerical predictions with the experimental results, the reader needs to be aware of the following sources of errors: 1. Assumptions, approximations and errors in the model development 2. Errors in specimen construction, experimental implementation and data measurement 3. Errors in basic material properties.

928 4.1. AS4/3501-6 laminates Investigations into the stress/strain behavior of the AS4/3501-6 carbon/epoxy material system under simple loading conditions were carried out by Swanson and his co-workers and are summarized in Table 1. A detailed description of these tests is presented in an accompanying paper by the organizers of the failure exercise. Let us consider a [90°/ + 45°/–45°/0°]S laminate under a combination of longitudinal or x (tensile or compressive) and transverse or y (tensile or compressive) loading. The biaxial failure stress predictions obtained from the strain energy based model using assumptions 1–5 for the complete range of loading combinations as presented by Wolfe and Butalia [2] are shown in Fig. 2. Experimental failure stresses reported by Swanson and co-workers [5–8] are also shown in Fig. 2. The experimental results were obtained by subjecting tubular specimens to a combination of axial loads (tensile or compressive) and internal/external pressures. The failure stress predictions reported by Wolfe and Butalia [2] were slightly lower than the experimental values for transverse tensile and longitudinal (tension and compression) quadrants. However, as seen in Fig. 2, by allowing for gradual unloading of failed plies (in about 50 unloading linear functions), slightly higher biaxial failure stresses, which are in better agreement with experimental observations for the quadrants mentioned above, are obtained. Agreement between numerical predictions and some experimental observations for

Fig. 2. Biaxial failure stresses for quasi-isotropic [90°/ + 45°/–45°/0°]s AS4/3501-6 laminates.

929 the biaxial tension quadrant is not good. The experimental results presented by Swanson and Colvin [9] for the longitudinal and transverse compressive quadrant are much lower than the numerical predictions because the failure of the tubular specimens in this quadrant is governed by buckling and it would be expected that the laminate crushing strength would be much higher than the buckling failure strength. The experimental laminate failure stresses reported by Swanson and Christoforou [6] for the tensile longitudinal (x) and transverse (y) quadrant are generally higher than those reported by Swanson and Trask [7]. The stress/strain response of a [90°/ + 45°/–45°/0°]S laminate under a transverse (y) tensile loading is presented in Fig. 3. Experimental results for a loading combination of y /x = 20/1 were reported by Christoforou [10] by subjecting tubular specimens with friction seals to internal pressure. The numerical results are presented for sudden unloading of plies [2] and gradual unloading of plies beyond initial failure. Gradual unloading of plies results in a much less amount of nonlinearity for the predicted curve compared to the sudden ply unloading condition. The transverse stress-transverse strain curve for the experimental observations and numerical predictions are almost identical provided the plies are allowed to unload gradually. The modified transverse stress-longitudinal strain curve also agrees well the experimental data. The small difference between the experimental and modified numerical model curves may be due the assumptions made for Poisson’s ratio. The initial failure occurred at 225 MPa due to transverse failure of 0° plies. Experimental observations indicated that an intermediate failure of the laminate occurred around 400 MPa and this is confirmed by the numerical model predicting an intermediate failure stress of 465 MPa due to matrix failure of the ± 45° plies. This is followed by longitudinal failure of 0° plies at 629 MPa. The ultimate transverse (y) failure stress obtained from the experimental tests was 718 MPa [10], while the model with gradual unloading of plies predicted a failure stress of 669 MPa due to longitudinal failure of 90° plies. An additional test conducted by Swanson and Trask [7] and Swanson and Colvin

Fig. 3. Stress/strain curves for quasi-isotropic [90°/ + 45°/–45°/0°]s AS4/3501-6 laminate (y /x = 1/0).

930 [9] on tubes with the stress ratio of 1 : 0 ( instead of 20 : 1) indicated a failure stress of 713 ( ± 42) MPa. Now consider a stress ratio of y /x = 2/1 for the [90°/ + 45°/–45°/0°]S carbon/epoxy laminate. The experimental stress/strain curves measured by Trask [11] using tubular specimens under tensile axial load and internal pressure are presented in Fig. 4. The numerical predictions of Wolfe and Butalia [2] with sudden unloading of plies and modified to allow for gradual unloading of plies are also shown in Fig. 4. The experimental and numerical model (gradual unloading of plies) transverse stress-axial strain curves are almost identical up to a load level slightly lower than the experimentally observed laminate failure stress. Initial transverse failure occurs in the 0° plies at 246 MPa, followed by transverse failure of ± 45° plies at 328 MPa. The 90° plies failed in transverse and longitudinal direction at 497 MPa and 722 MPa, respectively. Final failure occurred at 741 MPa due to longitudinal failure of ± 45° plies. The experimental results indicated a possible matrix cracking around 450 MPa, which may correspond with the model predicting a transverse (y) matrix failure of the 90° plies at 497 MPa. The predicted transverse (y) final failure stress using the gradual unloading ply mechanism is slightly lower than the experimentally observed value.

3.2. T300/BSL914C laminates Most of material property data listed in Table 1 for the T300/BSL914C carbon/epoxy material system were provided by Schelling and Aoki [12]. However, the shear stress/strain response of the material was assumed to be the same as that of T300/914 carbon/epoxy material as

Fig. 4. Stress/strain curves for quasi-isotropic [90°/ + 45°/–45°/0°]s AS4/3501-6 laminate (y /x = 2/1).

931

Fig. 5. Biaxial failure stresses for 0° unidirectional T300/BSL914C lamina.

reported by Panhwar and Scott [13], Ladevese and Dantec [14] and Lafarie-Fernot and Touchard [15]. Consider a 0° unidirectional T300/BSL914C lamina subjected to a combination of longitudinal or x (tensile or compressive) and shear loading. The tubular experimental biaxial failure stresses obtained by Schelling and Aoki [12] and numerical predictions using the strain energy based model [2] are shown in Fig. 5. Both the experimental results and the numerical predictions of Wolfe and Butalia [2] indicate the failure stresses to be increasing with even a small increase in the axial tensile stress. The comparison between the experimental and numerical results is good for the longitudinal tensile-shear quadrant except for a set of experimental data points which are in vicinity of the stress ratio x /xy = 6.25/1. For the longitudinal compressive-shear quadrant, the agreement is quite reasonable. As can be seen in Fig. 5, there is a large scatter in the experimental shear strength observations (ranging from about 50 to 100 MPa) when no longitudinal load is applied. A shear strength value of 80 MPa for the T300/BSL914C material was provided by the organizers. By increasing the shear strength of the material from 80 to 100 MPa (refer to Fig. 5), higher failure stresses which agree very closely with most of the experimental data provided by Schelling and Aoki [12] are calculated.

3.3. E-Glass/LY556 laminates Material property data for the E-Glass/LY556 system listed in Table 1 were obtained from Krauss and Schelling [16], Forster [17] and Hutter et al. [18]. The longitudinal or x (tensile

932 and compressive), transverse (y) compressive and shear strength for E-Glass/LY556 material are the lowest among the materials investigated in this study. Consider a 0° unidirectional lamina under transverse (tensile or compressive) and shear loading. Hutter et al. [18] conducted experiments on circumferentially wound tubes under torsion and axial loading and the results obtained from these tests are presented in Fig. 6. The numerical results presented by Wolfe and Butalia [2] differ from the experimental results largely because calculated shear strengths were higher and transverse (y) compressive strengths were lower than the experimental observations. Decreasing the shear strength from 72 to 61 MPa, increasing transverse compressive strength from 114 to 140 MPa, taking the critical LSER value to be 50%, allowing for elastic-perfectly plastic ply unloading mechanism and allowing calculated biaxial strains to exceed experimentally observed strains under simple loading conditions by 20%, result in much better agreement between the predictions obtained from the strain energy based numerical model and experimental observations. It needs to be noted that the modified unloading assumption and revised critical LSER value did not influence the results of Fig. 6 but were found to be parameters suitable for the E-Glass/LY556 material system investigations presented in Figs 7 and 8. Considering the [90°/ + 30°/–30°]S laminate under longitudinal (x) and transverse (y) loading, the numerical predictions presented by Wolfe and Butalia [2] did not agree well with the tubular experimental observations of Hutter et al. [18] (see Fig. 7). However, by modifying the strain energy based model and the input data as elaborated above, a significant increase in the failure stresses was calculated. It can be seen that the modified numerical results agree well with the experimental data for all quadrants except the longitudinal and transverse compressive loading region. The failure stresses obtained in this region are governed by

Fig. 6. Biaxial failure stresses for 0° unidirectional E-Glass/LY556 lamina.

933

Fig. 7. Biaxial failure stresses for [90°/ + 30°/–30°]s E-Glass/LY556 laminate (x  y).

buckling so it would be expected that the numerical crushing predictions would be higher than the experimental data. A comparison of the experimental and numerical results for the [90°/ + 30°/–30°]S E-Glass/ LY556 laminate under longitudinal (x) and shear (xy) loading is presented in Fig. 8. It is clearly seen that modifications of the assumptions made in the original strain energy based model and the input strength data result in much higher biaxial failure stress predictions and exhibit good agreement with the experimental results presented by Hutter et al. [18]. However, the longitudinal tensile strength of the laminate predicted by the modified strain energy based model is still somewhat lower than the experimental results. 3.4. E-Glass/MY750 laminates The material properties listed in Table 1 for the E-Glass/MY750 material system were provided by Hinton [19] to the organizers of the failure exercise. The longitudinal and transverse tensile stress/strain curves were obtained by subjecting thin walled circumferentially wound tubular specimens to internal pressure and axial tension respectively. The longitudinal compressive behavior was studied using a thick block of unidirectional lamina.

934

Fig. 8. Biaxial failure stresses for [90°/ + 30°/–30°]s E-Glass/LY556 laminate (x  xy).

The longitudinal compressive strength of the material changes with varying fiber content and a range of 520–1150 MPa was observed [19]. The organizers of the failure exercise interpolated the longitudinal compressive strength for 60% fiber volume fraction to be 800 MPa. Experimental verification of this interpolated strength value is lacking at this time. The transverse compressive stress–strain response of the material was studied by Harwood and Hinton [20] using thick hoop wound tubular specimens under axial compression and by Jones [21] using a thick unidirectional lamina block. The non-linear shear stress/strain response was investigated by subjecting tubular specimens to torsional loading. The numerical failure stress values presented by the authors [2] for loading a 0° unidirectional lamina of E-Glass/MY750 under longitudinal (x) and transverse (y) loading are shown in Fig. 9 along with the experimental results obtained by Al-Khalil et al. [22]. The experimental values were obtained by testing nearly circumferentially wound ( ± 85°) tubular specimens under axial load and internal pressure. As seen in Fig. 9, the author’s original predictions did not agree well with the experimental observations. In addition, the numerical model exhibited a sudden jump at y /x = 0/1 and 0/–1. The jump is largely a consequence of the application of assumption 4. As shown in Fig. 9, relaxation of this assumption, taking m1 = m2 = m6 = 2.0, and allowing for the elastic-perfectly plastic ply unloading mechanism, results in better numerical predictions. The sudden jumps predicted earlier by the numerical model are eliminated and good agreement is obtained between the modified numerical results and experimental data points. It also needs to be observed that the above modifications give rise to much larger failure stresses than those calculated by the authors in 1998 for the longitudinal and transverse tensile loading quadrant. This is largely due to the increased shape factor values as well as allowing the laminate failure strains to exceed lamina failure strains under simple loading conditions by 20%. Let us consider a [ + 55°/–55°]S angle ply laminate under longitudinal (x) and transverse (y) loading. Experimental investigations into biaxial failure strengths of the [ + 55°/–55°]S angle ply tubular specimens have been carried out by several researchers. Experimental failure

935 stresses obtained by Soden et al. [23,24] using thin and thick tubes with and without a flexible liner, and Kaddour et al. [25] using thick tubes are shown in Fig. 10 along with the numerical predictions. It can be seen that the original numerical predictions of the authors [2] did not agree very well with the experimental observations. Modifications to the strain energy based model as described above results in only somewhat better numerical predictions. Some of the modified numerical predictions agree quite well with the tests conducted by Soden et al. [23,24] without the use of a flexible liner. The use of a flexible liner in the tubular experiments results in much higher failure stresses. The thick tubular results of Kaddour et al. [25] also agree well with the modified numerical predictions (due to allowing of laminate failure strains to exceed lamina failure strains under simple loading conditions by 20%) except for stress ratios in the vicinity of y /x = –2/–1. In the tension–tension quadrant, numerical predictions generally do not agree well with experimental observations. Consider the [ + 55°/–55°]S angle ply laminate under a transverse (y) tensile loading. AlKhalil [26] conducted experiments on lined tubular specimens with several strain gauges at the outer surface by subjecting the tubes to internal pressure. It was observed that individual strain gauge readings differed up to approximately ± 20%. Axial loads due to seal friction and bulging of the specimen were neglected. The experimental stress/strain curve shown in Fig. 11 uses only one of these strain gauges. The original model predictions obtained by the authors (1998) and the modified model as explained above, are almost identical and agree well with the measured curves up to 280 MPa stress. Modification of the numerical models result in a

Fig. 9. Biaxial failure stresses for nearly 0° unidirectional E-Glass/MY750 lamina.

936

Fig. 10. Biaxial failure stresses for [ + 55°/–55°]s angle ply E-Glass/MY750 laminate.

failure stress of 280 MPa while Wolfe and Butalia [2] predicted it to be 253 MPa. Testing of unlined and lined tubes indicated failure strengths of approximately 400 MPa [27] and 600 MPa [26] respectively. Similarly, subjecting a [ + 55°/–55°]S angle ply laminate to a loading combination of y /x = 2/1 results in lower failure stress predictions (Fig. 12) than observed by [26] on lined tubular specimens and Soden et al. [23] for unlined tubes.

Fig. 11. Stress/strain curves for [ + 55°/–55°]s angle ply E-Glass/MY750 laminate (y /x = 1/0).

937

Fig. 12. Stress/strain curves for [ + 55°/–55°]s angle ply E-Glass/MY750 laminate (y /x = 2/1).

For a [ + 45°/–45°]S E-Glass/MY750 laminate under equal longitudinal (x) and transverse (y) tensile loading, the measured and numerical predicted behaviors are shown in Fig. 13. The experimental results were obtained by Reid et al. [28] using tubular specimens under axial tensile load and internal pressure. The strain gauge readings varied by as much as 22%. Ideally, the longitudinal and transverse strains for this laminate under y /x = 1/1 loading should be the same [29] however the experiments showed a consistently higher longitudinal

Fig. 13. Stress/strain curves for [ + 45°/–45°]s angle ply E-Glass/MY750 laminate (y /x = 1/1).

938 strain. Soden et al. [24] reported observing leakage of unlined specimens at around 200 MPa. The failure stresses predicted by the authors in 1998 and the modified model are identical but much lower than the failure stresses obtained from the testing of unlined and lined tubular specimens. The numerical model predicts failure at around 68 MPa which corresponds closely with the initial cracking of tubular specimens at stress levels as low as 50–70 MPa [28,29]. The experimental lined tubular results [30] and numerical predictions for a [ + 45°/–45°]S laminate under y /x = 1/–1 1oading are presented in Fig. 14. The stress/strain curves obtained from the strain energy based model and the experiments agree well up to a transverse tensile stress of about 50 MPa. The lined tubular specimens failed at a transverse tensile stress of approximately 95 MPa while the numerical model predicted a lower failure stress of 73 MPa. A [0°/90°]S E-Glass/MY750 laminate under longitudinal (x) tensile loading was also studied. Experimental results on tubular [31–33] and coupon [33,34] specimens for this laminate configuration were made available to the organizers of the failure exercise. The experimental results for the coupon specimens [34] along with the predictions of [2] and the modified model as explained earlier are shown in Fig. 15. It can be seen that the modified numerical model results in a much more accurate stress/strain curve as well as a higher failure stress compared with the original predictions. The modified numerical model predicts a stressstrain behavior that is in excellent agreement with the experimental coupon measurements. Similar to the experimental observations, the modified numerical model predicts initial failure due to transverse cracking of 90° plies at 77 MPa and final failure due to fiber fracture of the 0° plies at 510 MPa. The failure strengths predicted by Hinton [34] and Eckold [33] range from 520 to 660 MPa. The modified numerical model predicts a laminate failure strength of 510 MPa.

Fig. 14. Stress/strain curves for [ + 45°/–45°]s angle ply E-Glass/MY750 laminate (y /x = 1/–1).

939

Fig. 15. Stress/strain curves for [0°/90°]s cross ply E-Glass/MY750 laminate (y /x = 0/1).

4. Discussion It has been shown that improved predictions are obtained using the strain energy based model by making material specific modifications to the assumptions that the authors made in their model in 1998. The model parameters include strain energy shape factors in longitudinal and transverse tension and compression and shear as well as the critical longitudinal strain energy ratio (LSER). Additional numerical model options are various ply unloading mechanisms, maximum allowable ultimate strain for a lamina, and the variation of Poisson’s ratio with strain level. The model parameters are basic material properties for a given material system and are evaluated from reliable biaxial testing data for that material system. Once evaluated, these material properties apply to all laminate configurations for the material system under study. Hence, the model parameters and numerical modeling assumptions will depend only on the material system under study and not on the particular laminate configurations. From a designer’s perspective, the strain energy based model can be very useful in predicting stress/ strain response, initial failure, failure progression, and final failure for a variety of laminate systems and configurations. The numerical predictions and experimental observations agree well for all the material systems investigated in this study except for some E-Glass/MY750 angle ply laminate

940 configurations. The laminate failure strength predictions using the strain energy based model are generally lower than experimentally observed failure strengths. For E-Glass/MY750 angle ply laminates, typically initial and final failure are coincident and the failure generally occurs due to matrix cracking in shear or transverse tension. Investigations into the discrepancies between the experimental observations and numerical predictions for angle ply laminates, particularly due to LSER value and thermal stresses, are currently under way. In order to establish the effect of strain varying Poisson’s ratio on the failure strains, tests will need to be conducted to evaluate the variation of Poisson’s ratio with strain level. This may lead to more accurate prediction of laminate stress/strain response and ultimate failure strengths. The effect of strain varying Poisson’s ratio on biaxial failure of composite laminates is the subject of an ongoing investigation by the authors.

5. Conclusions Based on the original work of Sandhu [1], the authors presented a generalized strain energy based failure criterion for the response of non-linear composite laminates under biaxial loading [2]. The model coupled the cubic spline interpolation of non-linear lamina properties with an incremental iterative laminated plate theory to predict the stress/strain response, initial damage, progression and ultimate strengths for a wide variety of composite laminates. In this paper, several important issues relating to the strain energy based model were discussed. These issues included the evaluation of: (1) strain energy shape factors; (2) failure modes; (3) ply unloading mechanisms; (4) maximum allowable ultimate strains under biaxial loading; and (5) variation of Poisson’s ratio with strain level. Four material systems were studied for the failure exercise being coordinated by Soden et al. [3,4]. A comprehensive evaluation of performance of the strain energy based model was carried out by comparing numerical model predictions with experimental results obtained by a diverse group of researchers. Several of the assumptions made by Wolfe and Butalia [2] were relaxed for each material system to obtain improved predictions. The strain energy based model results agree very well with experimental observations for unidirectional and symmetric laminates made of AS4/3501-6, T300/BSL914C, and E-Glass/LY556 material. The model predictions for E-Glass/MY750 material for some laminate configurations do not agree well with experimental results. For these configurations, the strain energy based model generally predicts lower failure strengths than those from experimental testing. The laboratory measurement and subsequent inclusion of a strain varying Poisson’s ratio in the model may result in better ultimate strength predictions for composite laminates under biaxial loading.

Acknowledgements The authors would like to express their gratitude to Drs. Raghbir S. Sandhu and Ranbir S. Sandhu for their assistance. The authors acknowledge the help provided by graduate student Mr. Dong-Gyou Kim of The Ohio State University for his help in preparation of this manuscript. The patient efforts of Messers P. D. Soden, M. J. Hinton and A. S. Kaddour in organizing this exceptional composite failure exercise are appreciated.

941 References [1] Sandhu RS. Ultimate strength analysis of symmetric laminates. AFFDL-TR-73-137, AD 779927, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio, 1974. [2] Wolfe WE, Butalia TS. A strain-energy based failure criterion for non-linear analysis of composite laminates subjected to biaxial loading. Composites Science and Technology 1998;58:1107–24. [3] Soden PD, Hinton PD, Kaddour AS. Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates: Failure exercise benchmark data: V62, pp 1489–1514, 2002, see also Chapter 2.2 in this book. [4] Soden PD, Hinton MJ, Kaddour AS. Lamina properties, lay-up configurations and loading conditions for a range of fibre reinforced composite laminates. Composites Science and Technology, V58, pp 1101–1022, 1998, see also Chapter 2.1 in this book. [5] Swanson SR, Nelson M. Failure properties of carbon/epoxy laminates under tension-compression biaxial stress. In: Kawata K, Umekawa S, Kobayashi A, editors. Composites’86: recent advances in Japan and United States, Proceedings of Japan-US CCM-III Tokyo, Japan, pp. 279–86. [6] Swanson SR, Christoforou AP. Response of quasi-isotropic carbon/epoxy laminates to biaxial stress. Journal of Composite Materials 1986;20:457–71. [7] Swanson SR, Trask B. Strength of quasi-isotropic laminates under off-axis loading. Composites Science and Technology 1989: 19–34. [8] Colvin GE, Swanson SR. In-situ compressive strength of carbon/epoxy AS4/3501-6 laminates. Journal of Engineering Materials and Technology 1993;115:122–8. [9] Swanson SR, Colvin GE. Compressive strength of carbon/epoxy laminates under multiaxial stress, Final annual report to the Lawrance Livermore National Laboratory, UCRL21235, 1989. [10] Christoforou AP. An investigation of composite material response under tension-tension biaxial stresses. MS thesis, Department of Mechanical and Industrial Engineering, the University of Utah, Utah, USA, 1984. [11] Trask, B. Response of carbon/epoxy laminates to biaxial stress. MSc thesis, The University of Utah, Utah, USA, 1987. [12] Schelling H, Aoki RM. DFVLR, Germany, Personal communication with organizers of failure exercise, 1992. [13] Panhwar NM, Scott ML. Shear fatigue performance of post-buckling fibre composite panels. in ICCM-9 1993;1:527–34. [14] Ladevese P, Le Dantec E. Damage modeling of the elementary ply for laminated composites. Composites Science and Technology 1992;43:257–67. [15] Lafarie-Fernot MC, Touchard F. Comparative in-plane shear behaviour of long carbon-fibre composites with thermoset or thermoplastic matrix. Composites Science and Technology 1994; 52:417–25. [16] Krauss H, Schelling H. Kunststffe 1969;59:911–7. [17] Forster R. Kunststoffe 1972;62:57–62. [18] Hutter U, Schelling H, Krauss H. An experimental study to determine failure envelope of composite materials with tubular specimen under combined loads and comparison between several classical criteria. In: Failure modes of composite materials with organic matrices and other consequences on design, NATO, AGRAD, Conference Proceedings No. 163, Munich, Germany, 13–19 October 1974. [19] Hinton MJ. Personal communication with organizers of failure exercise, 1995. [20] Harwood CK, Hinton MJ. Investigation of the transverse compression properties of glass fibre/ epoxy resin system using circumferentially wound tubes, RARDE, Memorandum 38/79 (EM4), MoD, UK, 1979. [21] Jones RM. Mechanical properties of GRP in the through thickness. MSc Dissertation, UMIST, Manchester, UK, 1986.

942 [22] Al-Khalil MFS, Soden PD, Kitching R, Hinton MJ. The effects of radial stresses on the strength of thin walled filament wound GRP composite pressure cylinders. International Journal of Mechanical Sciences 1996;38:97–120. [23] Soden PD, Kitching R, Tse PC. Experimental failure stresses for + 55° filament wound glass fibre reinforced plastic tubes under biaxial loads. Composites 1989;20:125–35. [24] Soden PD, Kitching R, Tse PC, Tsavalas Y, Hinton MJ. Influence of winding angle on the strength and deformation of filament wound composite tubes subjected to uniaxial and biaxial loads. Composites Science and Technology 1993;46:363–78. [25] Kaddour AS, Soden PD, Hinton MJ. Failure of ± 55 filament wound composite tubes under biaxial compression. Journal of Composite Materials, V32, no. 18, pp 1618–1645, 1998. [26] Al-Khalil, M.F.S. Strength of filament wound structure under complex loading, PhD thesis, UMIST, England, UK, 1990. [27] Al-Salehi FAR, Al-Hassani STS, Hinton MJ. An experimental investigation into the strength of angle GRP tubes under high rate of loading. Journal of Composite Materials 1989;23:288–305. [28] Reid SR, Soden PD, Li S. Application of damage models to filament wound tubes, MoD final report, Contract No 2044/186, 1995. [29] Li S. Modelling damage in thin walled filament wound structures. PhD thesis, UMIST, Manchester, UK, 1993. [30] Kaddour AS, Soden PD, Hinton MJ. Unpublished work, 1986. [31] Eckold GC. The effect of damage on the mechanical properties of composites and its impact on design. IMechE Conference Designing Out Failure in Composites, London, 23 September, 1994. [32] Eckold GC, Hancox NL, Lee RJ. Application of micromechanics in the prediction of damage initiation and growth in structural composites. In: 3rd International Conference on Deformation and Fracture of Composites, Institute of Materials, University of Surrey, Guilford, UK, 27–29 March, 1995, pp. 66–76. [33] Eckold GC. Personal communication with organizers of failure exercise, 1997. [34] Hinton MJ. Personal communication with organizers of failure exercise, 1997.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

943

CHAPTER 5.11

A coupled analysis of experimental and theoretical results on the deformation and failure of composite laminates under a state of plane stress Peter A. Zinovieva,*, Olga V. Lebedevab and Ludmilla P. Tairovaa a b

Bauman’s Moscow State Technical University, 2nd Baumanskaya Str. 5, Moscow, 107005 Russia lnstitute of Composite Technologies, Orevo, Dmitrovskii raion, Moskovskaya obl., 141814 Russia

Abstract The paper presents a comparison of experimental and predicted data within the frameworks of the “Failure Prediction of Composite Laminates” exercise. The comparison was performed by using a wide variety of experimental data specially selected by the organisers of the exercise. Test problems included the data of 14 experiments carried out by different scientists with four types of reinforced plastics. The behaviour of six different composite laminates was studied under several types of combined state of stress. The present paper deals with the comparison of theoretical results and experimental data. It is stated that the theoretical model proposed by Zinoviev and Tarakanov and the STRAN computer program based on the model, adequately describe the proposed set of experimental data. The merits and demerits of the model are analysed as well as the ways for its potential improvement. A short description of further refinement of the theoretical model is given. Several possible directions for further investigations are proposed. Keywords: C. Deformation; B. Strength; Composite laminates

1. Introduction Initial data provided by the organisers of the “Failure Prediction of Composite Laminates” exercise [1] were used to predict the behaviour of a number of composite laminates. The theoretical model [2] developed by the authors was used to carry out the necessary calculations. Complicated processes take place during the deformation of laminated composites, and these processes result in going from the natural initial material state to the final material state – failure. Theoretical models must describe deformation and failure processes as adequately

* Corresponding author. E-mail address: [email protected] (P. A. Zinoviev).

944 as possible. Failure of laminated composites is not a single-step act of going from initial intact state to the state of complete failure. The processes of changing (degradation of) material properties inevitably progress during the deformation of laminated composites resulting in final failure of the materials. It is often believed that the above processes can be divided into two parts: the initial completely elastic (reversible) composite behaviour and the following stage of accumulation of irreversible damages – the degradation of composite properties. In case we study laminated composites, i.e. the materials that are non-uniform in the structure, the turn from elastic (reversible) deformation to the state of irreversible degradation of the properties can be ‘personally specified’, or in other words, can be related to each element of the material structure. This is a convenient engineering hypothesis that there exists a zone of ideally elastic reversible deformation for individual constitutive plies and laminated composites. This assumption rather adequately agrees with available experimental data (among those are experimental data considered below). At the same time, it should be admitted that experimental data exist (for example, stress/strain curves for unidirectional laminae under shear [3]) which make the assumption a doubtful one. The theoretical model applied for the calculations [2] assumes that the ply remains elastic when the following conditions are fulfilled F  1 ≤ 1 ≤ F + 1, F  2 ≤ 2 ≤ F + 2, | 12 | ≤ F12

(1)

Here F + 1, F + 2 are ultimate tensile stresses along and transverse to the fibres, respectively; F  1, F  2 are equivalent characteristics in compression, and F12 is the ultimate in-plane shear stress. Merits and demerits of assumption (1) are discussed in detail in Ref. [2]. The model assumes that the properties of reinforced fibres practically do not change at deformation, all processes of changing material properties are related to the development of the cracks in the matrix. Such changes are considered in the model by the modification of ply properties in the transverse direction and under shear. The model includes two groups of ply states under nonlinear deformation: the ply with ‘open longitudinal cracks’ (positive values of transverse stresses in the ply, 2, that is the ply is under transverse tension) and the ply with ‘closed longitudinal cracks’ (negative transverse stresses, 2, that is the ply is transversely compressed. Within the ‘open cracks’ sub-group, the model distinguishes four states of the ply: • active (‘pioneer’) deformation in transverse and shear directions, viz., the strains under transverse tension and the absolute magnitude of the strain under shear are maximum strains during the overall history of deformation; the increments of both strains are positive; • active (‘pioneer’) deformation in the transverse direction and unloading in shear; • active (‘pioneer’) deformation in shear and unloading in the transverse direction; • unloading in the transverse direction and in shear. Within the ‘closed cracks’ sub-group, we distinguish two states of the ply: • active (‘pioneer’) shear deformation; • deformation in the developed strain interval with the unloading shear modulus.

945 It is important that the above types of states are not determined once and for all but can vary in the deformation process. Another important feature is that the model considers the so-called ‘structural nonlinearity’, i.e. the possibility of reinforcing angles to be changed in the deformation process of composite laminates. As practice shows, such consideration is necessary, first of all for the prediction of the deformability of composite laminates including a small number of independent reinforcing directions (for example, angle-ply laminates). In our opinion, a test problem below provides a convincing illustration for such a necessity: this is the case of the ± 45° angle-ply E-glass/MY750 laminate under biaxial loading y /x = 1/–1 (see Fig. 10). Other examples of the ‘structural non-linearity’ can be found, for example, in Ref. [4]. Lastly, one more significant feature of the model is that the model considers the possibilities for ‘unstable’ deformation in composite laminates, i.e. a particular deformation mode in which the strains increase practically without any increase of the load, but the laminate shows no evidence of failure. Such deformation mode is typical, for example, for angle-ply composite laminates with reinforcing angles ±  > 45° under uniaxial tension y /x = 0/1. The present paper is a contribution to Part B of the “Failure Exercise” [1]. The paper analyses the comparison of experimental and predicted data. 2. Comparison between experimental and predicted results Below is given a successive consideration of experimental results [1] for two types of carbon fibre reinforced plastic and two types of glass fibre reinforced plastic and their comparison with theoretical predictions [2]. 2.1. Carbon fibre reinforced plastic AS4/3501-6 The organisers provided for the analysis the stress/strain curves for two types of stress states and the biaxial failure stress envelope of the laminate with the structure (0°/ ± 45°/90°). Stress/strain curves for the (0°/ ± 45°/90°) laminate under uniaxial tension (y /x = 1/0) are shown in Fig. 1. The theory predicts the four stages of deformation: (1) the initial elastic deformation of the laminate lasts up to y = 241 MPa when cracking begins in the 0° plies due to transverse tension; (2) the second stage of deformation lasts up to y = 466 MPa when the cracks appear in the ± 45° plies due to shear stresses; (3) the third stage of deformation, when only the 90° plies are undamaged, lasts up to y = 728 MPa; (4) lastly, let us note the fourth specific stage of the deformation – this is the stage of the final laminate failure. The fourth stage starts and finishes at y = 728 MPa, the complete failure of the laminates, as is predicted, occurs due to longitudinal tension of the 90° plies. One of the assumptions in the theoretical model is that the composite laminate is believed to be completely fractured, if only a single ply fails in longitudinal tension. Comparison between experimental and predicted data shows that the assumption is well justified in the case under consideration. Such assumption can be accepted for composite laminates consisting of a relatively small number of the plies (no more than 10 plies of various orientations). It seems likely that the correlation between theoretical and experimental results is adequate in this case. If one considers the main principles of the theoretical model [2], then it should

946

Fig. 1. Stress–strain curves for the (0°/ ± 45°/90°) AS4/3501-6 laminate under uniaxial tension (y /x = 1/0).

be mentioned that the assumption about the complete failure of composite laminates if only a single ply fails in longitudinal tension is quite justified for the above type of laminates. For the (0°/ ± 45°/90°) CFRP laminate under biaxial tension (y /x = 2/1), the model identifies the five stages of the deformation process (see Fig. 2). The first stage is the initial elastic deformation up to y = 243 MPa when the cracks appear in the 0° plies due to transverse tension; the second stage of the deformation lasts up to y = 322 MPa when the cracks appear in the ± 45° plies due to transverse tensile stresses; the third stage lasts up to y = 487 MPa (note that this stress well correlates to the point of “possible matrix cracking” mentioned in the experiments by Trask (1987) [1] where the cracking occurs in the 90° plies because transverse stresses reach their ultimate value F + 2. At the fourth stage of the deformation, which goes from y = 487 MPa up to y = 825 MPa, all the plies of the laminated CFRP have interior cracks. The fifth stage is the stage of the final laminate failure: as before, the failure occurs due to longitudinal tension of the 90° plies at y = 825 MPa. The comparison of experimental and theoretical curves demonstrates a rare case of surprisingly complete agreement. Such agreement may serve as circumstantial evidence that the procedure of reducing the stiffnesses of the plies is correct when the cracks appear in the plies due to transverse stresses. Experimental and theoretical values of failure stresses for the AS4/3501-6 CFRP laminate are given in Fig. 3. Theoretical data on the laminate strength quite adequately correlate with experimental data in the positive x ~ y quadrant. According to the theory, the laminate fails due to the fracture of the 0° plies in longitudinal tension. Note that the crosses in the first quadrant, mentioned as “matrix failure”, are very close to theoretical broken lines corresponding to the beginning of matrix cracking in the + 45 and 90° plies.

947

Fig. 2. Stress/strain curves for the (0°/ ± 45°/90°) AS4/3501-6 laminate under biaxial tension (y /x = 2/1).

Experimental points for axial compression of tubular specimens (y = 0 in Fig. 3) can be divided into two groups. The first one was obtained by Swanson and Nelson (1986) [1] with the use of relatively thin tubular specimens; this group is the result of tubes buckling and does not describe the compressive strength of the material. The group of the experiments (Colvin and Swanson (1993) [1]) was carried out with relatively thick-walled tubes. The average axial compressive strength of these specimens (x = 637 MPa) is rather close to the theoretical envelope which explains the laminate failure by longitudinal compression in the 90° plies. Pay attention to experimental points for the (0°/ ± 45°/90°) CFRP laminate under uniaxial compression for x = 0. The average ultimate stress determined by Swanson and Colvin (1989) [1] is 375 MPa. The magnitude is significantly lower that the ultimate stress for the laminate at y = 0 determined by Colvin and Swanson (1993) [1]: x = 637 MPa. As one and the same material was tested under the same types of the state of stress (uniaxial compression), it is likely that the tests by Swanson and Colvin (1989) [1] found buckling stresses for tubular specimens rather than the compressive strength of the quasi-isotropic AS4/3501-6 laminate under uniaxial compression. Experimental results in the “compression compression” quadrant also belong to Swanson and Colvin (1989) [1]. It is likely that these results describe buckling of the specimens rather than the laminate fracture. Those results can hardly be used for the present comparison. The data by Colvin and Swanson (1993) [1] for the unidirectional compression, which are beyond question, well agree with theoretical predictions.

948

Fig. 3. Biaxial failure stresses for the (0°/ ± 45°/90°) AS4/3501-6 laminate.

2.2. Biaxial failure of unidirectional CFRP T300/914C All experimental data on failure stresses of the CFRP T300/914C composite are for loading the unidirectional lamina with longitudinal and shear stresses or their combination (see Fig. 4). In our opinion, the information provided could be the subject of another collaborative investigation (“Structural Exercise”, for example) with the objective of comparing modern structural theories for predicting the properties of unidirectional laminae from the properties of the fibres and matrix. Considerable scatter of experimental results, which were obtained at different times and in different laboratories, makes it difficult to formulate definite conclusions.

2.3. Glass fibre reinforced plastic E-glass/LY556/HT907/DY063 Experimental data provided by the organisers begin with the biaxial failure stress envelope of the unidirectional GRP under transverse and shear loading, y ~ xy, (Fig. 5). Considerable scatter of experimental results is typical for such tests, and the scatter does not enable one to draw an unambiguous conclusion. It is believed that the magnitude of the transverse compressive stress (F  2 = 114 MPa) provided by the organisers describes the whole set of available data but not in the best way. The organisers explained the reasons of such discrepancy [1]. The magnitude of F  2 = 135–140 MPa seems to be more reasonable.

949

Fig. 4. Biaxial failure stresses for the unidirectional T300/914 lamina.

The biaxial failure stress envelope for the (90°/ ± 30°) E-glass/LY556 laminate under xy and x stresses is shown in Fig. 6. As is seen, the theoretical envelope as a whole qualitatively correlates with experimental data. Quite adequate quantitative agreement is observed in the positive quadrant of the stress space. A significant difference between theoretical and experimental results is observed for the states of stress close to pure compression along the x axis. The difference can be somewhat decreased in case one takes a more realistic magnitude

Fig. 5. Biaxial failure stresses for the unidirectional E-glass/LY556 lamina under transverse and shear loading, y ~ xy .

950

Fig. 6. Biaxial failure stresses for the (90°/ ± 30°) E-glass/LY556 laminate under xy and x stresses.

of F  2. Nevertheless, the experiments allow to suggest that one of the assumptions in the model [2] about the global laminate failure is too ‘severe’, namely, the assumption that the (i) composite laminate is believed to be immediately broken if the condition (i) 2 = F  2 is fulfilled only for a single constitutive ply. The above discrepancy between experimental and predicted results may be used as a reason for a modification of the model in order to predict the deformation of the laminates after transverse compressive stresses reach the ultimate magnitude in a single ply. Figure 7 illustrates the biaxial failure stress envelope for the (90°/ ± 30°) E-glass/LY556 laminate under combined x and y stresses. Experimental points in the positive x ~ y plane

Fig. 7. Biaxial failure stresses for the (90°/ ± 30°) E-glass/LY556 laminate under combined x and y stresses.

951 well correlate with the predicted envelope. Again, if one takes the corrected values of F  2, then theoretical values of the compressive strength along the x axis will be closer to experimental points. Unfortunately, our estimates carried out with the aid of the BarD program [5] show that experimental data in the negative x–y plane correspond to the buckling stresses rather than ultimate stresses of the laminate. That is why it is believed that theoretical magnitudes of the strength of the (90°/ ± 30°) GRP laminate in the negative half-plane are not experimentally justified but not ruled out with the experimental data.

2.4. Glass fibre reinforced plastic E-glass/MY750/HT907/DY063 The GRP E-glass/MY750 is the most completely investigated material in the exercise. Let us begin with the stress/strain curves for different E-glass/MY570 laminates. The stress/strain curves for the cross-plied GRP laminate (0°/90°) under uniaxial tension are shown in Fig. 8. One may conclude that the correlation between predicted and experimental data is quite adequate. The model predicts the twofold reduction of the laminate stiffness associated with cracking processes: first in the 90° plies (x = 77.8 MPa) and then in the 0° plies (x = 315 MPa). At the second stage, the cracks appear in the 0° plies as the result of constraining the

Fig. 8. Stress/strain curves for the (0°/90°) E-glass/MY750 laminate under uniaxial tension.

952

Fig. 9. Stress/strain curve for the ( ± 45°) E-glass/MY750 laminate under biaxial tension y /x = 1/1.

Poisson’s deformation in the plies. As is seen the model quite adequately shows the reduction of the laminate stiffness. The sole exception is the last segment of the stress/strain curve where the reduction of the laminate tangential stiffness was experimentally recorded: the model does not explain this fact. Figure 9 illustrates the stress/strain curve for the ( ± 45°) GRP laminate under y /x = 1/1. The organisers of the “Failure exercise” [1] correctly note that the reasons for the difference between hoop and axial curves are not clear. Our experience [8] shows that such a difference is possible due to technological deviations in the laminate reinforcement structure or (which is the most probable case) due to inexact ratio between the stresses y /x = 1/1. The authors of the experimental data recorded the scatter of strain gage readings up to 22%. The difference between predicted and experimental curves is well within the mentioned interval. Here we have several words about experimentally recorded leakage of unlined specimens. It is very useful to observe the leakage stage in the experiments. It is likely that leakage stresses (the stresses when the cracks appear) may be treated as the upper experimental estimation of first cracks, viz., the cracks have obviously appeared not later that those were observed with leakage. In real conditions, the moments at which leakage was recorded are likely related not only to cracking of the laminate, but the rate of diffusion of the pressure fluid throughout the tube wall. The stress/strain curves for the ( ± 45°) laminate under y /x = 1/–1 are drawn in Fig. 10. The curves are essentially non-liner. The strains near 10% were recorded in the tests. The model explains the final failure of the laminate by transverse compression of the plies for y = 90 MPa. This magnitude well correlates with the experimental failure stress. The nonlinear character of theoretical curves is related to the changes in the laminate stiffness due to cracking of the material and the appearance of so-called structural non-linearity, i.e. the changes in the reinforcing angle in the deformation process. The theory predicts that the final value of the reinforcing angle is ± 49.8° (y = 8.1%). The appreciable defect of the model,

953

Fig. 10. Stress/strain curves for the ( ± 45°) E-glass/MY750 laminate under y /x = 1/–1.

which is clearly seen from the stress/strain curve, is the angular character of the curve: the assumption about the instant changes of the stiffnesses with the beginning of shear cracks is too ‘severe’. It seems that one more condition is too ‘severe’: the assumption about the final failure of the whole laminate when a single constitutive ply fails in transverse compression, (i) (i) 2 = F  2, (see Fig. 6 as well) resulting in somewhat lower values of predicted strains. The stress/strain curves for the ( ± 55°) laminate under uniaxial tension, y /x = 1/0, are shown in Fig. 11. The model under consideration is qualitatively adequate for the description of the experimental data up to y = 383 MPa when the model predicts that compressive stresses in the plies reach their ultimate values. This prediction completely agrees with the experimental data: in the tests, “leakage of unlined specimens” was recorded at this very stress. It seems reasonable to say that a combination of the cracks appears in the specimens, which, in accordance with the model, is caused by transverse compressive stresses. Experimental data enable one to assume that the load carrying capacity of composite laminates retains to an extent after ultimate transverse compressive stresses have been reached in the plies. As was mentioned above according to the model, the ultimate transverse compressive stress (i) in a single ply ((i) 2 = F  2) results in the global failure of the whole laminate. As before (see comments to Figs 6 and 10), one may state that the theoretical assumption about the final (i) failure of the laminate after the condition (i) 2 = F  2 is fulfilled only for a single constitutive ply is very ‘severe’. In the case under consideration this results in somewhat underestimating the predicted ultimate strains as compared to the experimental ones. The discrete model of

954

Fig. 11. Stress/strain curves for the ( ± 55°) E-glass/MY750 laminate under uniaxial tension, y /x = 1/0.

reducing the stiffness also results in angular character of stress/strain curves while the experiment shows the smoother character of the curves. Experimental and predicted stress/strain curves for the ( ± 55°) laminate under y /x = 2/1 are presented in Fig. 12. The diagrams of such kind are well investigated. The state of stress y /x = 2/1 is typical for high pressure cylindrical vessels, and the ± 55° laminates have the structure close to the equilibrium one for this kind of state of stress. In case shear strains do not arise in constitutive plies of the laminates and linear strains y and x are equal to one

Fig. 12. Stress/strain curves for the ( ± 55°) E-glass/MY750 laminate.

955 another under a given state of stress, the structures are termed “equilibrium.” For the “net” composite model, which ignores shear and transverse stiffnesses of the plies, the equilibrium reinforcing angle is defined with the simple formula [8]: tg2eq = y /x whence it follows that eq = ± 54° 44, which is very close to  = ± 55°. Figure 13 illustrates the typical stress/strain curve for the angle-plied ( ± 55°) GFP [9] under biaxial state of stress y /x = 2/1. Following the first stage of deformation (when the matrix is monolithic, transverse and shear stiffnesses have initial values), the cracking of the laminate occurs resulting in reducing the shear and transverse stiffnesses to the magnitudes which are common for the “net” model. The equilibrium reinforcing angle for the given GRP is near  = ± 55°, which is why, following the cracking, the increments of the strains y and x are practically equal to one another. The organisers provided the experimental data (Fig. 12) in which the hoop stress/strain curve (y versus y) is close to the analogous curve in Fig. 13 while the curve in the axial direction (y versus x) significantly differs from the one in Fig. 13 on the stage after “leakage stress.” The experiments [1] were carried out with tubular specimens which were 100 mm inner diameter, 310 mm overall length, 60 mm gage length and 1 mm thick. The finite element numerical analysis [6] (with consideration of physical, structural and geometrical nonlinearity) shows that the zone of the non-linear edge effect can be equal to 1.0–1.2 of the diameter for the thin-walled GRP tube. Of course, these are tentative magnitudes depending on material characteristics, ply lay-up, loading conditions. Nevertheless, significant nonuniformity of the strains along the length of tubular specimens is very typical for many loading cases of angle-plied composite laminates. This means that the gage length in the experiments by Al Khalil (1990) [1] may be placed just in the zone of non-uniform (in the axial direction) edge strains. It is not improbable that this fact explains a considerable (up to

Fig. 13. Stress–strain curves for the (155°) GRP laminate under y /x = 2/1 (experimental and predicted data belong to the authors of [9]).

956 20%) scatter in the records of strain gages. It seems likely that experimental data need an additional analysis. Biaxial failure envelope of the unidirectional E-glass/MY750 GRP lamina under combined longitudinal and transverse loading is shown in Fig. 14. The experimental data are mainly placed in the half-plane of positive stresses x. Pay attention to the fact that the magnitude of the transverse compressive strength of the lamina was determined with the aid of the specimens which differ from the batches of the specimens used in the other tests of E-glass/ MY750 laminates. Biaxial failure stress envelope for ( ± 55°) E-glass/MY750 laminate is given in Fig. 15. The very comprehensive experimental information, obtained by Soden et al (1989, 1993), Kaddour and Soden (1996), Kaddour, Soden and Hinton (1997) [1], is of particular interest. First let us consider the results which lay in the half-plane of positive y . The results were obtained with thin-walled tubular specimens. The fibre volume fraction in the tubes was near 60% as the authors report [1]. Predicted data adequately correlate with experimental results in this halfplane. The dot-dash line in the theoretical envelope separates the zones of stable and unstable deformation of the ( ± 55°) laminate under biaxial tension [7]. In the case the state of stress corresponds to the point behind this line, the laminate becomes like a mechanism, and the strains may increase without appreciable increasing the stresses. Shear strains considerably grow in the plies with numerous microcracks, and final orientation angles may significantly differ from their initial values. Naturally, only the tubes with liner can be deformed in this zone.

Fig. 14. Biaxial failure stresses of the unidirectional E-glass/MY750 GRP lamina under combined longitudinal and transverse loading.

957

Fig. 15. Biaxial failure stresses for the ( ± 55°) E-glass/MY750 laminate.

The experiments in the compression-compression quadrant were carried out with the use of specially manufactured thick-walled tubes. The fibre volume fraction in the tubes was estimated by the authors as 68%. It seems likely that the relative difference of 14% in the fibre volume fraction between the specimens tested in the positive y half-plane and the ones tested in the compression-compression quadrant allows one to conclude that the results belong to the two different GRP laminates. Unfortunately, the ultimate longitudinal compressive strength of the unidirectional E-glass/MY750 lamina (see Fig. 14) was determined with the use of the third type of specimens. It is not surprising that there is not (and may not be!) a quantitative agreement between experimental and predicted results. Meanwhile, the general view of the predicted failure envelope and the clearly defined direction to the maximum failure stress along the trajectory y /x = –2/–1 give grounds for cautious optimism regarding the predictions with properly selected initial data.

3. Discussion Above comparison of the variety of experimental data provided by the organisers of the “Failure Exercise” with predicted results enables us to conclude the following. To our opinion, the generalised theoretical model by Zinoviev–Tarakanov [3] quite adequately describes the proposed set of 14 experimental data in a whole, and the experimental data belong to different types of composites, laminate lay-ups, and states of stress. In none of the cases under analysis does a dramatic, principal discrepancy between experimental and predicted results appear. Of

958 course this does not mean that obtained correlation between experimental and predicted data should be always estimated as satisfactory correlation. In a number of cases the model gives a qualitatively correct description of the phenomena, but a significant quantitative discrepancy is found between experimental and predicted data. For the most part, the cases are related to failure modes caused by ultimate transverse compressive stresses in a single constitutive unidirectional ply of the composite laminate (i) ((i) 2  F  2). It is likely that the applied theoretical assumption about the fatal impact of the ply failure on the whole composite laminate makes the picture too dramatising. The real content of the above assumption may be treated as follows: “at predicted level of the load, irreversible changes occur in one of unidirectional plies, the changes are caused by the transverse compression of the ply. The changes may result in the final failure of the composite laminate under analysis.” Another evidence defect of the model is related to its ‘angularity’. It is likely that the model includes too severe rules for reducing the stiffnesses when changing from the ideally elastic initial state to the main inelastic state of the material with the cracks. However, this demerit is marked in stress/strain curves only in the zones of changing the states of the unidirectional ply. The performed comparison between experimental and predicted data shows that the ‘angularity’ of the model practically has no impact on the accuracy of the prediction of final stresses and strains. The model postulates the abrupt uniform changing of all characteristics in the whole volume of the unidirectional ply. It is evident that this is in contradiction with the statistical nature of the ply properties. Somewhat ‘smoothing’ of angular points in stress/ strain curves may be also reached with the consideration of non-linear shear deformation of unidirectional plies. However, the details of the shear behaviour of the unidirectional ply under the unloading process and the connection of this behaviour with the processes of crack formation caused by transverse tension of the ply are as yet unclear. Separately stands a problem of the consideration of preliminary thermal loading of composite laminates. It is evident that significant initial stresses may arise in polymer composite laminates caused first of all by cooling the material from the polymerisation temperature down to the operation temperature. As the estimates show, residual stresses significantly depend on anisotropy of the coefficients of linear thermal expansion of the plies, i.e. residual stresses are less essential for the analysis of the behaviour of GFRP and more pronounced for the analysis of CFRP and OFRP. The total consideration of thermal stresses becomes possible only with the available detailed information on the behaviour of the stresses within the whole interval of temperatures and complete time-temperature history of the material preceding the force loading. Luckily, the strength of modern composites usually exceeds the level of potential initial stresses by orders, and relaxation processes of initial stresses make this difference even more pronounced.

4. Further development of the model Predicted data presented in the paper were obtained with the STRAN software [8]. The theoretical model [2] is the foundation for a series of computer programs for the analysis and optimal design of laminated composites – GeCAD [9], composite cylindrical bars and shellsBarD [5], as well as the ASSA [10] program for the nonlinear finite element analysis of multilayered branched shells. The programs mentioned are widely used in our investigations,

959 for example, in the experiments with the use of tubular specimens: BarD – for predicting critical loads of tube buckling, ASSA – for the analysis of the zones of edge effects, etc. At present, a modification of the STRAN program is under development, STRAN-Random; this is the program which operates with random characteristics of micro-plies constituting the plies of the laminates. This program will be free from the drawbacks of the STRAN program, which were revealed in the “Failure Exercise.”

5. Further investigations The problem of the strength is a major problem in mechanics of composite materials. “Failure Prediction of Composite Laminates exercise” is the timely and very well organised program for the generalisation and impartial analysis of extensive experimental results and theoretical models for predicting the strength of laminated composite materials. The organisers of the “Failure Exercise” carefully selected and very accurately described the unique experimental data, they well thought over the procedure to draw the authors of theoretical models into the exercise – this is the indubitable merit of the organisers. We are sure that the results of the “Failure Exercise” will serve a good basis for further refinements of existing models and development of new models for deformation and failure of laminated composites. In connection with this it seems reasonable to enable the contributors of the exercise to correct their models and check those using additional portion of the experiments. We have at our disposal an information on the experiments carried out at the Bauman’s Moscow State Technical University. The large set of experiments on biaxial loading of GRP specimens was conducted using a special-purpose test equipment [11,12]. A variety of GRP laminates ([ ± 30°], [ ± 45°], [ ± 50°], [ ± 55°], [ ± 60°], [ ± 75°], [ ± 30°/90°], [0°/(90°)2]) were tested under a variety of the state of stress (y /x = 1/0; 2/1; 1/1; 1/2; 0/1). A total number of test types was 40, stress/strain curves, ultimate loads and the modes of fracture were recorded in all the tests. To our opinion, the program of the “Failure Exercise” does not exhaust all the problems relating to the deformation and failure processes in laminated composites. Let us list several directions, which are very close to the “Failure exercise” and, at the time, of their own importance: • • • •

the processes of unloading and repeated loading of composite laminates complex trajectories of loading (when x, y, and xy are independent time functions) cyclic loading with consideration of asymmetric cycles and complex cyclic loading 3-D loading of composite laminates.

It seems to be quite reasonable to use the experience of the “Failure Prediction of Composite Laminates exercise” for the organisation of the international cooperation in the investigations of the above problems.

References [1] Soden PD, Hinton MJ, Kaddour AS. Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates: Failure exercise benchmark data: V62, pp 1489–1514, 2002, see also Chapter 2.2 in this book.

960 [2] Zinoviev PA, Grigoriev SV, Lebedeva OV, Tairova LP. Strength of multilayered composites under a plane-stress state. Composite Science and Technology 1989;58:1209–23. [3] Soden PD, Hinton MJ, Kaddour AS. Lamina properties and lay-up configuration and loading conditions of a range of fibre reinforced composite laminates. Composite Science and Technology 1998;58:1011–22. [4] Alfutov NA, Zinoviev PA. Deformation and failure of fibrous composites with brittle polymeric matnx under plane stress. In: Mechanics of composites. Moscow: Mir Publishers, 1982. pp. 166–185. [5] BarD (Bar Designer). Software and user’s manual. Institute of Composite Technologies, Moscow, 1994. [6] Alfutov NA, Zinoviev PA, Popov BG. Analysis of composite multilayered plates and shells. Moscow: Mashinostroenie Publisher, 1984 [in Russian]. [7] Zinoviev PA, Tsvetkov SV. Plastic instability of angle-plied cylindrical shells under biaxial tension. Mekhanika Kompositnykh Materialov. Mechanics of Composite Materials 1994;30:652–63 [in Russian]. [8] STRAN (Strength Analyzer of Multilayered Hybrid Composites). Software and user’s manual. Technomic Publishing Co. Inc. Lancaster–Basel 1995. [9] GeCAD (General Composite Analyzer and Designer). Software and user’s manual. Technomic Publishing Co., Inc., Lancaster–Basel, 1994. [10] ASSA (Non-linear Finite Element Analysis of Multilayered Anisotropic Branched Shells). Software and User’s Manual. Institute of Composite Technologies, 1994 [in Russian]. [11] Zinoviev PA, Tsvetkov SV. Mechanical properties of unidirectional organic-fibre-reinforced plastics under hydrostatic pressure. Composites Science and Technology 1998;58:31–9. [12] Zinoviev PA, Tsvetkov SV, Kulish GG. Van den Berg R.W., Ludo van Schepdael. Mechanical behaviour of high strength unidirectional composites under 3-D state of stress. In: Twelfth International Conference on Composite Materials, Paris, France, 5–9 July 1999 [CD].

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

961

CHAPTER 5.12

Predicting the nonlinear response and failure of composite laminates: Correlation with experimental results Travis A. Bogetti*, Christopher P. R. Hoppel, Vasyl M. Harik, James F. Newill and Bruce P. Burns US Army Research Laboratory; AMSRL-WM-MB, Aberdeen Proving Ground, MD 21005-5066, USA Abstract A comprehensive comparison of laminate failure models was established to assess the state-of-the-art in laminate modeling technologies on an international level (known as the Worldwide Failure Exercise) [Compos. Sci. Technol. 58(1998) 1001,1011,1225]. In our first contribution to this Exercise (Part A), we presented a complete theoretical description of an analysis methodology and documented predictions for the laminate response and failure behavior of various laminates under a broad range of loading conditions [see Chapter 3.13]. This paper represents our contribution to Part B of the Exercise where the laminate response and failure predictions for 14 different cases are presented and compared with actual experimental test data. The cases include prediction of the effective nonlinear stress vs. strain responses of laminates, as well as, initial and final ply failure envelope predictions under multi-axial loading. Correlation between the theoretical predictions and experimental results are discussed. While reasonable correlation was achieved, the failure analysis employed by the authors was not universally accurate in predicting the laminate failure response for the broad range of test cases considered. This statement, although not surprising, is likely true for any given failure methodology as it is applied to a wide range of laminate lay-ups and loading conditions. Keywords: Composite laminate

1. Introduction In our first contribution to the Worldwide Failure Exercise (Part A), we presented a methodology for predicting the nonlinear stress/strain response and failure behavior of composite laminates [1–4]. The theoretical analysis is an incremental formulation of a wellestablished three-dimensional laminated media analysis [5,6] coupled with a progressive-ply

* Corresponding author.

962 Table 1 Summary of the laminates and loading cases. Loading case

Laminate lay-up

Material

Description of loading cases

1

0

E-glass/LY556/HT907/DY063

2

0

T300/BSL914C

3

0

E-glass/MY750/HY917/DY063

4 5 6 7 8

90/ ± 30/90 90/ ± 30/90 ± 55 0/ ± 45/90 0/90

E-glass/LY556/HT907/DY063 E-glass/LY556/HT907/DY063 E-glass/MY750/HY917/DY063 AS4/3501-6 E-glass/MY750/HY917/DY063

9 10 11

± 45 ± 45 ± 55

E-glass/MY750/HY917/DY063 E-glass/MY750/HY917/DY063 E-glass/MY750/HY917/DY063

12 13

± 55 0/ ± 45/90

E-glass/MY750/HY917/DY063 AS4/3501-6

14

0/ ± 45/90

AS4/3501-6

Biaxial failure stress envelope under transverse and shear loading (y vs. xy) Biaxial failure stress envelope under longitudinal and shear loading (x vs. xy) Biaxial failure stress envelope under long. and transverse loading (y vs. x) Biaxial failure stress envelope (y vs. x) Biaxial failure stress envelope (x vs. xy) Biaxial failure stress envelope (y vs. x) Biaxial failure stress envelope (y vs. x) Stress–strain curve under uniaxial tensile loading for (y : x = 1 : 0) Stress–strain curves for (y : x = 1 : 1) Stress–strain curves for (y : x = 1 : –1) Stress-strain curves under uniaxial tensile loading for (y : x = 1 : 0) Stress-strain curves for (y : x = 2 : 1) Stress–strain curves under uniaxial tensile loading in y direction (y : x = 1 : 0) Stress–strain curves for (y : x = 2 : 1)

failure methodology. Nonlinear lamina constitutive relations for the composites are represented using the Ramberg–Osgood equation [7]. Piece-wise linear increments in laminate stress and strain are calculated and superimposed to formulate the overall effective nonlinear response. Individual ply stresses and strains are monitored to calculate instantaneous ply stiffnesses for the incremental solution and to establish ply failure levels. The progressive-ply failure approach allows for stress unloading in a ply and discrimination of the various potential modes of failure. The aforementioned laminate analysis and progressive ply failure methodology has been programmed into a FORTRAN-based software code entitled LAM3DNL. The LAM3DNL code employs a user-friendly database format for input of laminate architectures, lamina properties, and failure parameters [8]. Output from the code includes the effective laminate stress and strain files as well as a failure assessment summary file that identifies all ply failures that occur during a laminate response prediction program run. In this paper, we compare our theoretical predictions made in [4] with the experimental data for the 14 different laminate test cases described by Soden et al. [9]. These test cases have been grouped into three classes (a) biaxial failure envelopes of unidirectional lamina, (b) bidirectional failure envelopes of multidirectional laminates, and (c) stress vs. strain curves of laminates under uniaxial and biaxial loading. For completeness, a summary of the test cases investigated in the paper are presented in Table 1. It is also noted that four different materials were included in the study: (a) E-glass/MY750 epoxy, (b) E-glass/LY556 epoxy, (c) T300 graphite/BSL 914C epoxy, and (d) AS4 graphite/3501-6 epoxy. Correlation between the theoretical predictions and experimental results are discussed for each of the load cases.

963 2. Correlation of predictions with experimental results 2.1. Loading case 1: biaxial failure envelope of (y vs. xy) for [0] E-glass/LY556 epoxy A comparison of experimental results to theoretical predictions for this load case is presented in Fig. 1. The solid lines represent the predicted failure envelope. The predicted failure modes are indicated in bold print (Y12, Y2C, Y3T, and Y2T); the key for these symbols is given in Table 2. The dotted line on Fig. 1 indicates the predicted initial ply failure for the laminate (transverse tension in the through-the-thickness direction). The solid squares indicate the unidirectional strengths provided in the initial material property input (Table 1 in Ref. [2]), and the open circle indicate the test results. Reasonable correlation between experimental and theoretical predictions is observed. Key points for comparison are intersections with the load axes. Discrepancy between the transverse compression strength prediction and the test data can be attributed to the transverse compression strength originally provided as input not agreeing with the experimental data obtained to support this portion of the exercise. There appears to be some degree of interaction between the shear and transverse strengths that is not captured by the maximum strain criteria used in the present analysis. An interactive failure criterion such as Tsai-Wu may be better at capturing the biaxial load failure behavior.

Fig. 1. Biaxial failure stresses for 0° lamina made of GRP material. Material type: E-glass/LY556/ HT907/DY063.

964 Table 2 Designations for predicted failure modes. Designation

Predicted failure mode

Y1T Y1C Y2T Y2C Y3T Y3C Y23 Y13 Y12

Tensile failure in the fiber (1) direction Compressive failure in the fiber (1) direction Tensile failure in the transverse (2) direction Compressive failure in the transverse (2) direction Tensile failure in the through-the-thickness (3) direction Compressive failure in the through-the-thickness (3) direction Interlaminar shear in the 23 direction Interlaminar shear in the 13 direction In-plane shear (12 direction)

2.2. Loading case 2: biaxial failure envelope of (x vs. xy) for [0] T300 graphite/BSL 914C epoxy A comparison of experimental results to theoretical predictions for this load case is presented in Fig. 2. As with load case 1, reasonable agreement is observed for the biaxial failure envelope. The experimental data and the predictions both show no interaction between the applied shear stress and the fiber direction tensile stress when the predicted failure mode is tensile failure in the fiber direction. When the predicted failure mode is shear dominated (Y12) the experimental data shows potentially some interaction between the stresses that is not predicted by the theory, but it is difficult to draw a conclusion due to the significant scatter in the experimental data, seen especially at x = 0. When the predicted failure mode is fiber compressive failure (Y1C) there does not appear to be any interaction between the stress fields in the theory or experimental results, although this conclusion is again weakened by the scatter in the data.

Fig. 2. Biaxial failure stresses for 0° lamina made of CRFP material.

965

Fig. 3. Biaxial failure envelope of 0° GRP lamina under combined x and y stresses. Material: E-glass/ MY750 epoxy.

2.3. Loading case 3: biaxial failure envelope of (y vs. x) for [0] E-glass/MY750 epoxy A comparison of experimental results to theoretical predictions for the biaxial failure envelope for this unidirectional laminate is presented in Fig. 3. For the limited amount of test data, good correlation was found for this test case. The theory predicts that while the final tensile and compressive failures in the fiber direction (x) are almost independent of the transverse stressstate, the tensile and compressive failures in the transverse (y) direction are strongly influenced by the axial (x) stress due to the Poisson’s effects in the material. The theoretical predictions are found to be consistent with this general trend in the fourth quadrant of the failure envelope. The uniaxial compression strength prediction of 1 = 800 MPa and 2 = 0 MPa is due to a Y3T failure that changes the in-plane behavior of the lamina due to a drop in the transverse and through-the-thickness moduli. The theory over-predicts the transverse compressive strength of the laminate (Y2C) because it neglects the nonlinear behavior of the stress–strain curve under transverse compression. 2.4. Loading case 4: biaxial failure envelope of (y vs. x) for [90/ ± 30]s E-glass/LY556 epoxy A comparison of experimental results to theoretical predictions for this load case is presented in Fig. 4. The theoretical predictions for this load case indicate that multiple ply level failures occur prior to the “final” laminate failure. The first predicted failure modes are indicated by

966

Fig. 4. Biaxial failure envelope for (90°/ ± 30°) laminate under combined x and y stresses. Material: E-glass/LY556 epoxy.

the dotted line in Fig. 4. For the entire biaxial failure envelope, transverse tensile failures (either in the 90° or ± 30° layers) are predicted to occur first, followed by catastrophic of “final laminate failure.” Overall, the theoretical predictions match well with the test results, with the exception of the predictions made under load combinations involving transverse (y) compression. In this region (second and third quadrants) the predictions of failure overestimate the test results. The test results do not indicate the failure mode or the extent of damage in the specimens. It is also possible that the failure was dominated by the initial transverse tensile failure of the 90° plies.

2.5. Loading case 5: biaxial failure envelope of (x vs. xy) for [90/ ± 30]s E-glass/LY556 epoxy A comparison of experimental results to the theoretical predictions for this load case is presented in Fig. 5. Reasonably good agreement for this load case was achieved except in second quadrant where the predictions somewhat underestimated the test results. In this quadrant, failure is predicted to involve initial transverse tension failure (Y2T) in the –30° plies, followed by longitudinal compression failure (Y1C) in the –30° plies. In the analysis, the transverse tensile properties (modulus and Poisson’s ratios) are reduced to very small values when an initial failure is predicted. This approximation may be too severe for these experiments. In experiments, transverse cracking may occur, but it may not be extensive enough to completely reduce the mechanical properties in this direction. Thus, the strength

967

Fig. 5. Biaxial failure stresses for (90°/ ± 30°) laminate under xy and x stresses. Material: E-glass/ LY556 epoxy.

predictions are conservative. In the first quadrant, the predicted strengths are within the scatter of the experimental data. 2.6. Loading case 6: biaxial failure envelope of (y vs. x) for [ ± 55]s E-glass/MY750 epoxy A comparison of experimental results to theoretical predictions for this load case is presented in Fig. 6. The general appearance of the correlation between the predicted strengths and the

Fig. 6. Biaxial failure stresses for ( ± 55°) E-glass/MY750 laminates.

968 test results is good. The failure envelope is governed by longitudinal compression strain failure (YIC) in the third quadrant and by longitudinal tension strain failure (Y1T) in the first quadrant and in-plane shear (Y12) failure in the second and fourth quadrants. In the second, third, and fourth quadrants, the predicted strengths are within the experimental scatter. In the first quadrant, there is considerable variation in the experimental strengths depending on the test specimen geometry. The thin tubes tested without liner give results consistent with the predicted first ply failure (transverse tension or Y2T in the ± 55 plies). The thin tubes tested with a liner appear to follow the predictions for a final failure (Y1T) in the ± 55 plies. More data on the nature of the experimental failures for each of the different specimens would be helpful in interpreting these results. However, given the biaxial stress state and the predicted multiple failure modes, the predictions show reasonably good agreement with the experimental results. The underprediction of the Y1C failures in the third quadrant could be brought more in line with the experimental data if the through the thickness stresses were taken into account. Consideration of these through thickness stresses would increase the appearant Y1C value of the composite lamina. 2.7. Loading case 7: biaxial failure envelope of (y vs. x) for [0/ ± 45]s AS4 graphite/3501-6 epoxy A comparison of experimental results to the theoretical predictions for this load case is presented in Fig. 7. The predictions show good correlation with the experimental results in the

Fig. 7. Biaxial failure stresses for (0°/ ± 45°/90°) AS4/3501-6 laminates.

969

Fig. 8. Stress-strain curves for 0°/90° E-glass/MY750 laminate under uniaxial tension.

first quadrant where the predicted failure mode is tensile in the fiber direction of the 0 plies and in the fourth quadrant where the predicted failure mode is compressive in the fiber direction of the 90° plies. In the third quadrant, the experimental results appear to match more closely with the predicted initial failure mode (through-the-thickness tensile failure or Y3T) than the predicted final failure mode (Y1C in the 0° or 90° plies). This good correlation is likely coincidental as the model neglects the three dimensional through the thickness stresses due to the externally applied radial pressure. These through the thickness stresses, if taken into account in the predictions, would also alter the location of the Y3T line.

2.8. Loading case 8: stress–strain curves of (y : x = 0 : 1) for [0/90]s E-glass/MY750 epoxy A comparison of experimental results to the theoretical predictions for this load case is presented in Fig. 8. The predictions are in excellent agreement with the test results for this load case. A predicted transverse tensile failure in the 90° plies at x = 0.25% is associated with the observed initial cracking point on the stress versus strain curve. In the theoretical model, when the initial transverse tensile failure occurs in the 90 plies, the properties are reduced immediately (thus the sharp drop in the theory). In the experiment, the transverse cracking occurs progressively and the properties are reduced over a larger strain region (thus

970 the theoretical and experimental curves show a slight difference after the initial failure occurs). Although the model does not predict the second observed failure mode of “longitudinal splitting”, the ultimate load due to fiber tensile failure in the 0° plies was accurately predicted. The predicted Poisson strains (y) are also in good agreement with the test results.

2.9. Loading case 9: stress–strain curves of (y : x = 1 : 1) for [ ± 45]s E-glass/MY750 epoxy A comparison of experimental results to theoretical predictions for this load case is presented in Fig. 9. The theoretical predictions are in good agreement with the test results in the initial loading portion of the stress versus strain curve. The point where first cracks were observed correlates with the prediction of transverse tensile failure in the ± 45° plies. At this point, the model drops the transverse tensile properties, over-predicting the damage in the laminate. The experimental results show a more gradual reduction in properties, showing reasonably good correlation with the theory up until about 2% strain. Beyond this point the analysis predicts a higher stress at ultimate failure (predicted to be tensile failure in the fibers). It is possible that in the experiments the accumulated transverse matrix cracking caused the final failure before the predicted fiber tensile failure could occur. The following statement applies to load cases 9–12. The over-prediction of the stress responses for the laminates could be partially attributable to the fact that the current model does not account for changes in the tubular specimen geometry due to rotation (scissoring) of fibers and due to the radial expansion or contraction in the diameter.

Fig. 9. Stress–strain curves for ± 45° E-glass/MY750 laminate under y/x = 1/1.

971 2.10. Loading case 10: stress–strain curves of (y : x = 1 : –1) for [ + 45/–45]s E-glass/MY750 epoxy A comparison of experimental results to the theoretical predictions for this load case is presented in Fig. 10. The initial slope of the stress versus strain predictions matched well with the test results, but the data sets diverge at strains beyond 1% where the predictions are much stiffer than the test results. The predicted ultimate failure strain was far less than the experimental. Again, this large discrepancy between model predictions and the test data could be associated with the fact that the model does not account for fiber rotations and tubular specimen geometry changes.

2.11. Loading case 11: stress–strain curves of (y : x = 1 : 0) for [ + 55/–55]s E-glass/MY750 epoxy A comparison of experimental results to theoretical predictions for this load case is presented in Fig. 11. The predictions of the stress–strain response for this load case were in good agreement with the test data up to just over 2% strain, where ultimate laminate failure (shear failure in the ± 55° plies, Y12) was predicted. As with the previous load case, the test data indicates that the laminate was able to carry load well beyond the predicted point of ultimate failure. Model predictions for failure are at much lower strain levels than the test date for reasons explained above.

Fig. 10. Stress–strain curves for ± 45° E-glass/MY750 laminate (y /x = 1/–1).

972

Fig. 11. Stress–strain curves for ± 55° GRP laminate under uniaxial tension (y /x = 1/0).

2.12. Loading case 12: stress–strain curves of (y : x = 2 : 1) for [ + 55/–55]s E-glass/MY750 epoxy A comparison of experimental results to theoretical predictions for this load case is presented in Fig. 12. The predictions for the initial portion of both of the stress versus strain responses are in good agreement with the test results. At approximately 150 MPa, a transverse tensile failure (Y2T) is predicted in the ± 55 ply. In the model, the properties are reduced immediately. In the experimental results, the transverse damage accumulates in a progressive

Fig. 12. Stress strian curves for a ± laminate made of E-glass/MY750 epoxy material under y /x = 2/1.

973 manner so that the experimental results are initially stiffer, but become more compliant than the predictions. Better correlation with between the predicted and measure response in the tubular specimens could be achieved if the model was modified to account for fiber reorientation due to large deformation. 2.13. Loading case 13: stress–strain curves of (y : x = 1 : 0) for [0/ + 45/–45/90]s AS4 graphite/3501-6 epoxy A comparison of experimental results to theoretical predictions for this load case is presented in Fig. 13. In general, the predictions are in good agreement with the test results for this load case, for both the axial and Poisson strains. At approximately 0.44% strain, transverse tensile failure (Y2T) is predicted in the 90° plies. The experimental results do not show a reduction in properties at this point and therefore appear stiffer for the rest of the stress–strain curve. This difference could again be due to the immediate reduction in properties in the model. In the experiment, transverse cracking may have occurred in a more progressive manner (the experimental curve shows a slight deviation from linear behavior). Not surprisingly, ultimate laminate failure is dominated by longitudinal tension failure in the 0° plies, YIT. 2.14. Loading case 14: stress–strain curves of (y : x = 2 : 1) for [0/ + 45/–45/90]s AS4 graphite/3501-6 epoxy A comparison of experimental results to theoretical predictions for this load case is presented in Fig. 14. As with the previous load case, predictions for the quasi-isotropic graphite composite laminate are in good agreement with the test results. Transverse tensile failure in

Fig. 13. Stress–strain curves for (0°/ ± 45°/90°) laminate under uniaxial tension (y /x = 1/0).

974

Fig. 14. Stress-strain curves for (0°/ ± 45°/90°) laminate under biaxial stress (y /x = 2/1).

the 90° plies, Y2T, is predicted early in the load history (about 0.5% strain) and this seems to correlate with the nonlinear softening of the laminate stress–strain response. The predicted transverse tensile (Y2T) failure of the ± 45 plies occurs at the same location as a load-drop in the experimental results. As in Loading Case 13, ultimate laminate failure is dominated by longitudinal tension failure in the 0° plies, Y1T. The lower predicted failure strength could be due to assuming complete degradation of transverse properties in the 90° and ± 45 plies in the model, while experimentally only a partial degradation in properties occurs.

3. Conclusions Correlation between the theoretical predictions and experimental results were presented and discussed. While reasonable correlation was achieved for most of the case studies, the failure analysis employed by the authors was not universally accurate in predicting the laminate failure response for the broad range of test cases considered. This statement, while not surprising, is likely true for any given failure methodology as it is applied to a wide range of laminate lay-ups and loading conditions. In several cases examined in this exercise, over prediction of the stress responses for the laminates could be partially attributable to the fact that the current model does not account for changes in the tubular specimen geometry due to rotation (scissoring) of fibers and due to the radial expansion or contraction in the diameter. This is one example of the inherent variability or special circumstances that one may encounter in composites modeling that will ultimately contribute to difficulties in consistently correlating theoretical and experimental predictions. In general, a composite failure model is essentially a combination of assumptions, approximations and physical laws which are made to establish a tractable estimation of composite failure. The relationships between microstructural effects, statistical variations and

975 composite failure are vastly too complex to be completely addressed in the most comprehensive failure model. This is especially true in many of the cases presented in this study where multiple failure modes occur prior to the final laminate failure. It is the authors’ opinions that no truly universal composite laminate failure model or analysis exists. Even the most sophisticated “state-of-the-art” models are not capable of predicting the broad range behavior exhibited under a variety of materials, lay-ups and loading conditions. At best, those failure models that capture the “widest” range of behavior – with reasonable effort – are most valuable as predictive tools. The failure theories used should be relevant and proven within a given application. This is to say that the business of predicting composite laminate failure can perhaps be just as easily viewed from an engineering perspective than it can from a scientific one. It is worth pointing out that a comparison between the results of the present model and those of other models, employed in the failure exercise, is presented in Ref. [10].

References [1] Hinton MJ, Soden PD. Predicting failure in composite laminates: background to the exercise. Compos Sci Technol 1998;58(7):1001. [2] Soden PD, Hinton MJ, Kaddour AS. Lamina properties, lay-up configuration and loading conditions for a range of fibre reinforced composite laminates. Compos Sci Technol 1998;58(7): 1011. [3] Soden PD, Hinton MJ, Kaddour AS. A comparison of the predictive capabilities of current failure theories for composite laminates. Compos Sci Technol 1998;58(7):1225. [4] Bogetti TA, Hoppel CPR, Harik VM, Newill JF, Burns BP. Predicting the nonlinear response and progressive failure of composite laminates. Compos Sci Technol, V62, pp 329–342, 2004, see also Chapter 3.13 in this book. [5] Chou PC, Carleone J, Hsu CM. Elastic constants of layered media. J Compos Materials 1972;6:80–93. [6] Sun CT, Liao WC. Analysis of thick section composite laminates using effective moduli. J Compos Materials 1990;24:977. [7] Richard RM, Blacklock JR. Finite element analysis of inelastic structures. AIAA Journal 1969;7:432. [8] Bogetti TA, Hoppel CPR, Drysdale WH. Three-dimensional effective property and strength prediction of thick laminated composite media. ARL-TR-911, US Army Research Laboratory, Aberdeen Proving Ground, MD, October 1995. [9] Soden PD, Hinton MJ, Kaddour AS. Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates: Failure exercise benchmark data: V62, pp 1489–1514, 2002, see also Chapter 2.2 in this book. [10] Hinton MJ, Soden PD and Kaddour AS. A further assessment of the predictive capabilities of current failure theories for composite laminates: comparison with experimental evidence. Compos Sci Techno 2004;64:549–588, see Chapter 6 in this book.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

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CHAPTER 5.13

The predictive capability of failure mode conceptbased strength criteria for multi-directional laminates – Part B R. G. Cuntze* Main Department ‘Analysis’, MAN Technologie AG, Franz-Josef-Straup-Str. 5, D-86153 Augsburg, Germany Abstract This paper represents the author’s contribution to Part B of the worldwide failure exercise. An assessment is made of the correlation between the predictions, published in Part A, and experimental results provided for 14 test cases involving bi-axial initial and final failure envelopes and stress–strain curves of various unidirectional and multidirectional laminates. The theory, implemented into a computer code, employs 3D failure criteria for UD laminae together with non-linear modelling of laminae and laminate. The criteria are based on the author’s Failure Mode Concept (FMC). Some simplifications of the failure criteria have been made and an improvement in the computer code so that large strains to failure may be captured. Special emphasis has been put on the difference between an isolated and an embedded UD lamina. The correlation between the predictions and the test data is very satisfactory for the UD laminae, and is generally satisfactory for the laminates, especially where Fibre Failure (FF) is the dominant mode, which is the case for well-designed (according to netting theory) laminates. Discrepancy between predictions and test data was largest for the ± 55° angleply glass/epoxy laminates and also in areas of large deformations (involving post failure progression beyond the occurrence of the various Inter Fibre Failure (IFF) modes). Further steps, including testing and modelling, are identified for achieving a better understanding of failure prediction. Keywords: B. Non-linear behaviour; C. Failure criteria UD laminae; Failure of laminates

Nomenclature as, bs b⊥, b⊥储, b⊥储 E1( = E储), E2( = E3 = E⊥)

Ramberg/Osgood parameters in the softening reglme Curve parameters, see Eqs. (2) and (17) Elastic moduli of a unidirectional (UD) lamina in the x1, x2, and

* Tel.: + 49-821-505-2593; fax: + 49-821-505-2630. E-mail address: [email protected] (R. G. Cuntze).

977

E1(tan), E3(sec) E1f Eff (res) Eff (mode) = (mode) /R¯ mode eq maxEff (mode) et储, ec储 F 储 , F 储 , F ⊥, F ⊥, F⊥储 f (mode) = 1/Eff (mode) Res f (res) Res

G21, G21(sec) I1, I2, I3, I4, I5 m ˙ ¯ R R, Rp0.2 R t储 ⬅ X t, R c储 ⬅ X c R t⊥ ⬅ Y t, R c⊥ ⬅ Y c R⊥储 ⬅ S vf x1, x2, x3

1, 2, 3 v12 = –2 /1 1, 2, 3 c2, t1 储, ⊥ (mode) eq 1f ˆ

x3 directions Tangent and secant elastic moduli Fibre elastic modulus in x1 direction Resultant stress effort of all interacting failure modes Corresponds to Puck’s exposure factor fE Stress effort of a UD-lamina in a failure mode, e.g. 储eq / R c储 = Eff 储 corresponds to 1/f 储Res (for a linear behaviour), see Eqs. (6) and (24) Stress effort of the maximum stressed failure mode Tensile and compressive failure strains of a UD lamina in x1 direction Failure functions for two fibre failure modes (FF) and three inter-fibre failure (IFF) modes, see Eqs. (1), (16) and (30) = Reserve factor (or stretching factor) defined here as the ratio of the relevant strength value to the relevant ‘equivalent’ stress value of a given mode of failure Resultant reserve factor. In general for linear and non-linear cases the ratio of maximum load achieved in test or analysis divided by the design load Shear and secant shear moduli, respectively, of a UD lamina in the x2, x1 direction Invariants of the transversally-isotropic UD-material [see Eq. (10)] Mode interaction coefficient (rounding-off exponent) Mean strength and, strength ‘design allowable’ Stress value at 0.2% plastic strain = yield strength UD tensile and compressive (basic) strengths parallel to the fibre direction UD tensile and compressive strengths transverse to the fibre direction Shear strength of a UD lamina transverse/parallel to the fibre direction Fibre volume fraction Coordinate system of a UD lamina (x1 = fibre direction, x2 = direction transverse to the fibre, x3 = thickness direction) Normal (direct) strains of a UD lamina Major Poisson’s ratio in ‘failure exercise’ (corresponds to v⊥储 in Ref. [28]) Normal (direct) stresses in a UD lamina Compressive stress transverse to and tensile stress along the fibre direction Stresses parallel and transverse to the fibre direction 储 ⊥ ⊥ ⊥储 Equivalent stresses of a mode (储 eq, eq, eq , eq , eq ) include both, load induced and residual stresses, see Eq. (5) Stress in the fibres in x1 direction Mean (average) stress of the laminate

978 12 = 21, 13 = 31, 23 = 32

⊥储, ⊥⊥

12 = 21; 13 = 31;

23 = 32

Shear stresses of a UD lamina in the elastic symmetry directions. The first subscript locates the direction normal to the plane on which the shear stress is acting; the second subscript indicates the direction of the shear force Shear stressing transverse/parallel and transverse/transverse to the fibre direction Shear strains of a UD layer

Abbreviations and indices, signs CLT F FRP Jfp MS c, t f, m n Res (res) s – , 

Classical Laminate Theory Failure function Fibre-reinforced plastic Joint failure probability Margin of Safety Compression, tension (German guideline VDI 2014, [28]) Fibre, matrix Repetitions in stack Reserve Resultant Symmetric lay-up, softening Statistical mean indicate the failure induced by Mohr’s normal or shear stress.

1. Introduction Recently, and as a part of a worldwide failure exercise aimed at highlighting the current capability of failure prediction, Cuntze presented a methodology [1] based on the so-called Failure Mode Concept (FMC), for the prediction of failure in composite laminates. The exercise was carried out in two stages, referred to as Part A and Part B. The predictions, representing the Part A contribution, were made for 14 test cases and the results were published in Ref [1]. No experimental data was used. In the second part (Part B), the author was requested to submit a paper describing the correlation between those Part A predictions and a set of experimental results, provided by the organisers of the failure exercise. This paper represents the author’s submission for Part B. In order to ensure that the paper is self-contained, the author first provides some of the background to the FMC-based criteria and the associated non-linear modelling of UD laminae and multi-directional laminates. The correlation between Part A predictions and test data is then described. Based on the available test data, the paper then describes the modifications which were introduced in order to obtain a better fit between the predictions and the experimental results. Areas of potential problems in the test data and in the methodology are highlighted.

979

Fig. 1. Failure criticality.

2. Part A theory: comparison with experiment 2.1. FMC-based failure conditions for UD laminae Cuntze’s method of applying the FMC methodology is to propose a set of five equations describing five ‘pure’ failure modes in each lamina (ply) and then combine these equations in a suitable manner to predict failure in a lamina. Each failure mode is described by a distinct equation containing terms, generally, showing an interaction between the various active stresses (stress tensor components). The total number of modes [1] is five; two Fibre Failure (FF) modes and three Inter Fibre Failure (IFF) modes, see Section 2.3.1, Eqs 1 – and Figs 1 and 2. The interaction between the FF and IFF modes as well as between the various IFF modes is considered probabilistically. In order to account for this interaction between failure modes

Fig. 2. (a) UD lamina: stress and strength notations of 3D state of stress. t = tension, c = compression, (b) laminate and kth lamina subjected to a plane state of stress (mid-plane z = 0).

980 (referred to here as a Mixed Failure Domain MiFD), a rounding-off process in the interaction domains is employed, utilizing a suitable probabilistics-based ‘series spring model’ approach [2]. The method requires data for the pure failure mode domains, only and two curve parameters. Hence, data in the MiFDs, where the stress state influences various modes, is not requested.

2.2. Non-linear analysis of laminates composed of UD laminae 2.2.1. Input A maximum of five non-linear stress–strain curves are normally required to carry out a full non-linear analysis. However, for most conventional FRP materials, two non-linear curves c2(2) and 21( 21) are normally observed and have to be applied. The author believes that the non-linear behaviour of laminates composed of brittle laminae, similar to those used in the ‘failure exercise’, originates from damage development around inherent defects or flaws in the constituent matrix (a ductile matrix tensile specimen would show necking and so-called crazing, which appears in the case of glass fibre composites as whitening in a tensile test [3]) and at the interface between the fibres and the matrix. These defects grow to micro-cracks and later to cracks under increased stressing. 2.2.2. Procedure for progressive failure analysis of laminates The non-linear analysis procedure, used in the present paper, is based on ‘ply-by-ply’ analysis and uses a secant modulus approach for carrying out the computation. For performing the stress analysis the required non-linear stress–strain curves should describe both material hardening and material softening behaviour [1], see also Figs 3 and 4. The hardening

Fig. 3. The differences in the stress–strain behaviour of isolated and embedded UD-laminae. For the (b)- and (c)-curve Eq. (22) is applied. The softening parameters for (b) and (c) are different. Due to embedding, point + higher than 䊏).

981

Fig. 4. Transversally compression stress–strain curve c2(2); UD-lamina (softening parameters are ⊥c assumed). GFRP: E-glass/MY750/HY917/DY063 [8]. E¯ c⊥0 = 16.2 GPa; a⊥c s = –3.45%, bs = 0.47%, ⊥c T ⊥ ¯ n = 6.6. {R} = (1280, 800, 40, 145, 73) . The symbol eq shall exemplarily indicate that uni-axial and equivalent curve are the same.

behaviour is a load-controlled process, which describes the response of the lamina up to the point of maximum stress ( = strength, e.g. R¯ ⊥储) and this point corresponds to initial failure of IFF modes. The softening behaviour is a strain-controlled process, which describes the lamina response beyond that point and this is associated with progressive failure. However, the author believes that some minor damage (e.g. matrix yielding) may still take place even during the hardening response. The softening curve of a lamina in the laminate, when loaded beyond initial failure, is referred to as the effective stress–strain curve. This curve, e.g. beyond R¯ t⊥ in Fig. 3, is normally unknown and has to be determined. An example of the hardening and softening curve used in the calculations is shown in Fig. 4. In this figure, the effective curve c2(2) is assumed valid if a catastrophic failure of the ‘delamination-initiating’ wedge type is prevented by the laminate. In cases where oblique cracks do exist, the author believes that all these oblique micro-cracks remain closed under the compression but may show some compliance, caused by a possible movement of UD material in the thickness direction. A triggering approach is used to describe the effects of the stress state in the MiFDs. This approach increases the equivalent stress (which considers all influencing stresses) of each affected mode in the case of hardening (the secant modulus becomes smaller) and decreases the equivalent stress in the case of softening (the secant modulus becomes smaller, too), see Section 3.2.4 for more details. 2.3. Brief review of theoretical assumptions and remarks on the analysis 2.3.1. Failure conditions applied in non-linear analysis According to the Failure Mode Concept (FMC) theory [1], the characterisation of the strength of transversally-isotropic composites requires the measurement of five independent basic lamina strengths: R T储 , R c储 (tensile and compressive strengths parallel to the fibres), R t⊥, R c⊥ (tensile and compressive strengths transversal to the fibre direction) and R⊥储 (shear strength

982 transverse/parallel to the fibres), see Figs 1 and 2. The lamina is defined here to be the material or the building block that a laminate is made of. The following failure conditions, derived from the complete FMC-based invariant formulations, were employed in the non-linear analysis carried out in Part A [1] FF1, FF2:

1 · E t储 –1 = 1; =1 t 储 Eff · R 储 Eff 储 · R c储

IFF1, IFF2: IFF3:

2 321 + b⊥储22221 = 1; =1 Eff ⊥ · R t⊥ (Eff ⊥储 · R⊥储)3

(1)

(b⊥  1)(2 + 3) b⊥(2  3)2 + b⊥储 · 221 + =1 Eff ⊥ · R c⊥ (Eff ⊥ · R c⊥)2

where the three curve parameters b⊥, b⊥储 and b⊥储 are given as (see Figs 4 and 5) b⊥储 = b⊥ =

¯ 2 1  (⊥储 21 /R⊥储) , ¯3 2c2 · ⊥储2 21 /R ⊥储

c ¯ c 1 + (c 2 + 3 )/R ⊥ , c ¯ c c 2 ¯ c2 ( + 3 )/R ⊥ + (c 2  3 ) /R ⊥ c 2

(2)

 c ¯ ¯ 2 and b⊥储 = 1  (b⊥  1)c 2⊥储 /R⊥储  b⊥(2⊥储 /R⊥储)

As will be shown later, see Section 2.2.2.1, the last parameter b⊥储 will be not used in the analysis because it is an empirical fitting factor whose value depends greatly on the nature and accuracy of bi-axial test data for a lamina under combined transverse and shear stresses. In order to predict failure in the pure domains, i.e. in one of the above conditions, a stress effort Eff (mode), is computed by inserting the lamina stresses (1, 2, 0, 0, 0, 21) into Eq. (1). This will either lead to no failure, if Eff (mode) < 1, or to failure if Eff (mode) equals to 1, see also

Fig. 5. (Left): (B) and (A) Biaxial failure stress envelope (21, 2) and (31, 2). UD-lamina (no curing stress). GFRP: E-glass/LY556 epoxy. Eqs. (1), (23). Test data [4] of tube + . b⊥储 = 0.30, b⊥储 = 0, m ˙ = 2.5; ¯ = (1140, 570, 36, 138, 63)T. Not active parameters are given in b⊥储-calibration in 䊐, see Part A [6]. {R} parenthesis (Part A: b⊥ = 1.5, Part B: b⊥ = 1.09). Softening not applicable for isolated lamina. (Right): (B) with (A) Biaxial failure envelope for (TC1). Eqs. (1), (16), (23), Eq. (23) applied in all figures (1 ⬅ 储, ¯ = (1140, 570, 35, 2 ⬅ ⊥). *Part A ‘blind’ data set: b⊥储 = 0.13, b⊥储 = 0.4, m ˙ = 3.1, maxtau not active; {R} T   ¯ = (1140, 570, 38, 114, 72) . *Best fit Part B data set: b⊥储 = 0.56, b⊥储 = 0, m ˙ = 2.1, maxtau = 105 MPa, {R} ¯ = (1140, 570, 38, 135, 62)T. *Variations of fitting: b⊥储 = 0.13, b⊥储 = 0.4, m ˙ = 3.1. maxtau = 90 MPa. {R} ¯ = (1140, 570, 38, 135, 62)T. 135, 62)T. maxtau = 70 MPa. {R}

983 Eq. (23) for additional details of how to detect failure in the interaction zones. Beyond initial failure (IFF failure) the value of Eff remains constant and equal to 1. Both, the equivalent stress eq and the strength R are reduced according to the effective stress–strain curve. The equivalent stress is related to the effort according to Eff · R = eq({})

(3)

Final Failure (structural level) occurs after the laminate, and the structure, has experienced a stiffness reduction and has degraded to a level where it is no longer capable of carrying additional load. 2.3.2. Non-linear analysis in mode interaction zones The general relationships describing the equivalent stresses are 储 储 ⊥ ⊥ ⊥储 T {(modes) equiv. } = (eq, eq, eq ; eq ; eq )

with

储 ⊥ 储 eq = 1 · E t储, eq = –1, eq =

I2 + 兹I4 , 2

(4)

3/2 1/3 ⊥储 eq = (I 3 + b⊥储(I2I3  I5)) ,

⊥ eq =

2 · (b⊥I4 + b⊥储I3) (–b⊥  1)I2 + 兹(b⊥  1)2I 22 + 4b⊥I4 + 4b⊥储 I3)

(5)

The invariants I1, I2, I3, I4, I5 are as defined in Eq. (10). In the case of a non-linear analysis, the stress effort Eff has to be employed instead of the here ‘linearly’ (stress level not load level) defined reserve factor fRes. The stress efforts read

{Eff (mode)} =

冉

冊

储 ⊥ ⊥ ⊥储 储 eq eq eq eq eq , c, t , c , t R¯ 储 R¯ 储 R¯ ⊥ R¯ ⊥ R¯ ⊥储

T

(6)

Due to Eq. (3) the actual stress effort of a given mode, Eff (mode), is the ratio of the equivalent stress, eq, to the corresponding mode strength, R. The procedure of determining the resultant stress effort, Eff (res), in each lamina of the laminate is similar to that of f (res) Res shown in Eq (23a). 2.3.3. Residual stresses, Refs [7,14] Residual stresses are taken into account by adding their values to the stresses due to load {} = {}(L) + {}(R)

(7)

The residual stresses in the laminae of the laminate decay with decreasing stiffness, caused by the matrix degradation, which accompanies increasing non-linearity. In other words: In parallel to the decay of the stiffness the non-linear analysis releases matrix-dominated stresses. This applies for mechanical as well as for thermal stresses.

984 Chemical shrinking and thermal curing stresses do not affect the shear stresses. Micromechanical curing stresses (residual stresses of the second kind at filament/matrix level) could not be assessed and are not considered. They are usually assumed be respected in the values for the UD-strengths

2.4. Comparison between Part A theoretical predictions and experimental results The experimental results provided in Part B [4] datapack contained more information than was given in Part A for the ‘blind’ predictions. Therefore some parameter sets required further adjustment in order to improve the fit between the predictions of Part A and the test data. In order to distinguish between the original and modified predictions, all figure numbers with suffix A refer to the blind predictions (i.e. Part A prediction), and figure numbers with suffix AB refer to predictions with modified parameters. As will be seen later, figure numbers with suffix B refer to modified predictions introduced in this paper. For completeness, the captions of all of the figures will show the associated data set used. Those parameters, not relevant for the actual graphs, will be put into brackets. In addition, Test Case (TC) numbers, taken from Ref. [5], will be added in order to facilitate crossreferencing. 2.4.1. Stress–strain curves of the UD-lamina Typical stress–strain curves for a lamina made of E-glass/epoxy are shown in Figs 3 and 4. These show both the behaviour prior (hardening curve) and after failure (assumed softening). 2.4.2. Bi-axial failure envelopes of the UD-lamina 2.4.2.1. Behaviour of E-GFRP/LY556 lamina under (21, 2) and (21, 3) stresses (Test Case 1) The results in Fig. 5B with A for this Test Case (TC1) show the predicted Part A curve representing IFF-modes of failure and the stresses associated with these modes. Curve (31, 2) is obviously different to that of (21, 2) which indicates that 31 does not have the same action plane as t2 (known from the Puck–Hashin [6] model). The test data scatter is fairly large in the compression domain. The measured shear strength values in the compression quadrant were generally much larger than the predicted values. The measured shear strength reached a value of 99 MPa at –70 MPa compressive stress. This peak value appears to be unfamiliar or unique, as it has not been observed by other researchers. For instance, other test results, carried out recently in German R&D projects [7] on GFRP and CFRP materials, resulted in a shallower curve, see Ref. [1]. A prediction, based solely on the information provided in Part A for uni-axial cases, is not complete as one needs to estimate b⊥储 from biaxial test data. Therefore, a conservative value (giving predictions to be in the safe side), b⊥储 = 0.13, was taken. After receiving the test data, Fig. 5A curve was revised in Fig. 5A, B by taking a new value, b⊥储 = 0.3. This 0.3 value is

985 regarded by the author to represent the most extreme value or the upper limit, and may not be really reached in test. Although values of the other constant b⊥储 ≥ 0 are expected to produce a better fit with test data in the compression quadrant it was dropped from the analysis for two reasons: (a) to minimise reliance on test data and (b) to provide a maximum degree of simplicity in the analysis. Consequently, the value of this parameter was set to zero, b⊥储 = 0. The shape of the curve representing the test data is still not satisfactorily captured by the IFF conditions inclusively parameter set. In an attempt to capture the highest value of the shear strength observed under –70 Mpa compressive stress, the author has increased the value of b⊥储 from 0.13 to 0.56. The results are shown in Fig. 5B with A. Although the fit between the predicted and measured data is very good, the author believes that changing the value of that parameter from 0.13 to 0.56 is a purely curve fitting exercise. 2.4.2.2. Behaviour of T300/914C CFRP lamina under (21, 1) stresses (Test Case 2) Figure 6 shows the experimental (21, 1) results obtained from tests on unidirectional carbonfibre reinforced epoxy tubes superimposed on predicted results from Part A. In this case, the loading of this axially wound tube is torsion and axial loading. The test results (21, 1) achieved by different test specimens showed a wide scatter and ‘jumping’ in the positive domain. As this tube is heavily shearing under torsional loading the data given might not be the lamina stresses (⊥储, 储, ⊥) but other stresses belonging to the structure coordinate system (xy, x, y). In other words, the fibre direction angle of the lamina will change with the loading and this change in angle may have some effects on the lamina stresses. Therefore, an attempt is made to highlight how one could re-evaluate the experimental data (in MPa). In order to

Fig. 6. (TC2). Eqs. (1), (16). Bi-axial failure stress envelope (21, 1) in MPa. UD-lamina T300/ ¯ = (1500, 900, 27, 200, 80)T. BSL914C epoxy [8]. Axlally wound Tube. Eqs. (1), (16). m ˙ = 3.1 {R} Corrected test data due to computed shear deformation : 䊏 䉬. Transformation of (x ⬅ 1, y ⬅ 2, yx ⬅ 21) to real lamina stresses (储, ⊥, ⊥储). (Assumed, not active curve parameters b⊥储 = 0.13, b⊥储 = 0.4, b⊥ = 1.5.)

986 bring the test data to fit the predicted curve, a value is given to the shear angle . By applying a simple transformation, the stresses provided can be transformed from the structure coordinate system into the real or material (储-⊥) coordinate system of the lamina. Two examples are shown in Fig. 6; one for a single test data point in the tension side (with an angle of + 3°) and for another one in the compression side (with an angle of –2°). For the tension case, the transformation is given as follows: 储 = x(cos )2 + 2yx cos sin , ⊥ = x(sin )2  2yx cos sin , ⊥储 = –0.5x sin 2 + yx cos 2 , e.g. (x ⬅ 1, y ⬅ 2, yx ⬅ 21) = (1000, 0, 123), = + 3°⇒(储, ⊥, ⊥储) = (1010, –10, 70). This shows that, as a result of assuming an angle of 3°, the shear stress of 123 MPa can be reduced to 70 MPa. The latter value fits the predicted curve quite well. A similar set of calculations was carried on one test data, in the compression quadrant, where the shear strength was reduced from 94 to 76 MPa as a result of introducing an angle of –2°. The angles chosen above are arbitrarily so that the selected test data could be made to fit the predicted curve. It can be concluded that taking into account a small change in the winding angle could result in a large difference in the actual shear strength. 2.4.2.3. Behaviour of GFRP E/MY750 lamina under (2 , 1) stresses (Test Case 3) The test data were obtained from tests on ± 85° tubes. The effect of the actual winding angle of 85° instead of 90° will slightly reduce the strength values and was not considered as the results could increase the discrepancy between the measured and predicted data. In general, the correlation between predicted (Part B parameter set) and available measured data is reasonable for the limited range of test data provided. However, the data were not sufficient for a full validation of the present theory. Another missing feature in the test data is that the amount of scatter of each strength value was not given and only the mean values were provided.

Fig. 7. (TC3) Biaxial failure stress envelope (2, 1). UD-lamina. E-glass/MY750 epoxy [8]. Eqs. (1), ¯ = (1280, 800, 40, 145, 73)T, 1 = hoop, Part A or (16), Part B. Hoop wound tube data [4] + , m ˙ = 3.1. {R}  2 = axial (not active: b⊥ = 1.5, b⊥储 = 0.13, b⊥储 = 0.4).

987 2.4.3. Initial and final biaxial failure envelopes of the laminates (tube specimens) 2.4.3.1. Behaviour of [ + 55/–55/55/–55] E-glass/MY750 GFRP filament wound tubes (Test Case 9) Figure 8A shows the predicted and experimental initial and the final failure envelope of this GFRP-laminate. In the tension–tension quadrant some experimental data is larger than the predicted curve. At fracture, the material in the hoop (y) direction is essentially stronger than that in the axial direction. An explanation for this effect will be investigated in Sub-section 5.2.

Fig. 8. (A) (TC9) Initial and final failure envelope ˆ y(ˆ x). Filament wound tube, [ + 55/–55/55/–55]-laminate, E-glass/MY750 epoxy3. Eq. (1): m ˙ = 3.1, b⊥储 = 0.13, b⊥储 = 0.4, (b⊥ = 1.5). ˆ y = average hoop stress of the laminate, x = 0° direction. Limit of usage (lou) at = 4% (therefore, softening curve not utilized). (B) (TC9) Initial and final failure envelope ˆ y(ˆ x). [ + 55/–55/55/–55]-lam¯ = (1280, 800, 40, inate, E-glass/MY 750 epoxy [8]. Filament wound tube test data [4]: T = –68°C, {R} T   145, 73) , m ˙ = 3.1, b⊥储 = 0.13 [new F ⊥: Eq. (16)→newb⊥ = 1.09]. Bulging reported in experiment. Limit of usage (lou) at = 10% (softening curve applicable). Dashed curve: final failure of a full wedge failure-insensitive stack. 䉬 corrected value from 䉫.

988 In the bi-axial compression domain (third quadrant), the theoretical prediction gave values lower than those measured in the thick tubes. The author believes that the presence of external radial pressure has led to the discrepancy between the predicted and measured data. Section 5.2 gives examples of how the pressure may affect the behaviour of tubes. 2.4.3.2. Behaviour of [90/ + 30/–30/30/–30/90] E-glass/LY556 GFRP tubes under (ˆxy, ˆ x) stresses (Test Case 5) Figure 9A shows the comparison between the predicted and measured data for this laminate under (ˆxy, ˆ x) state of stress. The deficiencies are essentially located in the negative quadrant. Test experimental results are not well predicted by the model. The results of further modification of the theory are presented in Section 5.2.

Fig. 9. (A) (TC5) (Top) Initial and final bi-axial failure envelope ˆ xy(ˆ x). [90/ + 30/–30/30/–30/90]-laminate. E-glass/LY556 epoxy [8]. ˆ x is parallel to 0°-direction. Eq. (1). Filament wound tube test data [4] T = –68°C; b⊥储 = 0.13, b⊥储 = 0.4 (b⊥ = 1.5), m ˙ = 3.1. (B) (Bottom) (TC5) Modified initial and final biaxial failure envelope ˆ xy(ˆ x). [90/ + 30/–30]n-laminate. E-glass/LY556 epoxy [8]. ˆ x is parallel to ¯ = (1140, 570, 35, 114 72)T. b⊥储 = 0.13, m 0°-direction. Test data [4] T = –68°C. {R} ˙ = 3.1  (newb⊥ = 1.09). Eq. (16). Dashed line: increase due to R¯ c⊥ = 114⇒138 MPa.

989

Fig. 10. (a) (TC4) Initial and final bi-axial failure envelope ˆ y(ˆ x). [90/30/–30/30/–30/90]-laminate. E-glass/LY556 epoxy [8]. Filament wound tube, liner. Eq. (1). b⊥储 = 0.13, b⊥储 = 0.4, m ˙ = 3.1. (b⊥ = 1.5) T ¯ = (1140, 570, 35, 114, 72) . (b) (TC4) Revised initial and final biaxial failure envelopes ˆ y(ˆ x). {R} [90/ + 30/–30]n-laminate (n varies between 1 and 4). E-glass/LY556 epoxy [8]. T = –68°C. Tube Test ¯ = (1140, 570, 35, 114, 72)T. Eq. (16). b⊥储 = 0.13, m data [4]: {R} ˙ = 3.1, (newb⊥ = 1.09).

2.4.3.3. Behaviour of [90/ + 30/–30/30/–30/90] E-glass/LY556 GFRP tubes under (ˆ x, ˆ y) stresses (Test Case 4) Figure 10A is concerned with the prediction of the biaxial failure envelope for this laminate of a filament wound tube subjected to a (ˆ x, ˆ y) state of stress. The highest discrepancy is shown in the third quadrant. The author is unable to provide an explanation for the discrepancies observed in ‘extreme’ parts of the first and third quadrant. However, because the predicted failure in the third quadrant is F 储 and as the prediction is so far from test results, it is assumed that instability of the compressed tube might have taken place. Further justifications and improvements are discussed in Sub-section 5.2. 2.4.4. Stress–strain curves of the laminates The stress–strain curves in Part A were obtained from Eq. (1) and the data listed in Tables 3 and 6 in Part A [8]. The loading is assumed to be monotonic. A temperature drop from curing is considered in the reworked Part A’s graphs, however, the strain off-sets in these graphs are

990 shifted to the origin. Some curves in these figures are fibre-dominated. In such cases of welldesigned laminates, the fibre mesh controls the deformation. In contrast, other curves are matrix-dominated. No allowance was made in test evaluation for the bulging (barrelling) behaviour of tubes. A bulge is generated if there is a widening of the middle section of a tube in relation to the cross section at the reinforced ends. If the gauge length is small, then the behaviour at the middle cross section would be affected by the end constraint. Just one figure was taken from Part A where the difference to the corresponding Part B figure was obvious. The exact temperature drop used in the Part B predictions is indicated in these figures. As temperature drop in Part A and B the difference ‘stress free temperature minus room temperature’ as effective temperature difference was applied in order to consider the effect of curing stresses (are thermal residual stresses of the 1st kind). Moisture may be assumed here to have a balancing effect of 30°C. 2.4.4.1. Behaviour of [0/ + 45/–45/90]s AS4/3501-6 CFRP tubes under ˆ y stress (Test Case 7) Figure 11A shows the stress–strain curves of a laminate whose behaviour is fibre-dominated. The curves are fairly linear up to near fracture with a fracture load predicted to be less than 10% below the measured value. The behaviour near the final failure, where some softening in the predicted curve occurs, is thought to be due to problems associated with the computer code. The predicted stiffness was lower than the measured one.

Fig. 11. (a) (TC7) Stress–strain curves. Eq. (1). ˆ y : ˆ x = 1 : 0. [0/ + 45/–45/90]s-laminate AS4/3501-6 ¯ = (1950, 1480, 48, 200, 79)T. (b) Revised stress–strain curves for epoxy. Hand lay-up cylinder. {R} ˆ y : ˆ x = 1 : 0, (radial loading induced by pint + axial compression). ˆ y = hoop [0/ + 45/–45/90]s-laminate. ¯ = (1950, 1480, 48, 200, 79)T. T = –125°C. Test data [4]. Final Part A point AS4/3501-6/epoxy [8]. {R} 䊉.

991 There was another problem in the results of Part A. For ˆ y curve, the stiffness loss by the initial IFF degradation was not ‘subtracted’ but ‘added’ which was corrected here. However, this does not make the curves in total steeper.

2.5. General comments on correlation between theoretical and experimental results 2.5.1. Design of laminates and tubes The use of CLT-based analysis, as applied to flat plates, does not take into account some effects associated with the use of tubes as test pieces. In case of anti-symmetrical laminates an unconstrained laminate plate will twist under the action of in-plane direct stresses whereas a tube will not twist but experience in-plane shear straining. This will produce some effects on deformation and fracture behaviour on the different classes of laminates addressed. It was reported in Ref. [4] that “The tubes were designed by utilising linear elastic thin shell theory. Non-linear analysis of the 55° and the 45° tubes has indicated that the gauge length was too short for the 45° tubes. Hence, the so-called boundary constraints will influence the strains at the centre of the tube. In order to achieve a high level of verification of the current non-linear theory, a finite element code is required that takes into account large strain and large displacements as well as the actual state of stress, which includes softening or degradation. 2.5.2. Constraint effect on an embedded lamina (in situ behaviour) If applying test data from tensile coupons of isolated lamina to a lamina embedded in a laminate, one has to consider that tensile coupon tests deliver test results of the weakest link type (series model). However, an embedded lamina or even a lamina constrained on only one side, belongs to the class of the so-called redundant type behaviour (parallel spring model) [2]. In the latter case, and due to it being strain-controlled, the material’s flaws in a thin lamina cannot grow freely up to microcrack size in the thickness direction (this is called thin layer effect), because the neighbouring laminae act as micro-crack-stoppers [9]. In addition, the strain energy release rate responsible for the development of damage in the 90° plies from flaws into micro-cracks increases [10] with increasing ply thickness. Therefore, the actual thickness of a lamina in a laminate is a driving parameter for the initiation or onset of microcracks. This phenomenon is not considered in the present work. 2.5.3. Application of an effective stress–strain curve Due to the in situ strength effect, the author regards the peak value of the effective stress–strain curve of an embedded lamina to be slightly higher than the strength value R¯ of an isolated lamina, due to the change from the ‘weakest link behaviour’ to the real redundant behaviour (see Figs 3 and 4) of a laminate. However, due to a lack of knowledge and for the sake of simplicity, this ‘peak value’ is lowered down to the mean strength R¯ in the analytical

992 description of softening. Due to mapping reasons, 0.99 R¯ ⊥储 was taken instead of R¯ ⊥储 as the calibration point for the softening curve (see also Sub-section 3.2.6). 2.5.4. Application of mean properties Normally, the data used in any stress analysis are obtained from experiments and a certain amount of scatter is inevitable. Consequently, and from a statistical point of view, there will be at least three values representing minimum, average and maximum properties. In the nonlinear analysis, the mean values, determined from tests on unidirectional laminae, have to be used to carry out the stress and deformation analysis. The results correspond to the mean or average structural behaviour. This is regarded as the best approximation of the structure’s physical behaviour. Therefore, the execution of a non-linear stress analysis of the structure shall utilize a mean(,)-curve, whereby the associated secant moduli will be mean values, too, as performed in the work at hand. The application of a minimum(,) curve, which is sometimes required, will lead to lower stresses.

3. New ideas and refinements to Part A – theory 3.1. FMC-based set of lamina failure conditions In principle, when describing the various types of failures of a transversally-isotropic lamina, according to the author’s proposed extension of the FMC model, failure conditions that have to be provided include: (a) yielding (which represents one physical mechanism), and (b) fracture (which is described by five physical mechanisms). The following section describes a procedure for defining yield failure in a unidirectional lamina. 3.1.1. Yield condition of the UD lamina In conventional FRP materials the matrix is much more ductile (having typically 6% fracture strain) than the fibres (with typical fracture strains of 1.5–3%). It is probable that the initial yielding which is reflected in the shear (21, 21) as well as in the transverse compressive (2, 2) stress–strain behaviour of a lamina is dominated primarily by the matrix behaviour. The yield strength of the constituent matrix plays a role at relatively low strains of the lamina (model: smeared material). A practical approach to establish an initial yield condition is to assume the existence of a 3D yield failure condition in terms of macro-mechanical quantities such as the lamina stresses. It is furthermore assumed that a perfect bond exists between fibre and matrix. The proposed yield failure condition is derived as follows: It has been established by Beltrami [11] that, at initiation of yield, the strain energy (W ) in a cubic element of a material consists of two portions; one describing the strain energy due to a change in volume and another due to change in shape. The energy is given by W = 兰 {}{} d{} = WVol + Wshape with

{} = (1, 2, 3, 23, 13, 12)T.

(8)

993 Using generalised Hooke’s law, for a transversally-isotropic (UD) body, the above expression takes the following form, where sik are the compliance coefficients [12], WUD = [s1121 + s2222 + s3323 + s44223 + s55(212 + 213)]/2 + s12(12 + 13) + s2323 =

I 21 I 22(1  v⊥⊥) v⊥储 I1I2 I3 I4(1 + v⊥) +  + = 2E储 4E⊥ E储 2G储⊥ 4E⊥

volume volume

volume shape

(9)

shape

and the associated volume or shape change of the UD material element are indicated. The five invariants applied are: I1 = 1, I2 = 2 + 3; I3 = 331 + 221; I4 = (2  3)2 + 4223; I5 = (2  3)(231  221)  4233121 (10) The procedure how to apply Eq. (9) may be taken from the simpler well-known isotropic results, see Ref [4]. There, including Hooke’s law for isotropic material, it follows:

W=

冋

册

1  2v iso2 2 + 2v iso I1 + 3J 2 /2E 3 3

Volume

(11)

shape

with I iso I = f() = I + II + III, iso 2 2 2 Mises = 兹3J iso eq 2 and 6J 2 = f() = (I  II) + (II  III) + (III  I) ,

E: Young’s modulus, v: = Poisson’s ratio. The first term in Eq. (11) describes the volume change of the cubic material element (dilatation) and the second the change of its shape (distortion). Both quantities inside the bracket in Eq. (11) are used to formulate F = c1

(1  2v)I iso2 (2 + 2v)3J iso 2eq I 2 2 + = c = 2 3R¯ 2 3R¯ 2 R¯ 2

volume

(12)

shape

a failure function, where c1 is a shape determining curve parameter, c2 is the size governing parameter of the failure body, and R¯ is an appropriately chosen strength value that delivers a very practical normalization. For the fracture condition (final failure), R¯ is optimally taken as fracture tensile strength of the material, R¯ tm, whereas for the case of yield conditions, the stress at 0.2% plastic strain (R¯ p0.2) is employed. It should be noted that upon reaching the full-plastic domain, an appropriate value for the Poisson’s ratio v should be given. It has been suggested that at yield limit, the following is valid  = R¯ tp0.2⇒v = 0.5. In order to formulate a relatively simple yield failure condition the second term in Eq. (11), which represents the shape change, is chosen as the basic invariant. This means in the isotropic case, J2 is the appropriate choice, which leads to the (Hencky–von Mises–Huber model), called for short, Mises model 2 ¯ 3J iso 2 = eq. Due to normalization: If eq reaches Rp0.2, then the size parameter c2 equals 1. The result is a single yield failure surface, which represents initial yielding under an arbitrary combination of the three principal stresses or of the six structural stresses.

994 Similarly, for the transversally-isotropic material, it can be assumed that only those terms describing the shape change of the UD material can contribute to a failure function. Based on this, the approach gives I3 (R¯ ⊥储 p0.2)

2

+

I

(R¯

4 c 2 ⊥储 p0.2

)

= c2

(13)

with the size parameter c and the two yield strengths associated with the main two non-linear stress–strain curves c2(2) and 21( 21). This single (global) an-isotropic yield surface, depicted in Fig. 20a for (21, 2) stresses and in Fig. 20b for (2, 3) stresses is confined by the five partial fracture surfaces. In a yield condition the strength R changes whereas in a fracture condition R is constant. The subsequent yield surface is indicated by a vector normal to the actual global yield surface determined by its associated flow rule (normality criterion) that is not expressed here. The normality criterion is replaced by the ‘concept’ of proportional stressing, which means, the failure surface increases in the direction of the actual stressing which is seldom the normal direction. Of course, the loading of the laminate may be a proportional one (i.e. all loads increased in proportion to one another), however, the stressing in the laminae of the laminate may not be proportional due to nonlinearity. It has been observed that the fracture stresses of hoop wound tubes under combined (21, 2) stresses depend on the load path performed in the test (see Ref. [13]). This is obvious from the bi-axial failure envelopes for the UD composites as different failure modes may be passed on different ways to the ‘combined’ fracture stress point allowing for different degradation on these paths through the yield zone. The state of combined stresses at the onset of yielding may be obtained from Eq. (13). Furthermore, creep might have an impact on test results and on the yield limit. 3.1.2. Fracture conditions of a UD lamina In the application of the FMC, the decision as to “Which stress invariants are to be chosen?” is based on whether there are volume and/or shape changes of the material element, whereas the choice of linear or higher terms is determined by curve fitting considerations. For the sake of simplicity, the equation describing the IFF3 mode of failure has been modified. The aim is to obtain a homogeneous equation that can be easily used. This simplification is achieved by setting b⊥储 = 0 and by replacing I4 by 兹I4. The new equation reads as follows F ⊥ = (b⊥  1)

I2 兹I4 + b = 1. ¯R c⊥ ⊥ R¯ c⊥

(14)

It is be noted that in Eq. (14) the value of b⊥ is different from the former one in the Part A set, because F ⊥ is a new function. Now, the reserve factor is simpler (linearly) to compute   ¯ c ⊥ ¯ c f ⊥ Res = R ⊥ /eq = R ⊥ /[(b⊥  1)I2 + b⊥兹I4],

(15a)

995 as well as the stress effort ⊥ ¯c Eff ⊥ Res = eq = R ⊥

(15b)

in non-linear analysis. The visualisation of Eq. (14) is presented in Fig. A1 depicting a linearly running fracture curve. 3.1.3. Listing of fracture conditions employed Based on the simplifications outlined, the following revised set of failure conditions, F({}) = 1, was derived and used in Part B FF 1: F 储 =

I1 = 1, ¯R t储

FF 2: F 储 =

–I1 = 1, R¯ c储

IFF 1: F 储 =

I2 + 兹I4 = 1, 2R¯ t⊥

(16)

I2I3  I5 ¯3 IFF 2: F⊥储 = I 3/2 = 1, 3 /R ⊥储 + b⊥储 R¯ 3⊥储

IFF 3: F ⊥ = (b⊥  1)

I2 b⊥兹I4 + = 1. R¯ c⊥ R¯ c⊥

Two curve parameters (b⊥储, b⊥) are needed and these are determined from multi-axial test data or could be estimated by experience. In the above equations, the term I1 in the top equation describing F 储 is given as: I1 = 1 = vf · 1f = vf · 1 · E1f = 1 · E储 with 1f = tensile fibre stress. Homogenisation of the UD lamina as a ‘smeared material’ comes to its limit if a constituent stress governs the failure. This is the case for F 储 , where the macro-mechanical stress 1 has to be replaced by the actual fibre stress 1f . The two curve parameters (b⊥储, b⊥) have to be determined either from a single test at a given stress ratio (several measurements in this calibration point) or more accurately from curve fitting to the shape of various test data in the associated pure domain using a suitable regression analysis. Data from the calibration points (marked by ‘䊐’ symbol in Figs A1 and 5A, B) are inserted into the IFF conditions for F⊥储 and F ⊥ [Part A set, Eq. (16)], and the parameters are given by

b⊥ =

¯ 2 1  (⊥储 21 /R⊥储) ¯3 2c2 · ⊥储2 21 /R ⊥储

(c2, ⊥储 21 ),

(17a)

c ¯ c 1 + (c 2 + 3 )/R ⊥ c ¯ c c c 2 ¯ c (c 2 + 3 )/R ⊥ + 兹(2  3 ) /R ⊥

(17b)

b⊥储 =

from

These equations are used in Part B. The value of the parameter b⊥ differs to that of Eq. (2). Both parameters b⊥储 and b⊥ depend on the material behaviour and on the IFF formulation applied. Practical safe bounds for glass/epoxy (GFRP), carbon/epoxy (CFRP) and Aramid/ epoxy (AFRP) are 0.05 < b⊥储 < 0.15, 1.0 < b⊥ < 1.15 (instead of 1.5 used in Part A quadratic

996 equation). A value b⊥储 = 0 means ‘no bulge effect’ and b⊥储 = 1 means ‘no friction’ in the ⊥⊥-plane. In the absence of having actual calibrated data for these parameters, the author recommends the use of their lower limits in a pre-dimensioning of a component. With regard to the 3D nature of the IFF conditions, both IFFI (F ⊥: = transverse tensile failure and IFF3) (F ⊥: = wedge failure, local 3) may also serve as criteria for the onset of delamination generated by the interlaminar stresses (3, 32, 31).

3.2. Refinement of non-linear analysis procedure 3.2.1. Mapping 3.2.1.1. Hardening The degree of non-linearity essentially depends on the non-linear behaviour of the matrix material that manifests itself in the values of E c⊥ and G储⊥. For the non-linear stress analysis the secant moduli applied are derived from the Ramberg/Osgood equation

 = /E(o) + 0.002(/Rp0.2)n

(19)

It describes the non-linear stress–strain data very well for the materials analysed here (E(o) is the initial modulus). The so-called Ramberg/Osgood exponent n = ᐉn(p1(Rm))/ᐉn(Rm /Rp0.2)

(20)

is estimated from the strength point [Rm, p1(Rm)] in Fig. A3. Then, the secant moduli E⊥, G储⊥ may be obtained from Eq. (19) as (isotropic notation was used here for the sake of simplicity) E(sec) = E(o) /(1 + 0.002 · E(o) /Rp0.2 · (/Rp0.2)n  1)

(21)

3.2.1.2. Softening Beyond Initial Failure (IFF), an appropriate progressive failure analysis has to be employed. Occasionally, this process is termed as, Successive Degradation Modelling of Post Initial failure. Figure 4 depicts the hardening and the softening curve for a unidirectional lamina. Due to a lack of experimental data, the shape of softening curve (being strain controlled) has to be assumed as modelling of Post Initial Failure behaviour of a laminate requires some knowledge of the reduction or the degradation in the elastic properties of the embedded lamina [curve (c) in Fig. 4]. A rapid collapse (often named ‘ply discount method’) of E t⊥ is unrealistic and could lead to convergence problems in the computation algorithms. In order to overcome these problems, both E c⊥ and G储⊥ are assumed to decrease gradually rather than being suddenly annihilated. A simple exponential function was used to map this softening and this is given by s = Rm /(1 + exp[(as + )/bs])

(22a)

in which suffix s denotes softening. The two curve parameters as, bs are usually estimated from the data of at least two calibration points in achieving a reasonable fit, e.g. (Rm, (Rm))

and

(Rm · 0.1, (Rm · 0.1))

(22b)

997 or another suitable set. Rm corresponds generally to the strength of the isolated lamina under uni-axial loading. Eq. (22) models the softening part of the stress–strain curve of a lamina which is embedded in a laminate, and thus, it includes the effect of the altering micro-crack density up to a critical damage state (CDS), Refs [10,15,32]. Curve (c) is therefore termed an effective curve. Appendix C visualizes hardening (19) and softening function (22). 3.2.2. Interaction of failure modes Mechanical and probabilistic interactions cannot be clearly distinguished and therefore, the author models the failure mode interactions by a simple probabilistic series system model. This so-called ‘logical model of the failure system’ acts as a rounding-off procedure linked to (res) the determination of the desired values for f (res) Res or Eff The (resultant) Reserve Factor f (res)Res takes into account the interactions between all the various modes. It may be estimated by the rounding-off equation, describing the series spring model, Ref [2], m ˙ (modes) (1/f (res) ) . . . if linear state of stress Res ) = f ( f Res m ˙ ⊥储 m ˙ ⊥ m ˙ 储 m ˙ 储 m ˙ = (1/f ⊥ Res) + (1/f Res) + (1/f Res) + (1/f Res) + (1/f Res)

and

冘

(23a)

5

Eff

(res)m˙

=

Eff (modes) . . . if non-linear

1

储 ¯ c m ˙ ⊥ ¯ t m ˙ ⊥ c m ˙ ⊥储 ¯ m ˙ ¯ t m˙ = (储 eq /R 储) + (eq /R 储 ) + (eq /R ⊥) + (eq /R ⊥) + (eq /R⊥储) ,

(23b)

where m ˙ is the mode interaction coefficient (rounding-off exponent, the size of which is high in case of low scatter and vice versa). As a simplifying assumption, m ˙ is given here the same value, regardless of interaction zone! The value of m ˙ is obtained by ‘best fitting’ the course of test data, respecting the fact that in the interaction zones, micro-mechanical and probabilistic effects co-exist and cannot be discriminated. It is a compromise applying this value in all interaction domains. The author’s experience suggests that, m ˙ = 3.1 is appropriate. If a unidirectional fracture stress (i.e. the strength value R¯ t⊥) is inserted into the equation above, then a point on a 2D- or 3D-failure curve or failure surface (initial or final), described by f (res) Res = 1, is achieved. It is worth mentioning that a maximum of four failure modes could be interacting under three dimensional state of stress. 3.2.3. Determination of mode efforts (allowing for interaction of the lamina stresses) In the following set of formulae the effort of each mode is provided according to the general equation Eff (mode) = (mode) /R¯ mode, together with the so-called equivalent stress. An equivalent eq stress includes all load stresses and residual stresses that are acting together in a given mode. The following set of equations is employed in Part B (proportional stressing) ¯t Eff 储 = 1 · E t储 /R¯ t储 = 储 eq /R 储,

(24a)

Eff 储 = –1 /R¯ c储 = –储eq /R¯ c储 ,

(24b)

998 ¯t Eff ⊥ = [I2 + 兹I4]/2R¯ t⊥ = ⊥ eq /R ⊥

(24c)

1/3 ¯ ⊥储 ¯ Eff ⊥储 = [I 3/2 3 + b⊥储(I2I3  I5)] /R⊥储 = eq /R⊥储,

(24d)

¯c Eff ⊥ = [(b⊥  1)I2 + b⊥兹I4]/R¯ c⊥ = ⊥ eq /R ⊥.

(24e)

Note: If an Eff (mode) becomes negative, caused by the numerically advantageous automatic Res insertion (regardless of the sign of the stress) of the FEM stress output {} = (1, 2, 3, 23, 13, 12)T into all five failure conditions, a value of 0 shall replace the negative value. A negative value results, for example, if a positive 1 is inserted into Eq. (24b). The situation of an imaginary Eff (mode), which is only possible for Eff ⊥//, is bypassed by a query (see Appendix B). 3.2.4. Degradation ‘triggering’ In the laminae of a laminate, multi-axial states of stress are acting which in the interaction domains have an impact on more than one of the five failure modes. Adjacent failure modes are commonly affected. One has to pay attention to a proper interaction of the interacting modes in the stress and strain analysis in the following manner: In order to take into account the interaction of the failure modes the secant moduli E2(sec) and G21(sec) are taken from the 2(2)-curve or the 21( 21)-curve not just at the stresses 2 or 21 resulting from the stress and strain analysis for the actual level load. Their values are taken at a little higher stress in the ‘hardening branch’ with increasing stress and at a little lower stress in the ’softening branch’ with decreasing stress. This ‘stress correction’ is controlled by the so-called ‘triggering approach’. Therefore, a corresponding degradation (displayed by a stiffness reduction) has to be considered: • for increasing stress (Hardening)  > 0 corrected stress eq> (mode) = (mode) , TrF eq

(25a)

• for decreasing stress (Softening)  < 0 corrected stress eq< (mode) = (mode) /TrF eq

(25b)

The controlling parameter is the ratio of the resultant stress effort Eff (res) to the maximum mode effort max-Eff (mode). In Part B a revised (m’th ˙ root) trigger-factor is utilized m ˙

TrF = 兹Eff (res)/maxEff (mode)

(26)

that replaces the previous concept which resulted in a sudden decrease in the stiffness. In the equations above the stress effort of the maximum stressed mode governs the ‘triggering’, although TrF is dedicated to all IFF modes affected in order to really regard the combined degradation effect. By this triggering approach slightly lower secant moduli E2(sec) and G21(sec) are provided for the next calculation loop; lower than those that would result without the correction by the triggering approach. In other words: As adjacent failure modes mutually degrade their elastic stiffnesses, in order to apply correct secant moduli, equivalent stresses

999 储 ⊥ ⊥ 储⊥ T eq(modes){} = (储 eq, eq, eq ; eq ; eq ) corrected by TrF, have to be inserted into the equations for the secant moduli. Figure 19 visualizes Cuntze’s ‘triggering approach’. In this theory for the actual load the degradation of E2(sec) and G21(sec) is performed with the same trigger factor TrF. In contrast to Puck’s theory, if one of the corrected equivalent mode stresses has reached its strength level, a relatively rapid decrease of the mode’s average (smeared over the micro-cracks) equivalent stress will follow. The triggering approach is already active before the onset of IFF. In Section 5 the diminishing effect of triggering with increasing large strains will be considered.

3.2.5. Non-linear analysis The approach may be called a self-correcting secant modulus procedure. The laminate’s stiffness matrix is recomputed after each step. Then, the stresses 2 and 21 in the laminae of the laminate are computed by using secant moduli from the 2(2)- and 21( 21)-stress–strain curves and from them the equivalent stresses are determined. The solution procedure of the non-linear analysis aims to establish static equilibrium at each load step after material properties have been changed. For each iteration the procedure is repeated until convergence (equilibrium) is reached or a final failure occurs. A correction of the fibre angle in accordance with the change of the specimen’s geometry as consequence of large strain behaviour has been considered. In the non-linear computations for a small load increment, sometimes just one iteration step is needed in a secant modulus procedure in order to roughly consider stress-redistribution, that means, load from the weakening matrix (matrix-dominated modes) is transferred to a fibre (fibre mode). In the revised MATHCAD-based code ‘CLT FRP Non-linear’, generated for the Failure Exercise, the self-correcting secant modulus method, described in Part A [1] was applied to describe the successive degradation. 3.2.6. Determination of the degrading elasticity properties of the lamina By employing the equivalent stress reached in each failure mode the associated secant modulus of each mode was determined for the hardening and the softening regime. Considering a consistent stress concept for all (modes) an explicit dependency Esec((mode) ) has eq eq to be provided. For reasons of achieving such an explicit formulation two separate formulae are discriminated which are linked in the strength point. This automatically respects that the chosen non-linear calculation procedure demands the dependencies of the secant moduli on the corresponding equivalent stress. These dependencies are (see Fig. 4) • Pre-lFF analysis of lamina:  > 0 (increasing stress, hardening) E t⊥(sec) = E t⊥(o) ⊥ ⊥c n⊥  1 E c⊥(sec) = E c⊥(o) /[1 + 0.002 · (E c⊥(o) /R⊥c ] p0.2) · (eq /Rp0.2) c

⊥储 ⊥储 n⊥储  1 ] G储⊥(sec) = G储⊥(o) /[1 + 0.002(G储⊥(o) /R⊥储 p0.2) · (eq /Rp0.2)

(27)

1000 • Post-IFF analysis of lamina:  < 0 (decreasing stress, softening) ⊥ E t⊥(sec) = ⊥ eq /(eq )

冋冉

⊥t = (⊥ eq /bs )/ ᐉn

冊 册

R t⊥  ⊥ a⊥t eq s  ⊥ b⊥ts eq

(28)

The branches with increasing stresses (‘hardening’) of these stress–strain curves are found by the usual experiments with uni-axial 2-stress or pure 21-stress, respectively. Branches with rapidly decreasing stresses (called ‘softening’) are assumed, see Fig. 4. For the other modes the same formula is valid, however, the mode parameters are different. After reaching Eff (res) = 1, this value 1 is kept as constant during the degradation process. This causes a stress redistribution towards the fibres as long as the fibre net allows such behaviour. 3.2.7. Laminate failure Most engineers assume that FF in at least one lamina of a laminate means final failure of the laminate. Therefore, the biaxial failure envelopes for final failure of laminates predicted by the various authors do not differ that much, as long as the laminates are well-designed and have three or more fibre directions. The ‘strengths’ of these laminates are ‘fibre dominated’. Also, the predicted stress–strain curves of such laminates look very similar because the fibres that are much stiffer than the matrix carry the main portion of the loads. Different degradation procedures after the onset of inter-fibre failure (IFF) do therefore not influence the predicted strains very much. This is especially true for CFRP laminates. 4. Some comments on nature and effect of failures predicted 4.1. Comments on evaluation of experimental data A correct analysis of boundary conditions and stress state of the test specimen is mandatory before evaluating and applying the data. In this sense tubes instead of the flat coupon specimens will help to avoid problems associated with the ‘free edge effect’. A wide range of bi-axial stresses can be achieved. Real tri-axial stress states require refined specimens (see VDI report [7], p. 107). The tubular specimens may be subjected to internal and external pressure, to torsion and axial forces. However, the testing of tubular specimens is not free of problems such as bulging, caused by end constraints, or buckling of the cylinder. Further, tubes may exhibit non-linear changes in geometry during loading. In Ref. [4] it was stated: “Pressures and forces were usually increased continuously until fracture occurred. During the tests, the pressure was increased at a steady rate (2 MPa/min) and the axial load was increased continuously to maintain a constant predetermined stress ratio of the laminate’s mean stresses ˆ hoop /ˆ ax within the gauge length until fracture. During loading the effects of bulging were not considered. Fibre volume fraction vf in the laminates is approximately the same as for the UD laminae. Unless otherwise stated the given stresses applied in the test (dots) were calculated from measured pressure p and axial load F based upon un-deformed initial geometry, applying formula

1001 ˆ hoop = p · rint /t and

ˆ ax = p · rint /2t + F/(2 · rint · t )

(29)

in which rint is the internal radius of the un-deformed tube and t the laminate thickness. For ˆ ax, in case of large displacements corrections were sometimes reported (widening and/or barrelling) due to the fact that rint at the centre of the gauge length becomes greater than the radius at the ends of the tube. Whether the stress rate above might have caused some creeping in the highly loaded matrix-dominated cases is not made clear. Note: In order to accurately predict the behaviour of the tubular specimens, it is necessary to take into account material and geometrical non-linearities in both experimental and theoretical evaluation. The pressure loading produces a compressive stress (3 = –p) on the loaded surface whereas the other surface is free of radial stress. Despite of the fact that with increasing degradation the highest hoop stress in a vessel propagates beyond the IFF from the internal surface to the external surface, the most severe stress state will occur at the internal surface because there is no beneficial bi-axial compression, however, there may be some stabilizing by the curvature. The author envisages a difference between the behaviour of a metal and a FRP that consists of a relatively weak matrix and embedded stiff filaments. In contrast to metal where the failure is a strength failure and the metal may yield on the inside towards the centre (no barrier), for FRP, failure is caused by micro-buckling of the filaments and this depends on the magnitude of the curvature and on the quality of the embedment generating a spatial helical support. Such a constraint demands a higher energy and that increases the stability level. (Similarly: Under the same external pressure an internal liner in a fibre-reinforced shell will not buckle like an isolated liner shell). An estimation of the magnitude of this effect cannot be given by the author so far. The differences between uni-axially and bi-axially compression-loaded surfaces will have to be analysed. From Fig. A1, it can be concluded that uni-axial lateral pressure is more severe than having c2 bi-axially acting together with c3. In general, the pressure-stabilized surface lamina will not fracture first. 5. Application of refined theory Basically, the modifications or improvement introduced in this paper are concerned with the following: • • • •

Improvement of the non-linear analysis of the computer code Simplifications of the failure conditions Offering of some thoughts as to how to account for the effect of radial pressure on failure Test data evaluation.

In the results presented in Ref. [1], the author did not take into account the softening curves and hence the computation were terminated prematurely. However, in the results presented here the calculations continued to cover the softening regime in the lamina stress–strain curves. This turns out to have a significant effect on the predicted results. From the “refined” theory, which includes failure conditions and some convergence ideas in the non-linear analysis coding, the following impacts can be reported: The effect of the new F ⊥ is marginal. So, only the coding is of importance. Any difference between the revised

1002 graphs and the Part A graphs is essentially attributed to the improvements arising from a more stable numerical procedure.

5.1. Bi-axial failure envelopes of the UD-lamina Fig. A1 depicts the fracture curve F ⊥. It just outlines for this wedge failure mode (IFF3) the difference of the new linear F ⊥ to the former quadratic F ⊥. The curve 2(3) is now fully linear in comparison to the former one, the quadratic behaviour of the former function was pronounced in the first stress quadrant, only. Practically, there is no impact on the mapping capabilities, no change in the figures will take place.

5.2. Initial and final bi-axial failure envelopes for the laminates 5.2.1. Test Case 9, GFRP E/MY750 [ + 55/–55/ + 55/–55], wound tube Figure 8B incorporates the initial and the final failure envelope of this angle ply GFRPlaminate. There are three areas worth discussing: (a) behaviour under biaxial tension, (b) behaviour under biaxial compression and (c) pressure effects. (a) The predicted curve in the bi-axial tension, especially near the horizontal axis, is much lower than what the measured values [4] indicate. The largest discrepancy is probably on the ˆ y axis, which the author has no explanation for. Here, the know-how from the investigations on the (2 : 1) stress–strain curves (see Section 5.3.1 and Figs 13B and 14B) is to be included: The test data have to be shifted a little into the ˆ x direction. The simulation discussed in Section 5.3.1, confirms bulging and stress redistribution to the fibres, which are then loaded above R¯ t储! It indicates that after reaching F⊥储 (a non-critical condition for this laminate lay-up and loading) F ⊥ is achieved and then the numerical analysis fails. The failure envelope is closed. (b) The high test values (about one third higher than the theoretical predictions) in the biaxial compression domain could be partly explained by a higher R¯ c储 , effective in the externally pressurized tube. A description of the (ˆ y : ˆ x) = (–2 : –1) test including boundary constraints executed within this test campaign would have given much more insight in the interpretation of the Fig. 8B results. The load transfer or redistribution from the degrading matrix (softening, F ⊥) to the fibres (F 储 ) could not be managed by the actually utilized code for higher loadings. Computation stopped in the wedge mode, of course. (c) In order to demonstrate the strengthening effect of the normal pressure pext on the biaxial strength, a simple example is given which may explain the higher experimental results in the negative quadrant (biaxial compression quadrant). For a stress ratio of (–2/–1), the theoretical predicted laminate failure stresses are ˆ hoop = –436 MPa, ˆ ax = –218 MPa), taken at initial failure. The corresponding lamina stresses are collected in the stress vector: (1, 2, 3) = –pext), 0, 0, 21)T = (–524, –123, –pext, 0, 0, 43)T, where 3 is added to get an estimated 3D state of stress. Assuming that the thickness to radius ratio of the tube is t/r = 0.2 and using ‘thin shell’ theory, the failure pressure is estimated as pext = ˆ hoop · (t/r) = –436 MPa · 0.2 = –91.8 MPa = 918 bar. From this data set and utilizing R¯ c⊥ = 145 MPa, newb⊥ = 1.09, Eq. (16), an increase of fracture loading can be estimated by a simplifying linear analysis computation.

1003 (res) Without pressure on the outside of the tube, the reserve factors f ⊥ Res = 1.17 and f Res = 1.0 are computed. With pressure, a remarkable increase is achieved for

f ⊥ Res

R¯ c⊥ [(b⊥  1)(2 + 3) + b⊥兹(2  3)2]

= 9.9.

Regarding the other mode reserve factors f 储cRes = –R¯ c储 /1 = –800/(–524) = 1.53, f ⊥储 Res = 2.71, the m ˙ ⊥储 m ˙ ⊥c m ˙ 储c m ˙ resulting reserve factor, computed from (1/f (res) Res ) = (1/f Res) + (1/f Res) + (1/f Res) , becomes (res) f Res = 1.45, that is much larger than 1! This increase indicates considerable reserves. The value above demonstrates that, taking into account the radial pressure (via 3D analysis), the predicted failure stresses are much higher than those obtained using 2D analysis (in this it is not possible to take pressure effects into account). For the estimated 918 bar, the increased multi-axial strength is depicted in Fig. 8B by a filled rhombus sign. It can be seen that the consideration of 3 = –pex leads to an increase in the biaxial compression strength. Bulging could not be regarded in the CLT analysis, however the simple netting analysis indirectly considers it when taking the strains at failure. The calculation above assumed wedge failure might have taken place. In reality, for this pressure vessel stack wedge failure is not acting that catastrophically. Considering this, a further increase of the correction into the regime (–800/–400) can be estimated for the external surface. 5.2.2. Behaviour of [0/ + 45/–45/90]s AS4/3501-6 CFRP, hand lay-up, lined tube Figure 12 displays a symmetrical failure envelope for a CFRP laminate subjected to (ˆ x, ˆ y) states of stress. Coincidence with the experiment is poor in some places. The large scatter in the test data provided, especially the large scatter on the ˆ cx axis (Fax loading), makes correlation doubtful. Buckling in test will probably be responsible for the large discrepancy in the compression domain (ˆ cy , pext,-Fax) not allowing the material to exhibit the predicted mode of failure F ⊥. Hence, the available test results have only allowed for a partial validation of the theory and further tests are needed to confirm other aspects of the present theory. The failure envelope in Fig. 12 is slightly different to that in Part A due to the improvements with the programme, see Fig. 12B with A. According to joint failure probability of the laminae the sharp corners are smoothed artificially, but to be honest, such a ‘laminate smoothing process’ requires a high effort [16] and is not yet matured to an engineering tool. Shortcomings, not allowing for rapid improvements in this situation, are the normally poor knowledge about the uncertain design parameters (stochastic model of the strengths, loads) as well as the non-linear stress situation, need to be tackled. 5.2.3. Test Case 5, [90/ + 30/–30/30/–30/90], E-glass/LY556 GFRP tubes in torsion with axial load The lay-up used is not quasi-isotropic because the proportion of 90° plies is not 1/3 but 17.2%. The differences between Part A and Fig. 9B with improvement predictions are caused by the code improvements (which are still not fully robust for large non-linearity). Large bulging considering deformation cannot be considered in the simple 2D plate analysis, a non-linear

1004

Fig. 12. Right graph (b): (TC6) Modified initial and final failure envelope ˆ y(ˆ x). [0/45/–45/90]slaminate, AS4/3501-6 [8]. ˆ y: = average hoop stress of the laminate, x: = 0° direction. T = –125°C. ¯ = (1950, 1480, 48, 200, 79)T. (newb⊥ = 1.09). Hand lay-up cylinder. Test data [4]: b⊥储 = 0.13, m ˙ = 3.1, {R} Rounding by joint failure probability (estimated) of adjacent laminae. Left graph (a): Comparison between the original and modified initial and final failure envelope ˆ y(ˆ x) in MPa. [0/45/–45/90]slaminate, AS4/3501-6 [8].

cylinder analysis is required. On the positive ˆ x axis the mean value of about 550 MPa is now met whereas on the negative abscissa a gap remains. The UD lamina strength data provided by the organisers for this material seems to be low compared with those for the E-glass/MY750 (GFRP) material. There is some doubt about the accuracy of these low values. Utilizing TC1 allows for increasing R¯ c⊥ = 114–139 MPa and a second run delivers an improved matching of test data (dashed curve in Fig. 9B) in the compression domain. An idea for a possible and helpful higher R¯ c储 value was not found. Further work is needed to obtain a robust post failure prediction in all domains. 5.2.4. Test Case 4, bi-axial failure envelope for [90/ + 30/–30]n E-glass/L Y556 GFRP tubes subjected to combined pressure with axial load Figure 10B shows a comparison between the experimental results and the modified theoretical predictions for Test Case 4. Two curves are shown for the predicted results; one curve for the case of no wedge failure and the other for the case of a wedge failure. The dotted line would be valid if the wedge failure was not critical, which, the author believes, is not the case for this lay-up. The discrepancies between the predicted and measured results are essentially located in the negative quadrant. The few tests carried out (third quadrant) under external pressure and axial compression are reported [4] to be governed by buckling. As mentioned in Section 5.2.3, the compression strengths R¯ c储 as well as R¯ c⊥ seem to be too low. The ‘improved’ Part B computation did not reduce the discrepancies. Final failure, for the case of no wedge

1005 failure, has been essentially shifted (dotted line) in the third quadrant but this is not of importance. However, there might be some explanations for the discrepancies: 1. Due to the non-quasi-isotropy the achieved maximum load on the negative hoop axis (ˆ y) is about 50% lower than that on the negative ˆ x-axis, because the pure axial (x) loading of the cylinder is less buckling-critical according to the higher stiffness in axial direction than external pressure combined with axial tension load (failure caused by F ⊥90°). Minor influencing factors are: The external 90° layer (ˆ cy, pext, Fax) becomes stabilized by biaxial lateral compression (c2, c3 = –pext). The filaments of the internal 90° layer are a little stabilized by the curvature of the shell. 2. An increase of R¯ c储 would help to fit the test data on the ˆ x-axis. The author believes coupon buckling led to the low value of R¯ c储 = 570 MPa. Its effect on decreasing failure stress is the same in Fig. 10B (F ⊥ 90°) as in 9B (F ⊥ 30°, F ⊥ 90°). 3. An increase of R¯ c⊥ = 114 to 138 MPa (see TC 5) does the same in the fourth quadrant as may be seen from the dashed line in Fig. 10B. 4. A non-linear buckling analysis which considers the actual imperfections could help in reducing the discrepancy between theory and experiments, because the failure criticality of all affected modes is dependent on the imperfection geometry. With regard to the sharp corners of the failure envelope, the author advocates rounding these corners by the use of joint failure probability method. This method is not currently available due to computational difficulties.

5.3. Stress–strain curves of the laminates For the 45° tubes, bulging is reported in Ref. [4] and the same can be assumed for the 55° tubes. When strain gauges are attached to the surface of the specimens, they provide a measurement to the deformed shape. Hence, if bulging takes place, then the strains measured will be those of the specimens with an enlarged diameter. The difference between the nonlinear plane CLT prediction and test data can be reduced in some cases if the measured data is properly represented, e.g. the actual change in geometry is represented and the computed stresses are based on the actual dimensions, see also Table 1. An attempt is made in the present paper to consider some of these aspects. Table 1 Different meanings of theoretical and experimental data.

ˆ x, ˆ y ˆ x, ˆ y

Theory CLT (plane)

Experiment tube effect

Actual laminate, mean stresses Large strains No tube effect→no large deform. No tube effect→bulging missing –

Basis: small strains Large strains Tube effect→large deformation Tube effect→bulging included Creep??

1006 5.3.1. Test Case 10: [55/–55/55/–55] E-glass/MY750 GFRP tubes under hoop loading of tube ˆ y : ˆ x = 1 : 0 Details of the experimental results are presented in Ref. [4]. Figure 13 shows a comparison between the predicted and measured stress–strain curves. The predicted curve shows stiffer characteristics that the measured one in the hoop direction. The author believes that this is due to neglecting the bulging in the measured data. In other words, the measured hoop stress should have been computed from ˆ hoop = p(rint + r)/t rather than using Eq. (29) The theory was unable to predict the final strength of the laminates. The first IFF predicted by the theory is caused by shear fracture F⊥储 (an initial failure) and then, in the wedge failure mode F ⊥ followed by limit of usage (l.o.u.). The predicted final point seems to coincide with the stress at leakage. The improvement made in this paper in relation to that of Part A came as a result of allowing the computations to carry on to cover the softening regime in the lamina stress–strain curves. In Part A, the limit of usage l.o.u. value = 4% was applied. This corresponds to a final failure of the laminate at a hoop strain of around  = 2%, where the computation stopped as a result of a numerical instability. One idea to improve the prediction was to more correctly respect in computation the actual fibre angle that increases with increasing deformation. This has been done but not sufficiently well. Some other aspects are: (1) An improved large deformation/large strain correction (including > 10%) will follow the test curve longer. (2) In this application F ⊥ is not a ‘catastrophic’ failure mode, because the wedge cannot harm any existing load-carrying hoop layer Iying above. (3) Leakage cannot be predicted by the current theory.

Fig. 13. (B) with (A) (TC10) Stress–strain curves for ˆ y : ˆ x = 1 : 0 (radial loading by pint + axial compression load). Tube [ + 55/–55/55/–55]-laminate, E-glass/MY750 [8]; Test data [4]: T = –68°C. ¯ = (1280, 800, 40, 145, 73)T, max = 10%. ˆ y = hoop. b⊥储 = 0.13; >˙ m ˙ = 3.1, (newb⊥ = 1.09). Eq. (16). {R} Final Part A point 䊉.

1007 In order to better understand the failure behaviour a check utilizing the measured failure strains is performed: From measurement both strains on the external surface are known. They respect large strains (the real strains) and large deformation. The ‘test’ stresses given do not respect this [Eq. (29)]. Therefore, an attempt is made to close the gap between the non-linear plane CLT prediction, the measured strain data, and the given stresses, which – due to this fact – are not consistent w.r.t. the analytical stresses. First the failure strains are taken from the test curve. Secondly, a short analysis delivers the change of the fibre direction by a geometrical estimation using the measured fracture strains  = 55°, ˆ fracture ˆ fracture = –0.11, hoop = 0.09 ⬅ 9%,  ax fracture fracture ˆ = 55 · /180 ⇒ ˆ final = arctan(tan ˆ · (1 + ˆ hoop )/(1 + ˆ ax ) = 1.05 and final = 60°. Then, the lamina strains and associated secant moduli are estimated from the (assumed) softening curve. Eventually, via CLT computation the associated lamina stresses are derived and assessed. From these analyses a surprising fact seems to have been detected: Failure strain t储 and stress t储 determined in the tube are much higher than fracture strain ¯ fracture and ‘weakest 储 t ¯ link strength’ R 储 from the coupon test! Reason: A tube in contrast is a relatively redundant structure. Fibres turn into the hoop direction. They seek the optimum orientation and take more and more load. The stresses ⊥ and ⊥储 from this analysis were practically zero, as supposed. Netting theory (matrix stiffness zero), as a further analysis, however does not help here because the laminate is not designed by it. These results have to be investigated. The simple analysis above indicates that test data evaluation is a challenge and has to be performed very carefully by us. After the occurrence of the wedge mode of failure the ‘wedge degradation’ goes on and on. The numerical analysis eventually was not capable of performing further stress redistribution during degradation. The analysis of the MATHCAD code used might be made more stable by a better non-linear code with an improved softening curve for ⊥储 and c⊥ based on an improved understanding. These softening curve will have a steeper final decay, which better allows for stress redistribution to the fibres. It is to be hoped that tests in Germany will provide us with softening curves of a lamina in laminate in the next future, driven by the works cited in Refs. [17,18]. 5.3.2. Test Case 11, Stress–strain curves for [55/–55/55/–55] E-glass/MY750 GFRP pressure vessel loading under ˆ y : ˆ x = 2 : 1 The measured fracture strength in Fig. 14 is about 25% lower than the predicted one. Even in this case of a well-designed laminate, non-linear behaviour exists above initial failure. The author believes that fibre splitting occurs in the laminate. The axial strain curve ˆ x(ˆ y) intersects the hoop curve ˆ y(ˆ y) leading to very high strains. From theory a conclusion can be drawn: The curves also intersect but after intersection become parallel, because, respecting stiffness of the matrix it follows ˆ y ≥ ˆ x. If not, netting theory is applicable (multiple matrix cracking plus netting theory-designed laminate) and causes ˆ y = ˆ x which means parallelism. Furthermore, in netting theory, which should be increasingly valid after IFF has occurred, the ratio of x /y ≈ 1 is predicted, but that is not found in the experimental results. The reason for the discrepancy in the final strength prediction is believed to be as follows: As uniform increase in diameter of the cylindrical specimen cannot explain the difference between the theoretical and experimental curves, the effect must have come from bulging. The axial strain ax (at the gauge position) curve is affected by the end constraints more than the

1008

Fig. 14. (B) with (A). (TC11) Stress–strain curves for ˆ y : ˆ x = 2 : 1 ( pint), [ + 55/–55/55/–55]-laminate. ¯ = (1280, 800, 40, E-glass/MY750 [8]. T = –68°C. Test data [8]. Corrected maximum test values. {R} 145, 73)T. Final Part A point 䊉 ˆ y = hoop.

hoop strain. The boundaries are so close to the locations of the strain gauges that the readings of the strain gauges are affected by the end conditions. Fracture by bulging-based bending seems to be the main reason for the difference. The CLT plate analysis cannot directly treat any ‘bulge effect’ of the cylinder that might have taken place and a finite element analysis may be able to take this into account. A simple analysis is carried out below to consider the effect of scissoring (change in winding angle) on the strength of the laminate. According to the ‘short analysis’ applied in the discussion of Fig. 13 the measured laminate fracture strains are taken, again. From these T {ˆfracture outer } = ( + 4.2%, + 2.4%, 0) the final angle at fracture final = 54.6° (confirms netting theory was used in the design) is determined by CLT via the initial angle  = 55°, ˆ = 55 · /180, and ˆ final = arctan(tan ˆ · (1 + ˆ fracture ˆ fracture ). Then the lamina strains {fracture} = hoop )/(1 +  ax (3.1, 3.6, –1/6) are computed, further the secant moduli (degrading, softening regime) and fracture fracture eventually, the stresses {ˆ fracture ]{ˆfracture outer } = [A outer } and the lamina stresses {outer } = –1 fracture t [T] {ˆ outer }. This delivers for the deformed laminate ˆ y = 936 MPa, 储 = 1395 MPa. Simple netting analysis yields ˆ y = 882 MPa, t储 = 1395 MPa. Both sets of stresses are not too far from one another. This forces the author to propose a correction. The revised data is filled into Fig. 14 and marked by a filled rhombus in the graph. This confirms that the real test stresses are higher for a distinct strain level than those given in the test graph. Each interpretation of test results is helpful if it improves understanding. Here, netting theory is applicable for the failure strain state and therefore, gives some insight about the stress state in this benign acting redundant laminate. Note: This mass optimised, ‘two fibre direction’ laminate shows high sensitivity to little changes of the fibre winding angle . That means small deviations in winding angle cause large changes in stress–strain behaviour. At final fracture, netting theory-designed laminates can be simply examined by netting theory.

1009

Fig. 15. (B) with (A). (TC8) Stress–strain curves for ˆ y : ˆ x = 2 : 1 ( pint). [0/ + 45/–45/90]s-laminate. Tube ¯ = (1950, 1480, 48, 200, 79)T. test data [4]. AS4/3501-6 epoxy. T = –125°C. Final Part A point 䊉 {R} No curve parameters active.

5.3.3. Stress–strain curves for [0/ + 45/–45/90]s AS4/3501-6 CFRP pressure vessel loading under ˆ y : ˆ x = 2 : 1 (Test Case 8) Figure 15 shows a fibre-dominated behaviour, indicated by the almost straight curve up to fracture. The fracture load was predicted by about 10% higher than the measured value. The predicted value in the revised curves is also higher than that reported in Part A [1]. The reason for this is the improvement in the computer code. The correlation between the measured and predicted data is good for this type of laminates. 5.3.4. Stress–strain curves for {45/–45/45/–45] E-glass/MY750 GFRP lined tubes under biaxial stress (stress ratio ˆ y : ˆ x = 1 : –1) (Test Case 14) In this test case shear loading was generated by internal pressure and over-compressed in the axial direction. The tube had an internal radius rint = 50 mm and a wall thickness t = 5.9 mm. Thick cylinder theory was used in evaluation of the test results. Strains were measured at the inside and outside surface of the tube. However, the stresses, obtained from thick cylinder theory, are given for the inside surface. Bulging was reported. Figure 16B with A indicates a numerical instability beyond reaching failure F ⊥ of this maximally sheared laminate, after initially being too stiff in comparison to the test. Beyond this, a relatively poor mapping of the test data is recognized above 60% of the fracture load. According to Part B information and to the fact that the mutually strain-controlling layers are redundant the author increased the Part A value of 4% to the mentioned 10%. Following the plane CLT analysis the comparison of predictions and experimental results for this highly matrix-dominated laminate is a little disappointing because the analysis stops before reaching the fracture strain domain.

1010

Fig. 16. (B) with (A). (TC14) Stress–strain curves for ˆ y : ˆ x = 1 : –1 (shear by pint + axial compression), [ + 45/–45/45/–45]-laminate. E-glass/MY750 [8]. T = –68°C. Tube test data [4]. Bulging reported in experiment. ˆ y = hoop. Final Part A point 䊉.

Some reasoning is provided including the effect of the large deformations. The simple final angle estimation yields ˆ = 45°,

 = 45 · /180,

ˆ fracture ˆ fracture = –0.11 and final = 51°. hoop = 0.10, ax

This helps again to get information for the interpretation of the test data. The increasing angle stabilizes the laminate structure as far as the in this case not that catastrophically acting – wedge failure tolerates it. Regarding large strain/large deformation together with a correct softening curve the test curve will become stiffer in the upper region. Again it is obvious that bulging must have occurred. The author strongly supports the idea of carrying out more tests to clarify remaining problems and also executing a more thorough analysis which takes into account the following additional issues: stiffness reduction, crack density formation, softening curve and fibre failure. The utilized softening curve does not allow for further stress-redistribution to the fibres that seem, due to redundancy effects, still to act above the ‘weakest link value’ R¯ t储! 5.3.5. Stress–strain curves for [0/90]s E-glass/MY750 GFRP coupon test uniaxial stress (stress ratio = 0 : 1) (Test Case 12) In Fig. 17 the predicted and measured curves of a cross-ply coupon specimen are depicted. The measured curves show some higher stiffness. The correlation between the measured and predicted final strength is good, with the predicted value 10% higher than the measured one. The edge effect as well as the double centre layer is governed by the fracture toughness that affects the initial failure prediction, and the onset of the corresponding IFF modes. The final failure (fracture) is controlled principally by the fibre strength (mode FF1) and hence there is

1011

Fig. 17. (B) with (A). (TC12) Stress–strain curves for ˆ y : ˆ x = 0 : 1, (axial tension). [0/90]s-laminate. Coupon. E-glass/MY750 [8]. T = 68°C. Test data [4]. Final Part A point 䊉.

a lack of information on how the matrix cracking density could affect the FF1 mode of failure. Due to the error in stiffness addition – unfortunately in the Part A figure for ˆ y the stiffness loss by the IFF degradation was not ‘subtracted’ but added and this led to increasing stiffness – the initial failure stresses in Part A were a little different from those in Fig. 17. 5.3.6. Stress–strain curves for {45/–45/45/–45] GFRP E/MY750 tubes under biaxial stress (stress ratio ˆ y : ˆ x = 1 : 1) (Test Case 13) The experimental results were described in Ref. [4]. Among the main features of these results are the following: (a) In the derivation of the experimental results no allowance was made for large deformation. (b) The readings from the individual strain gauges varied up to 22%. Reasons for the divergence of the hoop and the axial strain for this symmetric lay-up are not clear. (c) The strength of the tubes – determined from several tests was 502 MPa, higher than shown in the experimental results, graph (Fig. 18). (d) Extensive data on crack spacing was recorded and bulging was reported. The results for this test case ‘tube subjected to internal pressure with axial tension’ showed that the predicted fracture strength was larger than the measured. Also, the observed shape of the stress–strain curve was not fully captured by the model. It is expected that bulging together with the relatively large deformation, will cause this difference. Possible explanations for some of the discrepancies are presented. The curves should lie on top of one another due to the symmetric geometry and loading. Different curves for ˆ x and ˆ y indicate that bulging took place during loading. Netting theory can be applied to assess the test data for this well-designed laminate. From the measured strains fracture = 2 18%, fracture = 2.48% and assuming ⊥ = ⊥储 (due to zero matrix stiffness), the x y following values are derived: ˆ x = ˆ y = 530 MPa and t储 = 1062 MPa. And from this, a reduction of the maximum theoretical fracture ‘load’ according to the fracture strength values is

1012

Fig. 18. (B) with (A). (TC13) Stress–strain curves for ˆ y : ˆ x = 1 : 1 ( pint + axial tension) [ + 45/–45/45/–45]-laminate. E-glass/MY750 [8]. T = –68°C. Tube test data [4]. Bulging reported in ¯ = (1280, experiment. 䉬 maximum test value after two corrections. ˆ y = hoop. Final Part A point 䊉. {R} 800, 40, 145, 73)T.

derivable down to a value ˆ y = 547 MPa = 660x(1062/1280) MPa, with 1280 MPa the given tensile strength. This matches very well. The test results show a leakage at around 210 MPa stress [4]. This structural failure mode cannot be predicted, because leakage is determined by the stochastic IFF-based micro-crack system of the laminate shell. 5.3.7. Stress–strain curves for [0/ + 45/–45/90]s AS4/3501-6 CFRP tubes under uni-axial stress (stress ratio ˆ y : ˆ x = 1 : 0) (Test Case 7) For this radially loaded tube, subjected to hoop tension induced by internal pressure plus axial compression, Fig. 11B shows fibre-dominated behaviour with an almost straight curve up to fracture with a fracture load predicted less than 10% too high. The predicted stiffness was lower than the measured one. The result is satisfactory. Again, the initial failure stresses are a little different to those in Part A due to the corrected stiffness error.

5.4. Application to stress–strain curves of distinct laminates As highlighted in Section 2.5.1 above and in Ref. [4], there are differences between the behaviour of tubular specimens and coupon specimens. Below are some other issues related to this subject.

1013 5.4.1. The symmetric laminate GFRP E/MY750 [ + 45/–45/–45/ + 45], (ˆ y : ˆ x) = (1 : 1) In comparison (see Figs 18a, 20a) to the following anti-symmetric case the inner lamina consists of two equal layers. This has an effect on micro-cracking which eventually has a deteriorating impact on FF as still mentioned in Sub-section 2.3.2. The applied theory cannot predict any difference between the magnitude of the failure stresses for this and the following case. Ply-by-ply FEA is necessary. 5.4.2. The anti-symmetric laminate [ + 45/–45/ + 45/–45], (ˆ y : ˆ x) = (1 : 1) Due to the sequence of winding angles some twisting is to be expected for a flat specimen composed of this stack (VDI 2014 guideline). In case of winding a positive winding angle normally follows a negative one and vice versa, if the winding process is not interrupted and the winding direction back-changed (increases winding costs). Then the fibre directions alternate and an anti-symmetric stack is manufactured with often many lay-up repetitions. However, in contrast to a flat plate specimen (experiences twisting under in-plane normal loading) a circumferentially closed, wound tube specimen will just turn under any ‘internal state’ of residual stresses the front sections to another. Furthermore it should be mentioned that the wound anti-symmetric tubes in contrast to the symmetrically stacked ones have no double central layer thickness. A double layer has

Fig. 19. Schematic illustration of Cuntze’s assumptions about the stresses 2 and 21 before and after IFF-initiation. Over-pronounced results of the ’triggering approach’.

1014 a detrimental impact on onset of IFF fracture [10] but that fracture mechanics aspect is not predicted by the author’s current theory. 5.5. Application of refined theory to the 55°-tube test specimen It is worth mentioning that the organisers [8] provided a description of the geometries, lay-up, and loading configuration for a ± 55° tubular specimen and asked those participants who possess numerical methods to provide a solution for the state of stresses. This extra test case was aimed at investigating how boundary conditions could affect the response of such laminates. Currently, MAN Technologie uses MARC finite element code for carrying out

Fig. 20. (a) Yielding zone (shadowed) in the (21, 2) domain; (b) yielding zone in the (3, 2) domain (rounding-off intentionally not applied).

1015 detailed stress analysis of composite structures. It is possible to implement a suitable subroutine for performing the analysis for the extra case posed by the organisers [4]. The author believes that some important steps are first needed to complete the current non-linear analysis proposed in this work prior to the implementation of such analysis into a FE code.

6. Concluding comments and recommendations 6.1. Designer’s need for ‘design according to failure modes’ There are a number of fundamental issues worth highlighting here regarding failure and design with composites: • According to some standards, e.g. Ref. [19], in any static design, a designer has in general to dimension a laminate against two main types of failure, namely, inter-fibre-failure (IFF) of the laminae and fibre-failure (FF). Occasionally, it may be required to prove that IFF does not occur when a laminate is subjected to the so-called ‘Design Limit Load (DLL)’ level. In practice, this corresponds to the Yield Proof of Design for isotropic materials. This means for composites ‘No IFF below DYL = DLL* jyield.’ • For Ultimate Proof of Design, then the rule is this: ‘No FF below DUL = DLL* jult.’ Here, further analysis is needed beyond that required to detect IFF modes. An IFF mode normally indicates the onset of failure in a laminate whereas the appearance of an FF mode in a single lamina usually marks the laminate final failure. In the case of brittle FRP composites, failure coincides with fracture. Fracture is defined in this article as a separation of material. The material is assumed to be initially free of damage such as technical cracks (size in the order of a mm) and delaminations but not free of tiny defects/flaws (size in the order of microns) prior to loading. • For the ‘higher’ Proof of Design at Design Ultimate Load (DUL), in cases of well-designed laminates, initial failure prediction is not important and does not have a large impact. If the common design strain of  ≈ 0.4% is required as a damage tolerance considering limit value, then this value is dealt with within the DUL Proof of Design [19]. • The IFF modes incorporate cohesive fracture of the matrix and adhesive fracture of the fibre matrix interface. Both fracture types are often termed ‘matrix failure’. • Loading a composite by a 储 stress will always induce a matrix stress acting in the fibre direction. The matrix stress is normally obtained from use of micro-mechanics equations, see for instance the Appendix A in Ref. [7]. However, as long as the fracture strain of the matrix is much larger than that of the fibre (e.g. fibre 2%, matrix 6%) one may neglect matrix stresses in the fibre direction of a 0° layer because their magnitude does not practically affect the failure of the fibres. • The ‘explosive’ effect of the so-called wedge shape failure (see Fig. A1) of a lamina in a laminate is an IFF mode caused by high transverse stress c [1,6,21]. The occurrence of this mode may lead directly to final failure, as for example in the case of Puck’s torsion spring [6]. It can also lead indirectly to a final failure of a laminate via the development of through thickness stresses and, local delamination and hence to buckling of the adjacent laminae. This IFF, where parts of a lamina move in the thickness direction may also initiate a catastrophic failure like FF (Fig. 1).

1016 • As a general rule in designing composites, it is a good practice to design a laminate to be stable as a truss (netting theory applicable) and stack the laminae at angles to generate a laminate robust against possible load changes. This will lead to a so-called ‘well-designed’ laminate. Hence, in composite structures composed of stiff fibres and well-designed, using netting theory, the fibre net will control the strain behaviour.

6.2. Comments on the FMC-based conditions The fracture conditions are now robust for automatic use in design after the F⊥储 convergence problem has been solved by a query. The use of Eqs. (30) will avoid this problem. The UD fracture conditions employed in the theory are proven to work in certain areas. However, multi-axial test data required for a full verification of the shape of the IFF failure body are still missing. It is to be hoped that the two dissertations of Knops [17] and Fischer [18] will close some of the remaining gaps. The complete failure surface consists of piecewise smooth regimes (partial failure surfaces). Each regime represents one failure mode and is governed by one single basic strength. The interaction (rounding-off) between adjacent failure modes is automatically considered when calculating the ‘resultant stress effort’ Eff (res) as a function of the ‘mode stress effort’ Eff (modes). The interaction coefficient m ˙ can be given a convenient conservative value. As suitable low value for m ˙ an odd number 3.1 is taken, for numerical reasons. The ‘mode fit’ avoids the shortcomings [20–33] of the ‘global fit’ which is normally used to fit the shape of test data covering the whole biaxial failure envelope by a single mathematical function (curve), regardless of the actual modes of failure. One typical shortcoming of the ‘global fit’ approach is that a reduction in one of the uniaxial strength values could increase the multi-axial strength in physically unrelated quadrant of the failure envelope. Homogenisation of the UD-material comes to its limit if a constituent stress governs the failure. This is the case for F 储 , where the macro-mechanical stress 1 has to be replaced by the actual fibre stress 1f . A fibre stress may be zero but the stress in the lamina along the fibre direction, 1, may still have a value. Therefore, 1f has to be estimated as 1f = 1 E1f . For the prediction of the level of final failure of the laminate, the initial failure (IFF of laminae) stresses may not be crucial, especially if wedge failure (caused by F 储 < 1), followed by delamination, does not occur or has no detrimental effect on load carrying capacity. Wedge failure was not regarded as catastrophic for the [ + 55/–55/55/–55] tube under, (ˆ hoop /ˆ ax = 1/0) loading. However, a catastrophic ‘wedging off’ behaviour could take place, if an extra hoop layer was wound onto the tube. In cases where either the scatter in the test data is not known or the test data are unavailable, one may resort to the use of the simplest set of parameters, given as b⊥ = 1, b⊥储 = 0. This will usually give a good approximation of final failure prediction of a laminate. For these parameters, the following set of failure conditions, derived from Eq. (16), is recommended as an engineering approach FF1:

F 储 =

I1 = 1*, * with I1 = 1 · E储 R¯ t储

1017 –I1 = 1, R¯ c储

FF2:

F 储 =

IFF1:

F ⊥ =

IFF2:

¯3 F⊥储 = I 3/2 3 /R ⊥储 = 1

IFF3:

F ⊥ =

I2 + 兹I4 =1 2R¯ t⊥

(30)

兹I4 = 1. R¯ c⊥

It should be noted here that the behaviour of a unidirectional lamina under biaxialcompression, in a plane perpendicular to the fibre direction, is important for designing structures such as submarine hulls [30]. Test data reported in Refs. [7,21] have shown an interesting trend of substantial enhancement in the biaxial strength under combined 3 and 2 stresses and this trend will determine the values of the curve fitting parameter b⊥ used for such applications.

6.3. Comments on the progressive failure analysis • An accurate failure prediction involves the application of (a) a physically-sound non-linear stress analysis to cover large shear strains, (b) a geometrically non-linear analysis to account for large deformation, and (c) a better understanding of the post failure behaviour via establishing the softening curves. • The lamina is the basic building brick (or basic computational element) for the prediction of laminate behaviour. The load, not the stress, was increased monotonically from zero to fracture. • Triggering loses its influence when approaching the large strain regime beyond the occurrence of an IFF mode. • The high internal redundancy of a multi-layered laminate is better modelled by a probabilistic code. The laminate acts as a so-called failure system of the stochastically failing laminae. Probabilistic tools should be applied, as has been done by the author [22,14], in order to improve the deterministic procedures and also to smooth (due to joint failure probability) the sharp corners of a laminate’s failure envelope, which now represents the ‘sum’ of all single lamina failure envelopes, only. Such a non-deterministic approach [14,16,22] should be employed to support the understanding of the subsequent failure behaviour, see also Ref. [23].

6.4. Industrial needs and future steps a. Industry seeks to replace the expensive ‘Make and Test’ design method by verified and benchmarked predictive tools that engineers could use with confidence. In other words, a practical ‘progressive failure analysis procedure’ has to be provided to designers. However, the limitations of the predictive tools ‘failure conditions’ should be clearly indicated. To this end, the organisers of the failure exercise (Hinton, Kadour and Soden) have provided an assessment of the maturity of the current methods, Refs [24,25].

1018 b. The 2D/3D-strength analysis, using the fracture conditions defined in this paper, is not yet fully validated/verified. Similarly, 3D-stress analysis of laminated shells obtained by commercial FEA codes (MARC, NASTRAN etc.) is not adequate, it is still too timeconsuming and pre-processing does not provide sufficient information regarding the design driving modes [26] as well as the reserve factors [26] demanded (see Fig. A1-A3, Ref [1]) for the ‘Proof of Design’. Hence, further improvement is needed. c. Verification of engineering approaches, based on qualified FEA-output level when analysing the test specimen, is necessary. d. Regarding the theoretical and experimental investigations carried out in Germany on the lamina material level in recent years [17,32,33] (and still going on) the understanding has improved greatly and seems to be a good basis to tackle laminates made up of UDlaminae or fabric laminae. For other ’textile pre-forms’ (3D, stitched etc.) engineering models have to be developed. The work in this field has been initiated. First steps indicate that the transferability to fabrics (rhombically orthotropic) composites should work [27]. e. Industry also has to cope with damage and the Proof of Design (justification) of damaged structures or laminates. Practical criteria for the assessment of damage size and criticality of delamination are needed. Sufficiently well working NDI methods for damage detection are desired in order to avoid unstable (sudden) delamination fracture. A design guideline for improving damage tolerance analysis has to be provided. The treatment of fatigue and stable damage growth has to be enhanced and may be better enhanced on failure mode basis. f. There is still a need for generating reliable multi-axial test data (not all section planes of the multiaxial fracture body are verified as one may conclude from Figs 5–7). This could be achieved through a coordinated and collaborative research programme between leading research institutions. g. Other areas include the development of probabilistic models and the encouragement of an improved worldwide standardization where manufacturers, technical associations and authorities are all involved. h. As the area ‘in situ behaviour of the embedded laminae’ has not attracted much attention, further work is highly recommended. i. For stress concentrations in the laminate such as bolt holes a suitable engineering approach for the strength assessment has to be provided and methods used for metals could be examined for ideas. This area lies outside the current failure exercise. j. In the case of stress intensity (where delamination has occurred) a practicable fracture mechanics tool has to be established to assess unstable delamination growth. Appropriate test specimens and test evaluation have to be discussed. k. Loading path effects on IFF have not been considered in the present work and require a thorough investigation. Deterministic failure path and probabilistic failure path of a laminate may not coincide with each other due to the possibility of having different scatter ranges of the design parameters [13,22]. l. A prediction of leakage as a mode of failure is not currently possible as one may need to calibrate the IFF behaviour by carrying out a sufficient number of tests with the aim of measuring critical values for micro-crack density and, as a characteristic quantity, a limiting strain. It depends on the grade of leak tightness that is required in the actual case. Design to avoid leakage covers a very wide range. Designing vessels and containers for cryogenic fluids like LH2, LOx and gases like Helium requires the use of liners as a

1019 barrier between the gas and the inner surface of the containers. In case of fluids, such as water, a rubber-sealing layer suffices if the pressure is not too high. Further efforts are needed to investigate the occurrence of this failure mode in the actual pressure vessels. m. An-isotropic yielding (see Fig. 20a,b) needs to be studied in order to get a better understanding of the behaviour of composite beyond the onset of yielding. The benefit may be more academic, but it could shed some light on load path dependencies in static and fatigue loading. n. Determination of the curing stresses (first kind) requires information on thermal shrinking (CTE and temperature difference) as well as on chemical shrinking of the matrix after the gel state in the solid state (data were not provided [1,4,8] in the original data-pack and the author did not request such data). The author recommends further work on dealing with residual thermal stresses, because there is lack of understanding of their magnitude and their possible decay with time. In addition, curing stresses of the second kind determines the UD strength but has not been evaluated here. This non-consideration implies that within the transfer of UD data to the structure it is assumed that the curing stresses of the second kind are approximately the same in lamina specimens and in laminated structures. 6.5. Comparison between theory and experiments 1. Two categories of laminate configurations have been investigated: (1) laminates made of plies where the fibres are oriented in three or four directions, and (2) laminates containing plies oriented in two directions. Correlation between theoretical prediction and experiment is expected to be the more imperfect the less the strength of a laminate can be predicted accurately by netting analysis. 2. The predicted initial and final failure envelopes did not match the test results in a number of instances. This can be attributed to (a) the current capability of the theory used, (b) the accuracy of the assumptions made in the interpretation of the test data, and (c) lack of test data and the accuracy of the data input provided. 3. Although a few examples were given in this paper on how to deal with a 3D state of stress, the full potential of the FMC-based IFF conditions has not been fully explored. 4. The comparisons between theoretical predictions and the experimental data helped to identify certain areas where further theoretical and experimental work is required for a final verification of theory. 5. Discrepancy between the predicted and measured results was observed to be high in areas where the high shear strains were encountered. This is because the computer software code developed in the present work for non-linear analysis requires further work to eliminate any convergence problems. However, high shear strains do not occur in well-designed laminates.

Acknowledgements The author gratefully thanks Mr Andreas Freund (now with ATENA) for performing all nonlinear analyses and for valuable discussions. He is also indebted to all reviewers and the organisers of the failure exercise.

1020 References [1] Cuntze RG, Freund A. The predictive capability of failure mode concept-based strength criteria for multidirectional laminates. Composite Science and Technology [in press]. [2] Thoft-Christensen P, Murotso Y. Application of structural systems reliability theory. Berlin: Springer; 1986. [3] Parvisi A, Bailey JE. On multiple transverse cracking in glass fibre epoxy cross-ply laminates. Journal of Material Science 1978;V13:2131–6. [4] Soden PD, Hinton MJ, Kaddour AS. Experimental failure stresses and deformations for a range of composite laminates subjected to uniaxial and biaxial loads: failure exercise benchmark data. Composites Science and Technology 2002;62:1489–514. [5] Hinton MJ, Kaddour AS, Soden PD. A comparison of the predictive capabilities of current failure theories for composite laminated, judged against experimental evidence. Composite Science and Technology 2002;62:1725–97. [6] Puck A, Schurmann H. Failure analysis of FRP laminates by means of physically based phenomenological models. Composite Science and Technology 1998;58. [7] Cuntze RG, Deska R, Szelinski B, Jeltsch-Fricker R, Meckbach S, Huybrechts D, et al. Neue Bruchkritenen und Festigkeitsnachweise fur unidirektionalen Faserkunststoffverbund unter mehrachsiger Beanspruchung Modellbildung und Experimente. VDI-Fortschritibericht, Reihe 5, Nr. 506, 1997. [New fracture criteria (Puck’s criteria) and strength ‘proof of design’ for unidirectional FRPs subjected to multi-axial states of stress – model development and experiments] [in German]. [8] Soden PD, Hinton MJ, Kaddour AS. Lamina properties, lay-up configurations and loading conditions for a range of fibre-reinforced composite laminates. Composites Science and Technology 1998;58:1011–22. [9] Schuelein, R. et al.: Verbesserung der Ausnutzbarkeit der neuen hochfesten C-Fasern in hochbelasteten Verbundstrukturen und Verbesserung des Dimensionierungsverfahrens. VDIFortschrittBericht, Reihe 5, Nr 186. VDI Verlag; 1989. [10] Peters P. The strength distribution of 90° plies in (0,90) graphite-epoxy laminates. J Comp Mat 1984;18:545–56. [11] Beltrami E. Sulle Condizioni di Resistenza dei Corpi Elastici. Rend ist d sci lett, Cl mat nat 1885;18:705–14. [12] Lekhnitskii SG. Theory of Elasticity of an Anisotropic Elastic Body. San Francisco: Holden Day Inc; 1963. [13] Choo VKS. Effect of loading path on the failure of fibre reinforced composite tubes. J Comp Mat 1985;19:525–32. [14] Cuntze RG. Deterministic and probabilistic prediction of the distribution of inter-fibre failure test data of pre-strained CFRP tubes composed of thin layers and loaded by radial pressure. Wollongong. Advanced Composites ’93. The Minerals, Metals & Materials Society, 1993. pp. 579–85. [15] Cuntze R, Heibel R. Increasing the limit of usability of CFRP tubes by built-in stresses. X. Int. Herman F. Mark-Symposium, Vienna, October; 1991. [16] Rackwitz R, Cuntze RG. System reliability aspects in composite structures. Eng Opt 1987;11: 69–76. [17] Knops, M. Sukzessives Bruchgeschehen in Faserverbundlaminaten (successive fracture behaviour of FRP laminates). Dissertation, Rheinisch-Westfalische Technische Hochschule Aachen, 2003. [18] Fischer, O. Zum Faserbruchgeschehen in Einzelschichten aus unidirektional kohlenstofffaserverstarktem Kunststoff (on the fracture behaviour in UD CFRP laminae). Dissertation, Rheinisch-Westfalische Technische Hochschule Aachen, 2003. [19] DORNIER Luftfahrt GmbH. Guidelines 728 JET: Structural analysis and certification documentation.

1021 [20] Tsai SW, Wu EM. A General theory of strength for an-isotropic materials. Journal of Comparative Materials 1971;5:58–80. [21] Collings TA. Transverse compressive behaviour of unidirectional carbon fibre. Reinforced Plastics Composites 1974:108–16. [22] Cuntze RG. Failure path analysis of multi-layered fibre reinforced plastic components with the reliability calculation programme FRPREL. ESA/ESTEC Conference Noordwijk, October, 1987. [23] Chamis C. Probabilistic composite design. ASTM Special Technical Publication 1997; 1242:23–42. [24] Kaddour AS, Hinton MJ, Soden PD. A comparison of predictive capabilities of failure theories for composite laminates: Additional contribution Composites Science and Technology, V64, pp 449–477, 2004, see also Chapter 4 in this book. [25] Hinton MJ, Kaddour AS, Soden PD. A further assessment of the predictive capabilities of current failure theories for composite laminates: comparison with experimental evidence, V64, pp. 549–588, 2004, see also Chapter 6 in this book. [26] Cuntze RG, Sukarie G. Effective dimensioning of 3D-stressed UD-laminae on basis of fracturetype strength criteria. Int Conf on Mechanics of Composite Materials. Riga, 1998 [Conference handbook presentation]. [27] Cuntze RG. Strength prediction for multi-axially loaded CMC-materials. In: 3rd European workshop on thermal protection systems. [ESA-ESTEC], Noordwijk, March 1998 [WP P141]. [28] VDI 2014: German Guideline, sheet 3, Development of FRP components analysis. (2003) [in German and English]. [29] Betten J. Recent advances in mathematical modelling of material behaviour. Math Comput Modelling 1990;14:37–51. [30] Rajapakse YPS. Mechanics of thick composites. New York: YDS [ASME-AMD]; 1993. [31] Hashin Z. Failure criteria for unidirectional fibre composites. J Appl Mech 1980;47:329–34. [32] Cuntze RG. Evaluation of multi-axial test data of UD-laminae by so-called “fracture type strength criteria” and by supporting probabilistic means. Gold Coast, Australia: [ICCM-11]; 1997. [33] Michaeli W, Knops M. Stress and strength analysis of structural components with inter fiber failure. Experimental and theoretical work. Proceedings SAMPE, Long Beach, 12–16 May, 2002.

Appendix A. Example for the FMC procedure with visualisation Fig. A1 refers to the (2, 3)-plane as one of the various failure planes. In the upper part it visualizes the evaluation of test data and in the bottom part the rounding-off (by the spring model) in the multi-fold (MfFD) and mixed failure domains (MiFD) as well as the shrunk design space (mean strength R¯ of mapping is replaced by a strength design allowable R) to be used by the designer in ‘dimensioning’ and in the ‘proof of design.’ In addition to the FMC-based ‘Mode Fit’ the ‘Global Fit’ (e.g. Tsai/Wu’s ‘single failure surface’ criterion [20] describes a global failure surface) is pointed out. The Global Fit interacts between the UD-stresses and the independent failure modes in one equation, achieving a description of the global (complete) failure surface. This procedure is simple, however error-prone in some domains, due to its physical shortcomings.

1022

Fig. A1. 2D case {} = (0, 2, 3, 0, 0, 0)T. MiFD = Mixed Failure Domain, MfFD = Multi-fold Failure Domain. Both MiFD and MfFD rounding-off is considered in the lower figure.

1023 1. Failure conditions F ⊥ =

I2 + 兹I4 =1 2R¯ t⊥

F ⊥ = (b⊥  1) I2 = 2 + 3; b⊥ =

IFF1 and IFF3

I2 兹I4 + b⊥ c = 1 R¯ c⊥ R¯ ⊥

with

I4 = (2  3)2 + 4223;

c ¯ c 1 + (c 2 + 3 )/R ⊥ = 1.09 c ¯ c c c 2 ¯ c (c 2 + 3 )/R ⊥ + 兹(2  3 ) /R ⊥

c from calibration point 䊐 (see figure) (c 2 , 3 ) = (–40 MPa, –191 MPa) and mean values ¯R t⊥ = 40 MPa, R¯ c⊥ = 144 MPa. Test data for this quasi-isotropic plane (2, 3) are assumed from isotropic knowledge. Note: Inherent to the FMC is a ‘Mode Fit’. This fit needs less data then the usually applied ‘Global Fit’ (such as with Tsai/Wu) and maps the MfFD, additionally! Appropriate fictitious data set. b⊥ 1 means angle 45° (45° = zero friction).

2. Rounding-off in MiFD, MfFD

冉 冊冉 冊冉冊冉 冊 1

f (res) Res

m ˙

=

1

f ⊥2 Res

m ˙

+

1

f ⊥ Res

m ˙

+

1

m ˙

f ⊥3

Means R¯ t⊥, R¯ c⊥⇒Design Allowables R t⊥, R c⊥, b⊥ remains because it is a physical property. t t c The mode reserve factors read (2D case) f ⊥2 f ⊥3 f ⊥ Res = R ⊥ /2, Res = R ⊥ /3, Res = R ⊥ /  (2b⊥2  2  3). • In the frame of an automatic numerical process a negative becoming f mode Res is set 10 • Automatic rounding without affecting adjacent modes.

Appendix B. Determination of max I 3/2 3 (Query F⊥储 for the case discrimination) in Eq. (24d). Then, A numerical problem exists with F⊥储 if b⊥储(I2I3  I5) becomes – I 3/2 3 visualized in the 2D space by the 21(2)-curve in Fig. B1, this curve turns asymptotically and an intersection with a proportional stress beam from the origin to determine f ⊥储 Res not achieved any more. The reserve factor becomes indefinite and the effort imaginary. To generally bypass this difficulty one has to put a query in the program and replace, if applicable, the formulation of the asymptotically becoming curve by a limiting value max I 3/2 3 . This is a ‘horizontal’ line in the 2D case of Fig. B1. Hence, the limit for the applicability of F⊥储 for the given 3D state of stress (marked by a tilde sign) is I 3/2 3 /(I2I3  I5) = –b⊥储. In order to have a safe distance the parameter b⊥储 is increased by a very small factor . Setting into the failure condition F⊥储 = 1,

1024

Fig. B1. Illustration of the cut-off in order to guarantee an intersection and to avoid imaginary or indefinite numbers. Eqs. (1), (16), IFF2.  = 1.01 is good guess [in case of Eq. (30d) no inter-section problem exists anymore].

3/2 ˜˜ ˜ Eq. (16), the new ratio maxI 3/2 3 /(I2I3  I5) = – · b⊥储 it can be deduced from maxI 3 + 3/2 3 b⊥储maxI 3 /(–b⊥储) = R¯ ⊥储 a bound

maxI3 =

冋 册 冋 册

maxtau =

 1

2/3

 1

· R¯ 2⊥储 with

 = 1.01

...

3D

case

(B1)

1/3

· R¯ ⊥储

maxI3 = maxtau2

...

2D

case

(B2)

In the 2D case the procedure can be visualized (see Fig. B1) by viewing at some distinct angles in the (21, 2) graph. These are, employing the limiting beam (21/–2) = 2 · b⊥储, the radiant cot)max = (21/–2) and the angle )°max = arcot(2b⊥储) · 180/ . To safely remain on the ‘intersection side’ a reduction of the angle )°max is introduced by the factor  by setting )° = arcot(2 · b⊥储) · 180/ . Hence, the sub-case of maxI3 is R¯ ⊥储 maxtau = 3 兹1  (2b⊥储 /cot ))

with cot) = 2b⊥储. And the reserve factors become f ⊥ Res = For the stress effort it is analogous.

R¯ c⊥ , –2

f ⊥储 Res =

maxtau . | 21 |

1025 Appendix C Visualisation of hardening and softening Table C1 Definition of the various lamina types. I: Tensile coupon, isolated lamina, load-controlled → weakest link type test results II: Tensile coupon, isolated lamina, strain-controlled (stiff test frame) III: Embedded (constraint) lamina, strain-controlled → redundant type test results

Fig. C1. Measured shear stress shear–strain curves. Mapping by Eqs. (19,22a). Strain-controlled demands for stiff test frame. (Upper bars  indicate mean or typical values are skipped in the windows here.)

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

1026

CHAPTER 5.14

A comparison of multicontinuum theory based failure simulation with experimental results J. Steven Mayesa,* and Andrew C. Hansenb b

a Mechanical Engineering Division, Alfred University, Alfred, NY 14802, USA Department of Mechanical Engineering, University of Wyoming, Laramie, WY 82071, USA

Abstract This paper is part of a broader exercise to determine the predictive capability of several currently accepted composite failure criteria. Blind failure and stress–strain predictions were originally submitted by us to the organizers of the exercise for 14 different cases involving four different composite materials. A finite element based analysis was used to simulate progressive failure of the various composite laminates under uniaxial and multiaxial loading. Under the assumption of the composite acting as a multicontinuum, fiber and matrix constituent stress and strain states were derived. This information formed the input to a quadratic, stress-interactive constituent-based failure criteria to predict damage at the composite level. Two-dimensional failure envelopes and stress–strain curves for the composite were developed. In what follows, the experimental results have been made available to us for comparison against our analytical failure simulations. Generally, these analytical predictions were in good agreement with experimental results. The presence of simultaneous normal and shear stresses in a composite laminate was found to significantly enhance shear strength which was not accounted for in the proposed failure criteria. Keywords. B. Microstructure; B. Strength; C. Damage mechanics; C. Failure criterion; C. Finite element analysis (FEA)

Nomenclature ± Ii Ii# ± Ki# ± Sij# ± kl Sij#

Indicates the appropriate tensile or compressive value is used depending on the constituent’s stress state. Composite transversely isotropic stress invariants (i = 1–5). Constituent # transversely isotropic stress invariants (i = 1–5). Constituent # failure parameter (i = 1–4). Constituent # strength in the i, j direction (i, j = 1–3). Constituent # stress in the i, j direction when stress ± Skl is applied (i, j = 1–3);

* Corresponding author.

1027 # ij ij#

(k, l = 1–3). Composite constituent (# = f (fiber), m (matrix)). Composite stresses referenced to the local lamina (i, j = 1–3) or global laminate (i, j = x–z) coordinate system. Constituent # stresses referenced to the local lamina coordinate system (i, j = 1–3).

1. Introduction Ultimately, the goal of a failure prediction methodology is to extrapolate results of a relatively easy-to-conduct laboratory experimental program, utilizing simple test specimen geometry and uniaxial loading, to real-world applications with complex structural geometry and multiaxial loading. Typical failure criteria are semi-empirical, phenomenological models attempting to describe experimental observations. Generally, only the occurrence of failure is predicted, not all possible modes of failure. For instance, in the case of continuous fiber composite materials, failure details such as fiber pullout, fiber breakage, fiber micro-buckling, matrix cracking, delamination, etc., are not discerned. In addition to multiple failure modes, failure of a composite material is further complicated by the interplay of load transfer between the constituents. The approach to composite failure analysis presented herein utilizes a multicontinuum theory (MCT) to illuminate this interplay by decomposing the composite stress and strain fields into those of its constituents. Hashin [1] states that most failure criteria that account for stress interaction are quadratic in stress and that choice of a quadratic polynomial is based solely on curve fitting considerations. A quadratic approximation, he notes, “is the simplest presentation which can fit the data reasonably well and in view of the significant scatter of failure test data it hardly seems worthwhile to employ cubic or higher approximations.” MCT uses a quadratic, stressinteractive failure criteria. However, in a major departure from traditional failure analyses, MCT applies the failure criteria at the constituent rather than the composite scale. The current MCT research is applied to composite laminates consisting of unidirectional continuous fiber lamina. Material symmetry considerations allow us to assume transverse isotropy for each of the unidirectional lamina. Therefore, the constituent failure criteria may be expressed in terms of transversely isotropic stress invariants [2] given by: I1 = 11, I2 = 22 + 33, I3 = 222 + 233 + 2223, I4 = 212 + 213, I5 = 22212 + 33213 + 2121323.

(1)

The constituent failure criteria are developed in detail in [3] and summarized below. The criterion for fiber failure is ±

K1f I 21f + K4f I4f = 1,

(2)

1028 where ±

K1f = ±

1 , S 211f

and

K4f =

1 . S 212f

The ± symbol indicates that the appropriate tensile or compressive ultimate strength value is used depending on the constituent’s stress state. For the matrix constituent, the failure criterion is ±K 3m 3m

(3)

I + K4mI4m = 1,

where ±

K3m = ±

S

2 22m

1 , + ± 22S 233m

and

K4m =

1

. S 212m

Constituent failure parameters, as well as constituent elastic constants, are extracted from experimentally determined composite results with the aid of finite-element micromechanics and MCT analyses [4] and are thus in situ rather than bulk values. We also note that thermal residual stresses were neglected in all analyses.

2. Simulation versus experimental results We are generally pleased with our failure simulations involving four different composite materials and 14 cases provided by the organizers of the failure exercise [5]. Generally, our results are conservative, i.e., falling inside the experimentally generated failure envelopes. We did not require any information other than that provided by the organizers, and the initial failure simulations were conducted by a single individual in less than 80 h. As in so many research endeavors, the experimental data raise as many questions as they answer. Questions such as: (1) the validity of extrapolating unidirectional lamina behavior to lamina that are part of a general laminate (cases 1, 3, and 3). (2) The effect of simultaneous shear and normal lamina stresses on lamina strength (cases 1, 2, 4, 5, and 9). (3) The constraining effects of adjacent lamina, in particular orthogonal orientations, on lamina strength (cases 10, 12, and 14). In the following sections, we compare our failure simulations with the experimentally generated data for each of the 14 cases and briefly discuss the correlation between them. We also correct our analyses for cases 9, 11, and 13 previously presented in Part A of the failure exercise.

2.1. Case I The analytically generated biaxial, y : xy, failure envelope for a 0° E-Glass/LY556/HT907/ DY063 lamina, shown in Fig. 1, was in good agreement with the experimental data. There is no doubt that the correlation would improve significantly if the compressive unidirectional strength, provided by the organizers, was in better agreement with the case 1 data set. This difference may be the result of using material properties derived from tests of finite-width test

1029

Fig. 1. Biaxial, y : xy, failure envelope for a [0°] lamina made from E-glass/LY556/HT907/DY063.

coupons that are influenced by free-edge effects which are absent in tubular specimens [6]. The net result of increasing the unidirectional lamina compressive strength value in the failure criteria would be to move the analytical failure envelope in quadrant II outward away from the axis origin and in better correlation with the experimental data. We also note that in this and other cases, most notably 2 and 9, lamina in situ shear strength appears to increase in the presence of simultaneous normal stress. The absence of an explicit shear/normal interaction term in the proposed MCT failure criteria may need to be addressed. There is an implicit normal-shear interaction term in the full failure criteria originally proposed by us [3, 4]. A truncated form of the full criteria which eliminated the implicit normal-shear term was used for the analyses conducted as part of this exercise because of the absence of material test data necessary to determine this term.

2.2. Case 2 The analytical biaxial, x : xy , final failure envelope for a 0° lamina made of CFRP T300/914C material, shown in Fig. 2, is again conservative without being overly so. As in case 1, the shear strength of the material appears to be enhanced in the presence of both tensile and compressive normal stresses. A question arises concerning this behavior as to whether it is peculiar to single UD lamina, i.e., does it also occur within a laminate of a general stacking sequence? In the single case (5) presented by the organizers of a general laminate under x : xy loading, there is some evidence of increasing shear failure stress in the presence of moderate axial tensile and compressive stress. In future research, we intend to conduct additional analyses and experiments to further investigate the effect of normal-shear interaction on laminate strength and the necessity of accounting for it in the analysis of structural laminates. Should it be proven to be a significant effect, it would provide the motivation for adding an explicit shear-normal stress interaction term to our failure criteria.

1030

Fig. 2. Biaxial, x : xy, failure envelope for a [0°] lamina made from T300/BSL914C.

2.3. Case 3 The failure envelope shown in Fig. 3 is for a biaxial, x : y , loading of a 0° lamina of E-glass/ MY750 epoxy. Data necessary to develop a full experimental failure envelope in all four quadrants are lacking. Hence, only partial conclusions can be drawn concerning correlation between analysis and experimental results. MCT is in good correlation with the data in quadrant I and not so in quadrant IV. The rectangular failure envelope predicted by MCT is attributed to a distinct transition from fiber to matrix failure and is similar to one that would be produced by a simple maximum stress failure criterion ( ± ij / ± Sij ≥ 1). The distinct transition from matrix to fiber failure is a result of assuming that the strength of the fiber constituent controls composite failure in the direction parallel to the fiber and the matrix strength controls failure in the transverse direction. The unidirectional (UD) strength data provided by the organizers tends to support

Fig. 3. Biaxial, y : x, failure envelope for a [0°] lamina made from E-glass/MY750/HT917/DY063.

1031

Fig. 4. Biaxial, y : x, failure envelope for a [90°/ ± 30°]S laminate made from E-glass/LY556/HT907/ DY063.

this assumption in the tension-tension regime where there is a two order of magnitude difference in the lamina’s longitudinal ( + S11 = 1280 MPa) and transverse strengths ( + S22 = 40 MPa). One might assume that this effect would also hold in the tension compression regime since there is still an order of magnitude difference in the lamina’s longitudinal tensile ( + S11 = 1280 MPa) and transverse compressive (–S22 = 145 MPa) strengths. The question is raised; is the experimental behavior in quadrant IV peculiar to a single UD lamina or does it also occur within a laminate of general stacking sequence? If further experiments indicated that the stress interaction in the + y : –x regime were important in the analysis of general structural laminates, it may be necessary to add a normal–normal stress interaction term to increase the accuracy of failure predictions for this material.

2.4. Case 4 We are very pleased with the correlation between analytical and experimental x : y failure envelopes, shown in Fig. 4, for the (90°/ ± 30°) E-glass/LY556 epoxy laminate. This case represents, to us, a general structural laminate under in-service loadings. Again, our results are generally conservative. The decrease in correlation in quadrant III, compression-compression loading, is likely due to buckling of the test specimen as noted by the organizers [7]. Buckling is not accounted for in the current MCT failure criteria. We note also that failure points A, D, E, F, and G were all compression induced and would be expanded outward if higher UD lamina compression values were used as indicated by case 1. In the region about point C, longitudinal (along the fiber) tension and shear coexist in the ± 30° lamina which ultimately causes laminate failure. But a shear strengthening effect, as seen in the single lamina test of case 1, does not appear to exist since MCT tends to slightly over-predict the ultimate laminate

1032 strength. Any discussion by the authors on the shear strengthening effect at this point would be pure speculation. As we stated previously, we were not aware of any shear/normal stress interaction on lamina failure before this exercise and we intend to conduct additional analyses and experiments to further investigate this phenomena. 2.5. Case 5 We are also pleased with the correlation between analytical and experimental failure envelopes, shown in Fig. 5, for the (90°/ ± 30°) E-glass/LY556 epoxy laminate under combined x : xy stresses. Again, our results are generally conservative. Because compression was the major contributor to failure at points A, B, and C, increasing the UD lamina compression strength would result in an improved analytical versus experimental correlation in the shear-compression region of quadrant II. As noted in case 1, there appears to be justification for increasing this strength. We note that in the region about points E, F, G, and H, longitudinal tension and shear stresses coexist and there is a lower correlation between analytical and experimental results. Likewise there is a lower analytical versus experimental correlation in the region about points A, B, and C where compression and shear stresses coexist. This information provides additional evidence that there may be a need to account for normal shear stress interaction with an additional term in the MCT failure criteria. 2.6. Case 6 Excellent correlation is observed between the analytical and experimental x : y failure envelopes in quadrants I and IV, shown in Fig. 6, for the AS4/3501-6, [0°/ ± 45°/90°]S

Fig. 5. Biaxial, x : xy, failure envelope for a [90°/ ± 30°]S laminate made from E-glass/LYS56/HT907/ DY063.

1033

Fig. 6. Biaxial, y : x, failure envelope for a [0°/ ± 45°/90°]S laminate made from AS4/3501-6.

laminate. Again, we are pleased with the results here because this case represents a widely used material and laminate configuration under realistic multiaxial loads. MCT appears to predict a particular flattening in the laminate failure envelope between points D and E but does so unconservatively. The reason for the overestimated laminate strength is not clear because the failure appears to be due to simple tension failure of the fiber constituent in the 0° lamina. The lack of correlation in quadrant III compression-compression loading is likely due to buckling of the test specimen, as noted by the organizers [7], which is not accounted for in the current MCT failure criteria. The MCT failure simulation also predicts non-catastrophic laminate damage, in quadrant I of Fig. 6, due to matrix tensile failure that correlates to results provided by the organizers [7]. Case 8 presents a stress–strain curve extracted from this data.

2.7. Case 7 The stress–strain curve for a AS4/3501-6, [0°/ ± 45°/90°]S laminate under uniaxial tension, y : x = 1 : 0, is shown in Fig. 7. MCT correlates exactly with the experimental results in the early part of the loading and conservatively under-predicts the ultimate laminate strength by 8%. Strain jumps in the analysis curve at approximately 250 MPa indicate initial laminate damage due to transverse matrix tensile failures in the 0° lamina. Intermediate damage, predicted around 400 MPa in the form of matrix failure in the ± 45° lamina, was caused by combined shear and tensile stresses. The organizers note [7] that a decrease in the slope of the

1034

Fig. 7. Uniaxial, y : x = 1 : 0, stress–strain curves for a [0°/ ± 45°/90°]S laminate made from AS4/ 3501-6.

experimental stress–strain curve around 400 MPa indicated a form of initial failure in the actual laminate. MCT predictions correlate with this behavior but slightly over-predict the softening of the laminate due to failed matrix induced damage. The overly soft predicted response is attributed to the damage model used in MCT which assumes a simple on/off constituent damage state variable. When a constituent fails, all its moduli are immediately reduced to a near zero value and an alternate set of composite properties, derived from micromechanics [3], is used. Actual damage evolves much more slowly and gradually. A more realistic damage model could be implemented but in keeping with the design tool philosophy inherent in the formulation of MCT, the strain jumps help visualize the occurrence of damage.

2.8. Case 8 The stress–strain response for the AS4/3501-6, [0°/ ± 45°/90°]S, laminate under y : x = 2 : 1 biaxial tension is shown in Fig. 8. The correlation with experimental data for the biaxial loading considered here is significantly better that the uniaxial stress–strain correlation shown in Fig. 7. MCT under-predicts ultimate laminate strength by 3% and failure strain values are virtually identical. The MCT simulation indicates that initial laminate damage occurs around 240 MPa due to transverse matrix tensile failures in the 0° lamina followed by matrix failure in the ± 45° lamina around 350 MPa caused by combined shear and tensile stresses. Additional intermediate damage is predicted at approximately 460 MPa due to tensile matrix failure in the 90° lamina. Correspondingly, the organizers note [7] that non-linear stress–strain behavior, possibly due to matrix cracking, occurs in the experimental stress–strain curve of unlined specimens around 480 MPa resulting in significant softening of the laminate. This

1035

Fig. 8. Biaxial, y : x = 2 : 1, stress–strain curves for a [0°/ ± 45°/90°]S laminate made from AS4/ 3501-6.

intermediate damage was visually manifested in the unlined specimens by fluid leakage at the 480 MPa stress level while the specimens tested with a liner failed at an approximate stress level of 600 MPa. We believe that the lower failure of the unlined specimen is possibly due to the corrosive and erosive effects of a high pressure fluid leaking through cracks in the lamina matrix. While the lined specimens may have experienced identical matrix cracking, the liner prevented any fluid damage to the composite. MCT does not model any fluid-structure interactions.

2.9. Case 9 We erroneously truncated our originally submitted x : y failure envelope presented for the E-glass/MY750/ HY917/DY063 [ ± 55°]S laminate by assuming that initial failure in quadrant I was also final failure. A new portion of the final failure envelope occurring in the tensiontension region has been added. Initial and final failure envelopes are virtually unchanged in quadrants II–IV. The corrected envelope is shown in Fig. 9. Revised results for the failure envelope are summarized in Table 1. We missed the final failure in quadrant I because a complete matrix tensile failure throughout the laminate caused the finite element program to conduct several hundred equilibrium iterations, which we previously mistook for catastrophic laminate failure, before convergence allowed continued loading. Final laminate failure in this quadrant is due to fiber shear : tension failure. The MCT results tend to provide conservative two-dimensional failure predictions in x : y space and are in good agreement with the data in the neighborhood of the origin. MCT significantly under-predicts the laminate failure strength at the positive and negative y

1036

Fig. 9. Biaxial, y  x, failure envelope for a [ ± 55°]S laminate made from E-glass/MY750/HT917/ DY063.

extremes. Failure around these regions (points “E” and “I”) was a result of combined shear and compressive or tensile stresses respectively. Again it appears that a shear-normal stress interaction term may be needed to account for enhanced shear strength in the presence of a normal stress. MCT predictions of intermediate damage conservatively correlate with the experimental intermediate failures, in regions close to the origin, of unlined tubes in which matrix cracking caused oil leakage. MCT predictions do not correlate well with experimentally determined intermediate damage states at higher x stresses. The lack of correlation here leads us to believe that a shear-normal interaction term may be needed for both fiber and matrix constituents. The lack of correlation in the presence of shear-normal stress is further discussed in the context of the stress–strain curves presented in cases 10 and 11. Table 1 Revised E-glass/MY750/HT917/DY063 failure envelope summary for a [ ± 55°]S laminate under biaxial, y : x, load. Point

Lamina

Primary term

Secondary term

Failure mode

A B C D

± 55 ± 55 ± 55 ± 55 ± 55 ± 55 ± 55 ± 55 ± 55 ± 55

K3m I3m = 1.0 K3m I3m = 0.90 K3m I3m = 0.62 K4m I4m = 0.79 K4f I4f = 0.86 K4f I4f = 0.61 K3m I3m = 0.87 K3m I3m = 0.69 K4f I4f = 0.96 K1f I1f = 1.0

K4m I4m = 0.0 K4m I4m = 0.10 K4m I4m = 0.38 K3m I3m = 0.21 K1f I1f = 0.14 K1f I1f = 0.39 K4m I4m = 0.13 K4m I4m = 0.31 K1f I1f = 0.04 K4f I4f = 0.0

Matrix-tension Matrix-tension/shear Matrix-tension/shear Matrix-shear/tension Fiber-shear/comp Fiber-shear/comp Matrix-comp/shear Matrix-comp/shear Fiber-shear/tension Fiber-tension/shear

E F G H I

1037 The organizers note that the specimens used in the compression-compression quadrant were made of thicker tubes. External pressures up to 200 MPa were needed to fracture the specimen which generated a radial stress component. Multicontinuum Theory is fully threedimensional as are the proposed failure criteria with the single assumption of lamina transverse isotropy. Radial stresses due to external pressure would be accounted for if they had been modeled as part of the load. We chose not to conduct the more complex analysis which would result in a three-dimensional (y : x : z stress space) failure envelope. 2.10. Case 10 Non-linear shear behavior characterizes the stress–strain curves of the E-glass/MY750/ HY917/DY063, [ ± 55]S laminate under uniaxial load, y : x = 1 : 0, as shown in Fig. 10. MCT is in good agreement with the experimental data to about 230 MPa ( | x | = 1.7%) where catastrophic laminate failure is predicted due primarily to fiber shear failure (K4f I4f = 0.96, K1f I 21f = 0.04). The lack of correlation between analysis and experimental data at higher stress levels is a direct result of the experimental data to which the failure criteria were fitted. In particular, shear strengths and non-linear shear stress–strain relations were taken from the pure shear stress–strain curve provided by the organizers [5]. These data specify a composite shear strength of S12 = 72 MPa and ultimate engineering shear strain of 12 = 4%. In contrast, linear classical lamination theory [8] predicts lamina shear stress levels, for a [ ± 55°]S laminate, on the order of 12 = 200 MPa at the experimental failure stress of y = 600 MPa. Using the generalized plane strain transformation equation,

12 x  y

xy = sin 2 + cos 2, 2 2 2

(4)

Fig. 10. Uniaxial, y : x = 1 : 0, stress–strain curves for a [ ± 55°]S laminate made from E-glass/MY750/ HT917/DY063.

1038 the local lamina shear strain, 12, corresponding to laminate strains | x | = y = 1.7% and xy = 0, is 3.1% which is close to the shear failure strain provided by the organizers. This result is expected since the MCT shear failure parameters were derived from these data. At failure (x = –10.93%, y = 8.78%) the experimental lamina shear strain, 12, is calculated to be approximately 18.5% which is over four times that provided by the organizers. Our analysis of this case predicts simultaneous shear and tensile lamina stresses of the same order of magnitude in the local (material) coordinate systems. Therefore, enhanced shear strength in the presence of normal stresses is again a possible contributor to the higher observed strengths. In addition to the effect of simultaneous shear-normal stresses, we believe the constraining influence of the laminate stacking sequence contributes to enhanced stress– strain ultimate values that cannot be predicted by extrapolation of the UD shear data. In case 12 matrix tensile strength of a [0°/90°]S laminate is enhanced without the presence of simultaneous shear stress and in case 14 the lamina shear strength of a [ ± 45°]S laminate is enhanced without the presence of simultaneous normal stress. 2.11. Case 11 Corresponding to the results presented in case 9, we originally presented a truncated stress– strain curve prediction for the E-glass/MY750/HY917/DY063 [ ± 55]S laminate under y : x = 2 : 1. Our original submission was just the initial portion (out to 0.4% strain) of the corrected stress–strain curve presented in Fig. 11. Initial damage in the form of complete matrix tensile failure throughout the laminate caused the finite element program to conduct several hundred equilibrium iterations, which we previously mistook for catastrophic laminate failure, before convergence allowed continued loading. Matrix tensile failure is predicted in the laminate around y = 120 MPa which is manifest on the analytical stress–strain curve as a strain jump. A corresponding strain increase can be

Fig. 11. Uniazial, y : x = 2 : 1. stress–strain curves for a [ ± 55°]S laminate made from E-glass/MY750/ HT917/DY063.

1039 seen in the experimental y–x curve but occurring at y = 260 MPa. Under-prediction of the stress level at which this intermediate damage occurs is again most likely due to the effect of enhanced shear strength in the presence of a normal stress which is not accounted for by the MCT failure criterion. Close examination of the initial portion of the stress–strain curves shows that MCT over-predicts the slope on both x and y stress–strain curves. This initial overprediction carries into the post-damage region. We cannot rationalize the lack of correlation with the experimental stress–strain curves especially in light of the relatively good curve correlations in case 10 (x : y = 1 : 0) for the identical laminate and in cases 13 and 14 for biaxial loading (y : x = 1:1 and y : x = 1 : –1 respectively) for a similar ([ ± 45°]S) laminate. The general dissimilarity between the non-linear experimental and relatively linear analytical stress–strain curve is again the result of a simple on/off constituent damage model used by MCT whereas actual damage evolves in a more gradual and continuous manner. The organizers note final fracture of the [ ± 55°]S laminates occurs at approximately y = 740 MPa which is higher than the levels of the experimental stress–strain graphs presented in Fig. 11. MCT over-predicts this laminate strength by approximately 11% but is within 1 % of similar y : x = 2:1 experimental results presented as part of case 9.

2.12. Case 12 The E-glass/MY750/HY917/DY063 stress–strain curves for a [0°/90°/0°] laminate under uniaxial load, x : y = 1 : 0, are shown in Fig. 12. There is excellent correlation between the analytical and experimental results. Initial laminate damage due to tensile matrix failure in the 0 lamina is predicted by MCT at approximately 80 MPa whereas the organizers note the onset

Fig. 12. Uniaxial, y : x = 1 : 0, stress–strain curves for a [0°/90°/0°] laminate made from E-glass/ MY750/HT917/DY063.

1040 of cracking at 117.5 MPa. Matrix tensile failure in the 90° lamina is predicted at approximately 260 MPa where a corresponding cracking of the experimental specimen is noted at approximately 400 MPa. The lower predicted value may be due to transverse lamina strength being taken from uniaxial tests of unidirectional lamina (or laminates) in which matrix damage is catastrophic failure. Conversely, in a uniaxial (0°) load test of a cross-ply there is a constraining effect of the 0° lamina on the transverse failure of the 90° lamina and vice versa for a uniaxial 90° load test. In situ properties for transverse lamina strength taken from a cross-ply test, and the MCT matrix strength parameter developed from it, would improve the analytical/experimental correlation for initial damage. The ultimate laminate strength for the [0°/90°/0°] laminate was predicted within 2% of the experimentally determined value. The stress–strain curves for this laminate presented in Ref. [3] (Part A submission, Fig. 18), for the case where thermal stresses are considered, showed a peculiar behavior at small strain values. This behavior was a result of the authors’ attempt to illustrate the thermal analysis capability of MCT and to reflect the test data which were generated on laminates where thermal stresses are likely to have been developed as a result of curing at elevated temperature. Our position regarding incorporating the effects of thermal stresses in a failure analysis is that our method of backing out constituent properties from experimental composite data mitigates the need to account for them in structural analysis or, at worst, reduces residual stress due to thermal mismatches to be secondary in their effect. The peculiar behavior seen in the graph with thermal loads is readily explained by considering what we used as the reference (zero) strain state. Specifically, in a room temperature experimental test, there may be significant thermal stresses in the composite due to the curing process. Hence, as the experiment begins we have residual stresses at what is arbitrarily defined to be the zero strain state. The above situation is not what we modeled. Instead we defined our zero strain state prior to introducing the thermal load. If we were to subtract out the strains caused by the thermal loads, the resulting graph would be our predicted response for an experiment where we include the effect of the thermal residual stress. This would eliminate the peculiar behavior seen in Ref [3]. In summary, for a direct comparison with experimental data, the curve we presented is not the most representative one. In retrospect, eliminating the thermal strains shown would have avoided any confusion interpreting the initial behavior of the laminate.

2.13. Case 13 We erroneously truncated our originally submitted stress–strain curves for a E-glass/MY750/ HY917/DY063, [ ± 45°]S laminate under a biaxial load, x : y = 1 : 1. Our original submission was just the initial portion (extending to only 0.2% strain) of the corrected stress–strain curve shown in Fig. 13. There is excellent correlation between the analytical and experimental stress–strain curves. Initial laminate damage due to tensile matrix failure in all lamina is predicted by MCT at approximately 70 MPa which corresponds to the initial cracking noted by the organizers between 50 and 70 MPa. The laminate exhibits a near linear stress–strain response until tensile failure of the fibers is predicted at approximately 620 MPa. MCT overpredicts the ultimate laminate strength by approximately 24%.

1041

Fig. 13. Biaxial, y : x = 1 : 1, stress–strain curves for a [ ± 45°]S laminate made from E-glass/MY750/ HT917/DY063.

2.14. Case 14 Stress–strain curves for an E-glass/MY750/HY917/DY063, [ ± 45°]S laminate under biaxial load, y : x = 1 : –1, are shown in Fig. 14. The lamina are in a state of pure shear in the local (material) coordinate system. MCT is in good agreement with the experimental data to about 2% laminate strain where shear failure of the laminate is predicted. Using the generalized

Fig. 14. Biaxial, y : x = 1 : –1, stress–strain curves for a [ ± 45°]S laminate made from E-glass/MY750/ HT917/DY063.

1042 plane strain transformation equation [Eq. (4)], the local shear strain, 12, corresponding to laminate strains x = y = 2% and xy = 0, is 4% which is equal to the shear failure strain provided by the organizers. This result is expected since the MCT shear failure parameters were derived from this data. At failure the experimental lamina shear strain, 12, is calculated to be approximately 20% which is five times that provided by the organizers. Again, the constraining influence of the laminate stacking sequence (in this case a [0°/90°] laminate rotated 45°) is believed to result in enhanced stress–strain ultimate values that cannot be predicted by extrapolation of the UD shear data. MCT under-predicts laminate failure strength by 22%. 3. General discussion We are pleased with our failure simulations of the four different composite materials and 14 cases provided by the organizers of the failure exercise. The generally good analytical versus experimental correlation validates the approach of predicting composite damage based on failure of the constituents. We modified our original predictions in only three of the 14 cases. In all cases, we simply failed to allow our non-linear finite-element analysis to reach equilibrium and prematurely stopped the analysis. The MCT approach provides a high resolution window on laminate behavior by highlighting initial and intermediate damage states. Initial failure is often difficult to delineate experimentally but is important in structural design because it reduces stiffness and produces nonlinear, inelastic laminate behavior. Constituent failure (typically the matrix) begins as scattered micro-ruptures that must multiply and coalesce to some threshold value before manifesting itself as a noticeable change in experimentally monitored metrics such as displacement, strain, or load. The predictive capability of micro/macro analysis techniques such as MCT can help the experimentalist discern subtle changes in the metrics when testing for initial and intermediate damage states. The experimental data, in particular cases 1, 2, and 9, indicate that shear strength may be enhanced in the presence of normal stress. Recognition of the magnitude of this phenomena is one of the lessons learned by the authors from this exercise. To account for increasing lamina strength, a normal-shear interaction term can be added to the failure criteria. The full constituent-based failure criteria proposed by the authors [3] K2m I 22m +

1 1 I3m + 2 I4m = 1 2S 223m S 12m

(5)

includes a term

±

K2m =

1 ± ( S22m + ± 22S33m)2

冉

±

1

冊

S 222m + ± 22S 233m 2S 223m

that acts implicitly as an interaction term by taking on a negative value for materials that exhibit strong shear-normal interaction (stress invariant contributions are always positive definite). The term K2m was omitted in the current research due to insufficient data to determine a value with confidence. In previous research [4], we saw a stronger interaction in an E-glass/vinylester (E-glass/8084) material (K2m was negative in the presence of tension)

1043 Table 2 Failure coefficients for K2m as defined in Eq. (4). Composite

+

E-glass/vinylester Carbon/epoxy Boron/epoxy

–5.61E-4 16.75E-4 6.205E-5

K2m

–

K2m

7.67E-4 4 84E-4 1.19E-4

than in the carbon/epoxy (AS4/3501) or boron/epoxy (boron/5505). ± K2m values for these materials are presented in Table 2. Incorporating the effects of K2m in the present exercise would certainly be of interest. As mentioned in our contribution to Part A [3] of the failure exercise, data from off-angle, balanced, symmetric laminates, [ + ]S, provide an excellent basis for determining a best fit of failure parameters K12m, K23m, and K12 f . Hence, some of the laminates analyzed as part of this exercise, e.g., [ ± 55°]S and [ ± 45°]S, could be used as inputs to “tune” the failure analysis. Additionally, the fifth transversely isotropic stress invariant, I5 = 22212 + 33213 + 2121323,

(6)

which was not included in the quadratic failure criteria proposed, may provide an explicit term to account for normal : shear stress interaction. Modifying the MCT failure to include this term may be the subject of future research. The experimental data, in particular cases 10, 12, and 14, indicate that lamina strengths may be enhanced by the constraining action of adjacent lamina, especially if they are oriented orthogonally. The most effective way to account for this effect may be to derive failure parameters from cross-ply rather than unidirectional laminate tests. In previous research [4], the lamina shear stress–strain behavior was taken from uniaxial compressive tests of [ ± 45°]S laminates with good success. Likewise, lamina transverse strengths (and associated constituent strengths) could be taken from uniaxial tests of [0°/90°]S laminates. This method has been used to determine the compressive strength of carbon/epoxy lamina [9]. The reader is reminded that a comparison between the predictions of the present method and those of other methods is made in Ref. [10]. 4. Concluding remarks There is a severe lack of stress–strain and failure data on composite laminates subjected to multiaxial loads. The scarcity of data is due in large part to the difficulty and expense in generating it. There is also a bewildering variety of approaches to predicting failure of composite structural laminates. The need for a venue to objectively evaluate the composite community’s collective ability to predict failure of structural composites under in-service load conditions has been painfully missing. This exercise should prove to be a seminal work in this area and the organizers (Hinton, Kaddour and Soden) are to be commended for their hard work and valuable contribution to the body of engineering knowledge. The MCT approach to conducting composite failure analysis was intended from the beginning to be used as a mainstream design tool. The method was implemented into the finite

1044 element frame work, formulated to minimize user input, be computationally fast, completely three dimensional, and generally produce conservative failure values. Constituent-based failure criteria form the basis for a progressive failure analysis allowing for accurate composite laminate structural analysis in pre-, ongoing and post- damage conditions. Because information is developed on a level at which failure initiates, load redistribution to other parts of the structure as well as the remaining constituents can be efficiently included. This load redistribution allows a designer to track failure as it occurs, region by region, and reduce the stiffness and strength of damaged areas without necessarily declaring total structural failure. Changes in structural design, to remedy weaknesses, can be concentrated to specific areas leading to a more optimized design. We view capturing improved failure parameters for the MCT analysis as the best way to improve this design tool. We have made multiple comments throughout the paper on how to improve upon these failure strength estimates. However, we believe the fundamental MCT methodology represents a simplistic, elegant, and accurate approach to failure analysis of composites. Acknowledgements The research reported herein was supported by the Naval Surface Warfare Center, Carderock Division under purchase order N00167-01-M-0246 and by the Office of Naval Research under Grant N00014-97-11081. References [1] Hashin Z. Failure criteria for unidirectional fiber composites. J Appl Mech 1980;47:329–34. [2] Hansen AC, Blackketter DM, Walrath DE. An invariant-based flow rule for anisotropic plasticity applied to composite materials. J Appl Mech 1991;58:881. [3] Mayes JS, Hansen AC. Composite laminate failure analysis using multicontinuum theory, Comp Sci Technol, V64, pp 379–394, 2004, see also Chapter 3.15 in this book. [4] Mayes JS. Micromechanics based failure analysis of composite structural laminates. Naval Surface Warfare Center, Carderock Division Report, NSWCCD-65-TR-1999/15, September 1999. [5] Soden PD, Hinton MJ, Kaddour AS. Lamina properties, lay-up configurations and loading conditions for a range of fibre-reinforced composite laminates. Compos Sci Tech 1998;58:7 1011. [6] Herakovich CT. Laminate failures. In Mechanics of fibrous composites. New York: John Wiley & Sons; 1998. [7] Soden PD, Hinton MJ, Kaddour AS. Experimental failure stresses and deformations for a range of composite laminates subjected to uniaxial and biaxial loads: Failure exercise benchmark data. Comp Sci Technol 2002;62(12–13):1489–514. [8] Hyer MW. Stress analysis of fiber-reinforced composite materials. New York: WCB McGrawHill; 1998. [9] Welsh JS, Adams DF. Unidirectional composite compression strengths obtained by testing minisandwich, angle-, and cross-ply laminates. Composite Materials Research Group Report No. UW-CMRG-R-95-106, Laramie, WY, April 1995. [10] Kaddour AS, Hinton MJ, Soden PD. A further comparison of the predictive capabilities of current failure theories for composite laminates, judged against experimental evidence. Comp Sci Technol 2002;62(12–13):1725–97.

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

1045

CHAPTER 5.15

Correlation of the bridging model predictions of the biaxial failure strengths of fibrous laminates with experiments Zheng-Ming Huang* Department of Engineering Mechanics, Tongji University, 1239 Siping Road, Shanghai, PR China Abstract In my other paper (in this issue), the bridging micromechanics model has been combined with the classical lamination theory to predict the progressive failure strengths or the entire stress–strain curves of a set of typical polymer resin based composite laminates subjected to different biaxial loads. The predictions were performed only using the constituent fiber and resin properties and the geometric parameters of the laminates specified independently. Comparison of the predictions with the experimental measurements provided by the failure exercise organizers is carried out in this paper. As a whole, the overall correlation between the theory and the experiments is reasonable. Some additional comments regarding the applications of the bridging model to the simulation of ultimate behavior of fibrous laminates are provided. Comparison of the predictions of each other with and without thermal residual stresses is also made. It is demonstrated that for most of the present epoxy resin based composites, the effect of the thermal residual stresses is grossly insignificant. Thus, a general conclusion may be that in most cases the thermal residual stresses can be neglected for a thermoset polymer resin based composite.

1. Introduction A recently developed micromechanics model, the bridging model [2], has been incorporated with the classical lamination theory to predict the progressive failure behavior and ultimate strength of 14 multi-directional laminates subjected to biaxial loads [1]. Those laminates constituted the failure exercise problems proposed by the organizers [3]. The predictions were made only using the constituent fiber and resin properties and the laminate geometric parameters provided by the organizers, without knowing any experimental information of the laminates. The prediction efficiency is shown in this paper, by correlating the predictions with

* Corresponding author. Tel.: + 86-21-65985373. E-mail address: [email protected] (Z.-M. Huang).

1046 the experiments. Additional comments regarding this correlation are thus made. Further predictions without considering any effect of thermal residual stresses on the laminate responses are carried out, and are compared with the experiments. It has been found that for the majority of the problems under consideration, the predictions with and without the thermal residual stresses nearly coincide one another. However, a maximum difference as high as 70% between the predicted ultimate strengths with and without the thermal residual stresses for a laminate problem has also been recognized. These results indicate that although in most cases the effect of thermal residual stresses can be neglected, cautions should also be kept in mind that the neglect may cause a significant error in some other case.

2. Overall correlation Reference [3] specified in detail the information of all of the 14 exercise problems (referred to occasionally as Test Cases) including the laminate lay-up sequences and angles, the fiber volume fractions, the lamina thicknesses, the constituent properties, and the stiffness and strength parameters of the individual unidirectional (UD) laminae. From these results, the input data required for the bridging model predictions were determined, as illustrated in Ref. [1]. Those data can be classified into two groups (see Ref. [1]). The first group is the constituent fiber and resin properties and the second group is the laminate geometric parameters. In the predictions made in Ref. [1], all of the fiber materials were considered as linearly elastic until rupture for which only elastic and ultimate properties are required, whereas the resins were regarded as elastic–plastic materials. Thus, the material properties consisted of the fiber and resin elastic constants, the resin plastic parameters, and the tensile and compressive strengths of the fibers (along longitudinal direction) and the resins. The second group data, i.e. the laminate geometric parameters, included the laminate lay-up arrangements, laying angles, and thickness of each lamina in the laminates. While the second group data can be obtained from the condition as-fabricated (in the predictions in Ref. [1], those provided by the organizers were directly employed), the first group data have to be prepared with more care. In Ref. [1], the elastic properties of both the fibers and the resins for all of the 14 problems were taken exactly the same as those provided by the organizers [3], whereas the remaining constituent properties were retrieved from the responses of the individual UD laminae. The resin plastic parameters were back calculated against the stress– strain responses of the corresponding laminae subjected to in plane shearing up to failure. Having determined the fiber elastic and the resin elastic–plastic parameters, the constituent strength data were retrieved against the ultimate strengths of the UD laminae subjected to different kinds of uniaxial loads (longitudinal tension, transverse tension, longitudinal compression, transverse compression, and in-plane shearing) up to failure, i.e. the ultimate uniaxial loads. The retrieval of the constituent strengths was accomplished in such a way that when all the ultimate uniaxial loads had been applied to the individual UD lamina respectively, the comparative maximum tensile and compressive stresses generated in the fiber and the resin materials were taken as their respective tensile and compressive strengths. For example, when the ultimate longitudinal tension, transverse tension, and the in-plane shearing were applied to the UD lamina respectively, the resin material would generate three different maximum tensile stresses. The resin tensile strength was defined as the largest of these three tensile stresses. It has been found that the tensile strengths of both the fibers and the resins

1047 were determined against the ultimate longitudinal tension of the UD laminae. The fiber and the resin compressive strengths were obtained based on the ultimate longitudinal and transverse compressions of the UD laminae, respectively. More details are provided in Ref. [1]. The theoretical results taken from Ref. [1] are replotted in curves designated as “theory with thermal residual stresses” in Figs 1–14 for all of the 14 exercise problems, respectively, which are compared with the test results provided by the exercise organizers [4]. It is seen that except for the problems 3, 4, 6, and 8, the correlations between the theory and the experiments for all of the other 10 problems are reasonable. The correlation for the problem 3 is slightly less satisfactory. However, relatively large differences have been found for the problems 4, 6, and 8. Discussions on the correlations of the specific problems are given below. Problem 1 (UD E-glass/LY556/HT907/DY063 lamina subjected to combined ␴yy and ␴xy loads). The theoretical and experimental data for this problem are graphed in Fig. 1. It is seen that the predicted strength envelope in variation trend from the tensile to the compressive transverse stress component is similar to the measured one. However, discrepancy in magnitude exists. In general, the prediction overestimated a point on the failure envelope with a tensile transverse stress component involved whereas underestimated a point with a compressive transverse stress component involved. This is mainly attributed to the fact that the resin tensile strength used is somewhat “higher” than the in situ resin tensile strength whereas the resin compressive strength used is somewhat “lower” than the in situ resin compressive strength for this lamina. As mentioned earlier, the fiber and resin tensile strengths were

Fig. 1. Measured and predicted biaxial failure stresses for 0° lamina subjected to combined yy and xy . Material type: E-Glass/LY556/HT907/DY063.

1048 retrieved from the longitudinal tensile strengths of the UD lamina, whereas the fiber and resin compressive strengths were retrieved from its longitudinal and transverse compressive strengths respectively. The used resin tensile strength, 56.5MPa, was higher than the resin tensile strength, 33.1 MPa, retrieved from the transverse tensile strength of the UD lamina. In other words, the predicted transverse tensile strength of the lamina based on 56.5 MPa resin tensile strength was higher than the 35 MPa measured value, as indicated in Table 1 of Ref. [1]. On the other hand, the provided transverse compressive strength of the lamina, 114 MPa, was lower than the compressive strength data shown in Fig. 1, which are around 138 MPa, resulting in a lower resin compressive strength used (55.7 MPa). If the lamina transverse strength were specified as 138 MPa, the retrieved resin compressive strength would be 69.3 MPa. As the lamina failure when subjected to the combined transverse and in-plane shear loads was initiated from the resin failure, the predicted lamina failure loads, if resulted in resin tensile failure, would be higher than the measured values and the predicted lamina failure loads, when resulted in resin compressive failure, would be lower than the measured counterparts, giving rise to the over- and under-estimations respectively, as indicated in Fig. 1. It may be noted that both the test data and the theory showed that the shear strength increased with the application of a moderate transverse compressive stress component. However, the test result indicated that the maximum shear strength occurred at (–70.5) MPa of the applied transverse stress, whereas the predicted such stress was smaller (–21.1 MPa). The reason is the same as the above mentioned. If the resin compressive strength used is higher, the predicted maximum shear strength will occur at a larger applied compressive transverse stress. Problem 2 (UD T300/BSL914C lamina subjected to combined ␴xx and ␴xy loads). This is the only composite made of T300/BSL914C material system amongst the 14 exercise problems. Correlation between the theory and the experiments for this lamina is grossly satisfactory, as shown in Fig. 2. The figure clearly indicates that the experimental deviation for composite strengths can be quite large. For the present lamina when the longitudinal load was small, the difference in the measured failure strengths can be as large as 80% of a relative error. Thus, sufficient experimental data should be obtained when they are used to verify theoretical predictions. Problem 3 (UD E-glass/MY750/HY917/DY063 lamina subjected to combined ␴yy and ␴xx loads). Results of this problem are shown in Fig. 3. It may be noted that according to the organizers [4] the problem 3 was actually not an exact UD lamina, but an angle plied laminate with a ply angle of 5°, i.e. [ ± 5°]s. Thus, the predicted curve (for the laminate) plotted in Fig. 3 is slightly different from the corresponding figure shown in Ref. [1]. Similarly to Problem 1, the correlation in the longitudinal (x-directional) tension and compression and the transverse (y-directional) compression is good. This is because the lamina strengths in these directions were used to determine the constituent strengths. The resin tensile strength was also retrieved from the lamina longitudinal tensile strength, resulting in a higher predicted transverse tensile strength for the lamina. It is seen that the predicted failure envelope on the fourth quadrant is approximately a rectangle, whereas the measured data could be better represented by a monotonically increased curve as the longitudinal tensile stress component decreased from its maximum to zero. However, compared with those for problem 2, not enough measured data for this problem have been available. More experiments are required, especially to determine the data points of the failure envelope on the other three quadrants.

1049 Problem 4 [(90°/ ± 30°/90°) E-glass/LY556/HT907/DY063 laminate subjected to combined ␴yy and ␴xx load]. Relatively large discrepancy exists between the theory and the experiments for this problem, as indicated in Fig. 4. This is one of the few exercise problems for which the bridging model predictions did not correlate reasonably with the measured data. The discrepancy became the largest when the combination of the applied biaxial loads, yy and xx, was near to the linear line as shown in the figure, i.e. when xx = 1.5yy + 34.4 (MPa). The resulting laminate response could not be accurately estimated using the procedure of Ref. [1], see the discussions given for problem 10 for more detail. Furthermore, the resin plastic parameters used may also have played an important role. Figure 21 of Ref. [1] clearly indicated that for this laminate structure subjected to the combined yy and xx loads, different use of the resin plastic parameters resulted in very large difference in the predicted failure envelope. The present resin plastic properties were retrieved from the in-plane shear response of the UD lamina, which could be somewhat different from those measured directly on monolithic resin specimens. It would be useful by comparing the predictions based on the resin properties retrieved from the lamina overall response and measured from the monolithic specimens. However, no measured data from the monolithic specimens were available and no attempt for this comparison has been made. Problem 5 [(90°/ ± 30°/90°) E-glass/LY556/HT907/DY063 laminate subjected to combined ␴xx and ␴xy loads]. Relatively good correlation between the theory and the experiments for this problem has been found, as shown in Fig. 5. The theory under-predicted the uniaxial tensile strength, which is dominated by the lamina longitudinal tensile strength.

Fig. 2. Measured and predicted biaxial failure stresses for 0° lamina subjected to combined xx and xy . Material type: T300/914C.

1050

Fig. 3. Measured and predicted biaxial failure stresses for [ ± 5°]s laminate subjected to combined xx and yy . Material type: E-Glass/MY750 epoxy.

Fig. 4. Measured and predicted biaxial failure stresses for [90°/ ± 30°]s laminate subjected to combined yy and xx . Material type: E-Glass/LY556 epoxy.

1051

Fig. 5. Measured and predicted biaxial failure stresses for [90°/ ± 30°]s laminate subjected to combined xx and xy . Material type: E-Glass/LY556 epoxy.

Fig. 6. Measured and predicted biaxial failure stresses for [90°/ ± 45°/0°]s laminate subjected to combined yy and xx . Material type: AS4/3501-6.

1052 Problem 6 [(0°/ ± 45°/90°)s AS4/3501-6 laminate subjected to combined ␴yy and ␴xx loads]. The predicted and test results for this problem are graphed in Fig. 6, and poor correlation has been found. This may be the only exercise problem for which large discrepancy exists without apparent reasoning. It is not very clear why significant difference could exist between the theory and the experiments for this problem. The difference for the part of the failure envelope where biaxial compressions were applied is understandable. During the measurement, the material buckling was observed with the biaxial compressions that would reduce the composite load carrying ability distinctly, whereas the prediction [1] did not account for the material buckling. Moreover, the present problem involves a symmetrically arranged laminate (quasi-isotropic in the laminate plane) subjected to all the possible in-plane biaxial loads. Ideally, the failure envelope of this laminate must be symmetric with respect to some in-plane axis. The predicted curve shown in Fig. 6 confirms this symmetry. Nevertheless, from the given experimental data, this symmetry cannot be seen. It would be useful if more experiments had been performed, especially with load combinations of negative yy and positive xx. Problem 7 [(0°/ ± 45°/90°)s AS4/3501-6 laminate subjected to uniaxial ␴xx load]. The correlation between the theoretical and test results for this problem is better than that for the preceding problem, but still, not very satisfactory, as shown in Fig. 7. It is noted that without considering thermal residual stresses, the prediction seemed to be much better. In the figure,

Fig. 7. Measured and predicted yy vs. yy and yy vs. xx curves for [90°/ ± 45°/0°]s laminate under uniaxial tension (yy /xx = 1/0). Material type: AS4/3501-6. Only the failure information without thermal effect is shown, in which “TT ” = strength, “MT ” = resin tensile failure, and “FT ” = fiber tensile failure.

1053 the progressive failure information of the predictions without the thermal residual stresses was indicated. The similar information with the thermal effect has been reported in Ref. [1] (Fig. 24 of Ref. [1]). It may be noted that the third ply failure of the laminate without the thermal residual stresses was due to the fiber tensile failure, which is different from the failure mode, i.e. the resin tensile failure, identified in Ref. [1] where the thermal residual stresses were taken into account. It is also noted that after the first ply failure the predicted stress– strain curve in the x-direction began to change its slope to more ductile, whereas the curve in the y-direction had little variation in slope. On the other hand, the second ply failure resulted in an apparent stiffness reduction in the y-directional response, but gave rise to a stiffness increase in the x-directional curve. The same features in stiffness variation have been also observed with the results without the thermal residual stresses, as indicated in Fig. 7. Problem 8 [(0°/ ± 45°/90°)s AS4/3501-6 laminate subjected to combined ␴xx and ␴yy loads (␴yy /␴xx = 2)]. Quite large difference of the theory from the test results exists for this problem, as shown in Fig. 8. This is the problem for which the poorest correlation has been found. It can be clearly seen from Fig. 8 that the influence of thermal residual stresses for this problem was significant. Large deviation exists between the predictions with and without the thermal residual stresses. The prediction procedure without the thermal residual stresses is described in the next section. In Ref. [1], an exact formula for calculating the thermal residual stresses generated in the laminate was elaborated and utilized throughout. However, that formula can be applied only to an ideally fabricated laminate. No stress concentration due to any fabrication defect can be incorporated. Moreover, when the predicted thermal residual stresses

Fig. 8. Measured and predicted yy vs. yy and yy vs. xx curves for [90°/ ± 45°/0°]s laminate under biaxial tensions (yy /xx = 2/1). Material type: AS4/3501-6.

1054 have been large enough so that the resin material in the composite has been caused to yield, the resin instantaneous compliance matrix should be defined more carefully. One should know whether the resin is in the status of loading or unloading when a subsequent increment of a mechanical load is applied. If it is in unloading, the compliance matrix should be specified using Hooke’s law. Otherwise, the matrix is defined using Prandtl-Reuss theory. However, the knowledge on loading or unloading is usually not straightforward in the simulation because of the multiaxial stress state generated in the resin material (both the thermal residual stresses and the mechanical load generated stresses were generally multiaxial for any exercise problem where a laminate is involved). It remained a question in Ref. [1] that according to which criterion the material should be regarded as loading or unloading when both the existing and the additional stresses are multiaxial. Due to this difficulty, essentially no unloading was applied in Ref. [1] to any of the 14 exercise problems. Namely, the mechanical load generated stresses were simply superimposed with the cooling down generated thermal residual stresses and the resin instantaneous compliance matrix was defined using the Prandtl-Reuss theory according to the resulting total stresses without considering occurrence of any specific unloading. It is believed that such treatment could probably have introduced some prediction error. It must be pointed out that such error could only possibly occur to those laminates where the resin materials had already been in yielding status before the application of any mechanical load. More specifically, only problems 6, 7, and 8 could possibly involve such an error, see Figs. 23a, 24, and 25 of Ref. [1]. For all the other problems, the composites were still in elastic deformation after the incorporation of the respective thermal residual stresses, and hence no such error was involved. Furthermore, it may be noted that in the present prediction (without the thermal residual stresses) the failure mode of each ply failure is the same as that recognized in Ref. [1] (see Fig. 25 of Ref. [1]), i.e. due to the resin tensile failure. However, the stress levels shared by the fiber material are different at different cases. Consider, for example, the stress levels corresponding to the third (i.e. the last) ply failure. With the thermal residual stresses incorporated (using the data provided in Table 3 of Ref. [1]), the three principal stresses in the fiber material are: 1f = 953.8 MPa, 2f = 58.4 MPa, and 3f = 0, whereas these stresses are: 1f = 1752.7 MPa, 2f = 60.0 MPa, and 3f = 0 when the thermal residual stresses are not taken into account (using the data in Table 3 of the present paper). The stress level in the latter case is much higher than that of the previous case. Although in both the cases the fiber material has not been loaded to its ultimate level, the estimated overall composite strengths are different due to the different load levels being sustained by the fiber material. Problem 9 [( ± 55°)s E-glass/MY750/HY917/DY063 laminate subjected to combined ␴yy and ␴xx loads]. Good agreement has been found between the theory and the experiments for this problem, as indicated in Fig. 9. It is seen that the predicted results are close to the measured ones without liners. Problem 10 [( ± 55°)s E-glass/MY750/HY917/DY063 laminate subjected to uniaxial ␴yy load]. Figure 10 plots both the theoretical and experimental stress–strain curves for this problem. In the experiment, the angle plied laminates, both the ( ± 55°)s laminates (used in problems 9, 10, and 11) and the ( ± 45°)s laminates (used in problems 13 and 14), were also tested in tubular forms, by applying liquid pressure combined with axial load. For comparison, liners were again placed on the inner surface of some of those laminate tubes

1055

Fig. 9. Measured and predicted biaxial failure stresses for [ ± 55°]s laminate subjected to combined yy and xx . Material type: E-Glass/MY750.

Fig. 10. Measured and predicted yy vs. yy and yy vs. xx curves for [ ± 55°]s laminate under uniaxial tension (yy /xx = 1/0). Material type: E-Glass/MY750.

1056 before applying the liquid pressure. The liners significantly increased the load carrying capacity of the cylinders if subjected to the internal pressure. On the other hand, the simulation procedure described in Ref. [1] is apparently applicable only to the specimens without the liners (or without the interaction with the liners). Although the liner used is a nonstructural membrane, the liner and the cylinder may constitute a combined structure, the analysis of which could be achieved by considering the interaction between each other through the use of an FEA (Finite Element Analysis) software package. Thus the liner’s properties (including strength behavior) and geometry (thickness) must be provided. As both the liner’s parameters are not available and the incorporation of the bridging model into an FEA software package has not been accomplished yet, the prediction for the responses of the specimens with the liner was not made. It is seen that the predicted curves agreed reasonably with the measured ones (without the liner). However, at a later stage, the measured curves were somewhat stiffer than the predicted ones. As only one set of stress–strain relationships were provided, it is difficult to make a further judgment. Problem 11 [( ± 55°)s E-glass/MY750/HY917/DY063 laminate subjected to combined ␴yy and ␴xx loads (␴yy /␴xx = 2)]. The predicted failure strength agreed with the measured one for the cases where no liner was used, as shown in Fig. 11. On the other hand, the measured

Fig. 11. Measured and predicted yy vs. yy and yy vs. xx curves for [ ± 55°]s laminate under biaxial tensions (yy /xx = 2/1). Material type: E-Glass/MY750.

1057

Fig. 12. Measured and predicted xx vs. xx and xx vs. yy curves for [0°/90°/0°] laminate under uniaxial tension (yy /xx = 0/1). Material type: E-Glass/MY750.

stress–strain curves showed slightly non-linearity before failure, whereas the predicted ones were essentially linear up to failure. As far as the final strength of the lined specimens is concerned, the present theory significantly under-estimated the measured value by a factor less than 3. Therefore, this test case is one of those cases where the theory was unable to match the test data accurately. Problem 12 [(0°/90°/0°) E-glass/MY750/HY917/DY063 laminate subjected to uniaxial ␴xx load]. For this problem, the predicted stress–strain curves up to failure are in reasonable agreement with the test results, as shown in Fig. 12. However, the measured ultimate strength is about 20% higher than the predicted value. Considering that large deviation exists in the measurement of composite strengths (see Figs. 4, 5, and 9), this difference should be acceptable in engineering. For this problem, the experiment showed an initial cracking stress of 117.5 MPa and a longitudinal splitting stress of 402 MPa. These specific failure modes could not be recognized using the procedure of Ref. [1]. However, the prediction of Ref. [1] indicated that the laminate yielded at 64 MPa and had its first-ply failure strength of 212.3 MPa. According to these observations, it appears as if the initial cracking occurs between the laminate yielding and the first ply failure. Problem 13 [( ± 45°)s E-glass/MY750/HY917/DY063 laminate subjected to combined ␴yy and ␴xx loads (␴yy /␴xx = 1)]. Somewhat similar to problem 11, the predicted curves for this

1058

Fig. 13. Measured and predicted yy vs. yy and yy vs. xx curves for [ ± 45°]s laminate under biaxial tensions (yy /xx = 1/1). Material type: E-Glass/MY750.

problem were also essentially linear up to failure, in reasonable agreement with the experimental data (without the liner) which showed slightly non-linear before failure (Fig. 13). The laminate yielding was estimated at yy = 47.2 MPa [1], whereas the observed initial cracking was at 60 MPa. Again, the initial cracking occurred after the laminate yielding. The theory was unable to predict leakage failure and also under-predicted the measured fibre tension failure strength. Problem 14 [( ± 45°)s E-glass/MY750/HY917/DY063 laminate subjected to combined ␴yy and ␴xx loads (␴yy /␴xx = –1)]. The theoretical predictions and test results for this problem are graphed in Fig. 14. Grossly, the predictions agree reasonably with the experimental data. However, the predicted stress–strain curves in the non-linear region up to failure were stiffer than the counterparts of the experiment. It seemed that a higher yield strength had been used in the prediction. The largest difference occurred in the ultimate strength, the predicted one being about 30% higher than the measured value. Furthermore, the majority of the predicted curves can be approximated using two linear segments, whereas the measured curves are more ductile especially in the region before failure. This may be attributed to the fact that other factors, in addition to the resin plasticity, also contributed to the composite non-linear response. Apparent such factors include: resin microcracking and fiber-resin interface debonding. More over, the glass fiber bundle can also display some non-linear deformation before failure [5]. However, in the present simulation, only the resin plasticity has been incorporated. The other factors, which are more difficult to be described quantitatively, were neglected in the simulation.

1059 2.1. Summary comments It has been pointed out that the thermal residual stresses might have affected the prediction accuracy for some problems (see the discussions for problem 8). On the other hand, such error could possibly occur only when a significant difference exists for the predictions with and without thermal residual stresses. Thus, for example, the difference between the theory and the experiments shown in Fig. 4 cannot be attributed to the effect of thermal residual stresses. Further research is required to establish a criterion to make sure that an additional application of the external load is in loading or is in unloading to the resin when both the existing and the additional load caused stresses in the material are multiaxial. Except for problems 6 and 8, the bridging model predictions for all of the remaining exercise problems were reasonable, either in reasonably good agreement with the test results (for most of the remaining problems) or having considerable reasons in the discrepancy between the theory and the experiments (for, e.g., problem 4). Considering that the thermal effect might have not been correctly accounted for in problem 8, there remains only one problem, problem 6, for which poor correlation of the theory with the provided experimental data has been found and no apparent reason can be attributed to. More test results for this problem are apparently required. Except for problem 12, the experimental data for all of the other 13 exercise problems were obtained based on tubular specimens by applying internal liquid pressure combined with either axial or torsional load. Thus, edge delamination was generally not observed in such experiments. The problem 12 was characterized using cross-plied laminate coupons subjected to uniaxial tension. For such kind of laminates, no delamination has been found [6]. Hence,

Fig. 14. Measured and predicted yy vs. yy and yy vs. xx curves for [ ± 45°]s laminate under biaxial stresses (yy /xx = 1/–1). Material type: E-Glass/MY750.

1060 it is not necessary to account for the effect of any delamination in the present 14 exercise problems. Although some other defects such as those listed in the discussions for problem 14 may have influenced the prediction accuracy for some composites, the incorporation of them into the present bridging model is not easy. A main reason is that those defects are case-sensitive, dependent on the fabrication of the composites. Therefore, except for a proper criterion should be added to check whether an incremental application of the external mechanical load to the composite will result in a loading or unloading to the resin material when it has already been yielded due to the thermal residual stresses, the bridging model simulation procedure described in Ref. [1] can be regarded as fairly mature and robust for use by designers. Herein, the constitutive description of the Prandtl-Reuss theory for the constituents especially for the resin material of a specific composite has been implicitly assumed to be correct. If another theory is more suitable, an adaptation must be made.

3. Free of thermal residual stresses As aforementioned, the thermal residual stresses can possibly affect the overall response of a resulting composite. On the other hand, the exact incorporation of those stresses into the analysis still remains a question. Thus, it is natural to ask in which case the effect of the thermal residual stresses will be estimated significant? A simple answer can be obtained by comparing the predictions of each other with and without the thermal residual stresses. The prediction with the thermal residual stresses can be made using the procedure described in Ref. [1], whereas that without their influence can also be simply achieved using the same procedure (the same software). If both the predictions for the same problem are comparable, one can make a conclusion that the thermal residual stresses have no remarkable effect for the considered problem. In this section, the predictions without thermal residual stresses for all of the 14 exercise problems are made, and are compared with the experiments and with the predictions having the thermal residual stresses involved. It must be pointed out that the input data for the prediction without the thermal residual stresses are different from those with the thermal residual stresses, not only in the stress-free temperature value but also in the constituent strength data in general. This is especially true if the constituent strengths are retrieved from the ultimate strengths of a composite, such as a UD lamina. The reason is that the lamina must be considered to have the same stress free temperature as that of the composite laminate under consideration. Thus, for example, either both the lamina and the laminate should be considered as being fabricated at a high temperature (with the thermal residual stresses involved) or both of them should be regarded as being made at room temperature (without accounting for the thermal residual stresses). This also confirms the fact that the constituent in situ properties may be somewhat different from those measured using monolithic specimens. Suppose that the stress-free temperature of the UD lamina made of each of the four material systems, i.e. the E-glass 21
K43 Gevetex and LY556/HT907/DY063 epoxy (the first material system), the Silenka E-glass 1200tex and MY750/HY917/DY063 epoxy (the second material system), the graphite AS4 and 3501–6 epoxy (the third material system), and the graphite T300 and BSL914C epoxy (the fourth material system), is the same as the room temperature at which the laminate responses are obtained. Further, let all of the other material

1061 data except for the ultimate strengths as well as the bridging parameters be the same as used in Ref. [1]. The strengths of each fiber and resin materials can be retrieved from the given lamina uniaxial strengths [3]. The retrieval is accomplished by applying the ultimate uniaxial loads to the respective UD lamina. For example, when applying a longitudinal tension of 1140 MPa, a longitudinal compression of (–570) MPa, a transverse tension of 35 MPa, a transverse compression of (–114) MPa, and an in-plane shearing of 72 MPa to the UD lamina made of the first material system with a fiber volume fraction of Vf = 0.62, the calculated maximum or minimum stresses in the fiber and resin materials without the effect of a temperature variation are:  fmax = 1808.9 MPa and mmax = 48.7 MPa (corresponding to the longitudinal tension),  fmin = –898.2 MPa and mmin = –34.6 MPa (to the longitudinal compression),  fmax = 43.8 MPa and mmax = 20.7 MPa (to the transverse tension),  fmin = –90.1 MPa and mmin = –66.6 MPa (to the transverse compression), and  fmax = 94.9 MPa and mmax = 34.7 MPa (to the in-plane shearing). Choosing the largest and the smallest values of them, the fiber and the resin tensile and compressive strengths are determined as:  fu = 1808.9 MPa,  fu,c = 898.2 MPa, mu = 48.7 MPa, and mu,c = 66.6 MPa. The fiber and resin strengths of the other three material systems are determined similarly. Results are summarized in Tables 1–4, respectively. It deserves mentioning once more that the constituent strengths determined above were based on free of thermal residual stresses in the lamina, which are different from those determined in Ref. [1] using the same procedure but with the incorporation of thermal residual stresses in the lamina. For the first material system, for example, the presently determined constituent strengths are:  fu = 1808.9 MPa,  fu,c = 898.2 MPa, mu = 48.7 MPa, and mu,c = 66.6 MPa (Table 1 of the present paper), whereas those determined in Ref. [1] are:  fu = 1804.1 MPa,  fu,c = 908.9 MPa, mu = 56.5 MPa, and mu,c = 55.7 MPa (Table 1 of Ref. [1]). The differences are apparent. Comparing Tables 1–4 with those in Ref. [1], it can be seen that the fiber strengths are slightly affected by the thermal residual stresses, to only a little bit extent. On the other hand, without the thermal residual stresses the retrieved resin tensile strengths were lower and the resin compressive strengths were higher than the corresponding counterparts with the thermal residual stresses. These results suggest that if the composite failure is essentially governed by the fiber failure, the composite strength will be almost independent of the thermal residual stresses. However, if the composite failure is essentially controlled by the resin fracture, the composite strengths with and without the thermal residual stresses may possibly be distinctly different. The material data of Tables 1–4 were first used to estimate the stress–strain responses of four types of UD laminae made from the respective material systems. The estimated longitudinal tensile responses of the four UD laminae were exactly the same as the longitudinal tensile stress–strain relationships plotted in Fig. 7 of Ref. [1], which were obtained based on the constituent parameters with the thermal residual stresses. This is unsurprising, because the lamina longitudinal strengths had been used to determine the constituent strengths in both the cases (with and without the thermal residual stresses). However, the estimated transverse tensile and in-plane shear responses of the laminae with and without the thermal residual stresses were slightly different, as shown in Figs. 15 and 16 for the laminae made of the first material system, in Figs. 17 and 18 for the laminae made of the second material system, in Figs. 19 and 20 for the laminae made of the third material system, and in Figs. 21 and 22 for the laminae made of the fourth material system. In the figures, the stress–strain curves designated as “with thermal effect” were obtained based on

1062 Table 1 Properties of E-glass 21
K43 Gevetex and LYS56/HT907/DY063 epoxy UD lamina (fiber volume fraction: Vf = 0.62; stress-free temperature: 25°C; working temperature: 25°C; bridging parameters used: # = 0.45 and  = 0.35). Properties

E11 (GPa) E22 (GPa) G12 (Gpa) v12 (Y )1 (MPa)a (Y )2 (MPa)a (Y )3 (MPa)a (Y )4 (MPa)a (Y )5 (MPa)a (Y )6 (MPa)a (Y )7 (MPa)a (Y )8 (MPa)a (ET )1 (MPa)a (ET )2 (MPa)a (ET )3 (MPa)a (ET )4 (MPa)a (ET )5 (MPa)a (ET )6 (MPa)a (ET )7 (MPa)a Lu(MPA)b Lu,c(MPa)c Tu(MPa)d Tu,c(MPa)e Su(MPa)f 11 (%)b 11,c (%)c 22 (%)d 22,c (%)c 212 (%)f

Lamina

Fiber

Resin

Provided

Predicted

Provided

Used

Provided

Used

53.48 17.7 5.83 0.278 – – – – – – – – – – – – – – – 1140 570 35 114 72 2.132 1.065 0.197 0.644 3.8

50.87 14.38 5.72 0.257 – – – – – – – – – – – – – – – 1140 570 81.5 114 102.1 2.262 1.124 0.691 1.680 11.861

80 80 33.33 0.2 – – – – – – – – – – – – – – – 2150 1450 – – – – – – – –

80 80 33.33 0.2 – – – – – – – – – – – – – – – 1808.9 898.2 – – – – – – – –

3.35 3.35 1.24 0.35 – – – – – – – – – – – – – – – 80 120 – – – 5 – – – –

3.35 3.35 1.24 0.35 31.9 38.4 44.7 49.9 53.6 56.1 58.1 60.0 1566 1337 944 584 338 245 197 48.7 66.6 – – – – – – – –

a

Assumed to be the same in both tension and compression. Longitudinal tension. c Longitudinal compression. d Transverse tension. e Transverse compression. f In-plane shearing. b

the material data specified in Tables 1–4 of Ref. [1], whereas those designated as “without thermal effect” were estimated using the data in Tables 1–4 of the present paper. It is seen that the predictions without the thermal residual stresses in the non-linear region were slightly stiffer than those with the thermal effect for the lamina transverse tension and in-plane shear responses. The differences for the transverse responses are comparably more whereas for the

1063 Table 2 Properties of Silenka E-glass 1200tex and MY750/HY917/DY063 epoxy UD lamina (fiber volume fraction: Vf = 0.60; stress-free temperature: 25°C; working temperature: 25°C; bridging parameters used: # = 0.45 and  = 0.35). Properties

E11 (GPa) E22 (GPa) G12 (Gpa) v12 (Y )1 (MPa)a (Y )2 (MPa)a (Y )3 (MPa)a (Y )4 (MPa)a (Y )5 (MPa)a (Y )6 (MPa)a (Y )7 (MPa)a (Y )8 (MPa)a (ET )1 (MPa)a (ET )2 (MPa)a (ET )3 (MPa)a (ET )4 (MPa)a (ET )5 (MPa)a (ET )6 (MPa)a (ET )7 (MPa)a Lu(MPA)b Lu,c(MPa)c Tu(MPa)d Tu,c(MPa)e Su(MPa)f 11 (%)b 11,c (%)c 22 (%)d 22,c (%)c 212 (%)f

Lamina

Fiber

Resin

Provided

Predicted

Provided

Used

Provided

Used

45.6 16.2 5.83 0.278 – – – – – – – – – – – – – – – 1280 800 40 145 73 2.087 1.754 0.246 1.2 4

45.74 13.45 5.31 0.26 – – – – – – – – – – – – – – – 1280 800 88.9 145 111.2 2.834 1.761 0.829 4.569 15.845

74 74 30.8 0.2 – – – – – – – – – – – – – – – 2150 1450 – – – – – – – –

74 74 30.8 0.2 – – – – – – – – – – – – – – – 2097.3 1303.4 – – – – – – – –

3.35 3.35 1.24 0.35 – – – – – – – – – – – – – – – 80 120 – – – 5 – – – –

3.35 3.35 1.24 0.35 32.6 39.9 46.8 52 55.6 58 60.1 62.0 1698 1387 918 542 317 244 186 54 85.6 – – – – – – – –

a

Assumed to be the same in both tension and compression. Longitudinal tension. c Longitudinal compression. d Transverse tension. b

in-plane shear responses are very little. The measured in-plane shear stress–shear strain relationships for the four laminae provided in Ref. [3] are also shown on the respective figures. The predicted curves with and without the thermal residual stresses essentially coincided with the measured ones. However, the predicted curves were longer and stopped at higher failure stresses. The reason is, as pointed out earlier, that the lamina failure under the in-plane shear load was controlled by the resin tensile strength, which had been retrieved from the lamina longitudinal tensile strength.

1064 Table 3 Properties of graphite AS4 and 3501-6 epoxy UD lamina (fiber volume fraction: Vf = 0.60; stress-free temperature: 25°C; working temperature: 25°C; bridging parameters used: # = 0.45 and  = 0.3). Properties

E11 (GPa) E22 (GPa) G12 (Gpa) v12 (Y )1 (MPa)a (Y )2 (MPa)a (Y )3 (MPa)a (Y )4 (MPa)a (Y )5 (MPa)a (Y )6 (MPa)a (Y )7 (MPa)a (Y )8 (MPa)a (ET )1 (MPa)a (ET )2 (MPa)a (ET )3 (MPa)a (ET )4 (MPa)a (ET )5 (MPa)a (ET )6 (MPa)a (ET )7 (MPa)a Lu(MPA)b Lu,c(MPa)c Tu(MPa)d Tu,c(MPa)e Su(MPa)f 11 (%)b 11,c (%)c 22 (%)d 22,c (%)c 212 (%)f

Lamina

Fiber

Resin

Provided

Predicted

Provided

Used

Provided

Used

126 11 6.6 0.28 – – – – – – – – – – – – – – – 1950 1480 48 200 79 1.38 1.175 0.436 2.0 2

136.7 9.23 5.54 0.256 – – – – – – – – – – – – – – – 1950 1480 70 200 108.3 1.430 1.084 0.788 3.742 3.77

225 15 15 0.2 – – – – – – – – – – – – – – – 3350 2500 – – – – – – – –

225 15 15 0.2 – – – – – – – – – – – – – – – 3216.5 2438.3 – – – – – – – –

4.2 4.2 1.567 0.34 – – – – – – – – – – – – – – – 69 250 – – 50 1.7 – – – –

4.2 4.2 1.567 0.34 38.1 41.8 46.1 50.1 54.0 57.6 61.2 64.6 2507 2530 2072 1721 1409 1202 991 50.3 130.1 – – – – – – – –

a

Assumed to be the same in both tension and compression. Longitudinal tension. c Longitudinal compression. d Transverse tension. e Transverse compression. b

Next, predictions for all of the 14 exercise problems were made, are graphed in Figs. 1–14, respectively, designated as “theory without thermal residual stresses (or without thermal effect)”, and are compared with the results obtained in Ref. [1]. It is seen from those figures that distinct differences between the predictions with and without the thermal residual stresses only occurred in problems 6–8. Even for the problem 6, majority of the failure envelope with the thermal residual stresses almost coincided with that without the thermal residual stresses. The biggest difference occurred in problem 8, where the predicted strength, 480.3 MPa,

1065 Table 4 Properties of graphite T300 and BSL914C epoxy UD lamina (fiber volume fraction: Vf = 0.60; stressfree temperature: 25°C; working temperature: 25°C; bridging parameters used: # = 0.45 and  = 0.35). Properties

E11 (GPa) E22 (GPa) G12 (Gpa) v12 (Y )1 (MPa)a (Y )2 (MPa)a (Y )3 (MPa)a (Y )4 (MPa)a (Y )5 (MPa)a (Y )6 (MPa)a (Y )7 (MPa)a (Y )8 (MPa)a (ET )1 (MPa)a (ET )2 (MPa)a (ET )3 (MPa)a (ET )4 (MPa)a (ET )5 (MPa)a (ET )6 (MPa)a (ET )7 (MPa)a Lu(MPA)b Lu,c(MPa)c Tu(MPa)d Tu,c(MPa)e Su(MPa)f 11 (%)b 11,c (%)c 22 (%)d 22,c (%)c 212 (%)f

Lamina

Fiber

Resin

Provided

Predicted

Provided

Used

Provided

Used

138 11 5.5 0.28 – – – – – – – – – – – – – – – 1500 900 27 200 80 1.087 0.652 0.245 1.818 4

139.6 9.09 5.04 0.26 – – – – – – – – – – – – – – – 1500 900 58.8 200 81.8 1.075 0.645 0.65 7.938 4.525

230 15 15 0.2 – – – – – – – – – – – – – – – 2500 2000 – – – – – – – –

230 15 15 0.2 – – – – – – – – – – – – – – – 2471.8 1482.8 – – – – – – – –

4.0 4.0 1.481 0.35 – – – – – – – – – – – – – – – 75 150 – – 70 4 – – – –

4.0 4.0 1.481 0.35 41.6 29.6 55.8 59.9 63.1 66.3 68.9 71.4 2015 1384 769 548 457 324 275 42.3 127.2 – – – – – – – –

a

Assumed to be the same in both tension and compression. Longitudinal tension. c Longitudinal compression. d Transverse tension. e Transverse compression. f In-plane shearing. b

without the thermal residual stresses was 72% higher than the predicted strength, 279 MPa, with the thermal residual stresses. Even though this extreme difference has been found, a gross conclusion would be that the thermal residual stresses can be neglected in most cases for the composite laminates made of glass/graphite fibers and epoxy resin materials. If we recognize the fact that in and only in problems 6–8 were the laminates made from the third material system, i.e. graphite AS4 and 3501-6 epoxy, a more concise conclusion can be

1066

Fig. 15. Stress–strain responses of UD composites made of the material system 1 (given in Table 1) with and without thermal residual stresses subjected to transverse load up to failure. Different fiber volume fractions were used.

Fig. 16. Stress–strain responses of UD composites made of the material system 1 (given in Table 1) with and without thermal residual stresses subjected to in-plane shear load up to failure. Different fiber volume fractions were used. Measured data [3] for the lamina of Vf = 0.62 are also shown.

1067

Fig. 17. Stress–strain responses of UD composites made of the material system 2 (given in Table 2) with and without thermal residual stresses subjected to transverse load up to failure. Different fiber volume fractions were used.

Fig. 18. Stress–strain responses of UD composites made of the material system 2 (given in Table 2) with and without thermal residual stresses subjected to in-plane shear load up to failure. Different fiber volume fractions were used. Measured data [3] for the lamina of Vf = 0.6 are also shown.

1068

Fig. 19. Stress–strain responses of UD composites made of the material system 3 (given in Table 3) with and without thermal residual stresses subjected to transverse load up to failure. Different fiber volume fractions were used.

Fig. 20. Stress–strain responses of UD composites made of the material system 3 (given in Table 3) with and without thermal residual stresses subjected to in-plane shear load up to failure. Different fiber volume fractions were used. Measured data [3] for the lamina of Vf = 0.6 are also shown.

1069

Fig. 21. Stress–strain responses of UD composites made of the material system 4 (given in Table 4) with and without thermal residual stresses subjected to transverse load up to failure. Different fiber volume fractions were used.

Fig. 22. Stress–strain responses of UD composites made of the material system 4 (given in Table 4) with and without thermal residual stresses subjected to in-plane shear load up to failure. Different fiber volume fractions were used. Measured data [3] for the lamina of Vf = 0.6 are also shown.

1070 drawn. Only for these laminates, the predicted results with and without thermal residual stresses have shown significant differences. No remarkable effect has been found for the laminates made from the other three material systems. It may be noted that the other three material systems had a fabrication temperature of 120°C (or, a temperature variation of T = 95°C) whereas the third material system experienced a temperature variation of T = 152°C. Thus, a reasonable conclusion might be that if a composite undergoes a temperature variation no more than 100°C, the resulting thermal residual stresses can be neglected. Otherwise, a comparison with and without the thermal effect should be estimated, before going into accurate calculation for the composite response. Finally, it should be pointed out that predictions without thermal residual stresses were also made in Ref. [1] for some composites. However, those predictions were incorrect/inaccurate, because the constituent strength parameters used there were obtained from the lamina uniaxial strengths with the thermal residual stresses. For Problem 1, for example, the composite strength subjected to the combined transverse and in-plane shear loads, yy and xy , is essentially controlled by the resin tensile or compressive strength. Thus, with mu = 48.7 MPa and mu,c = 66.6 MPa (Table 1) and by setting T = 0, the predicted failure envelope was plotted in Fig. 1 in dashed line designated as “without thermal residual stresses.” On the other hand, taking mu = 56.5 MPa, and mu,c = 55.7 MPa (Table 1 of Ref. [1]) and by setting T = 0, the predicted failure envelope had been graphed in Fig. 16 of Ref. [1] also in dashed line designated as “without thermal stresses”. The envelope in Fig. 1 of the present paper is correct whereas the envelope (“without thermal stresses)” shown in Fig. 16 of Ref. [1] is wrong. The reason is that the resin strengths, mu = 56.5 MPa, and mu,c = 55.7 MPa, used there were not the strengths “without thermal residual stress effect”. As a result, the constituent (resin) strengths measured from monolithic specimens should not be directly used in simulation unless the thermal effect has been fully and correctly taken into account. As a final note, a comparison study between the current predictions and those of other theories has been made in Ref. [7]. The study highlights the maturity of the various existing models, including the present one.

4. Conclusion Comparison of the theoretical predictions with experiments for the biaxial in-plane strengths of a variety of composite laminates indicates that the bridging micromechanics model is a useful tool for composite failure analysis. The most potential feature is that this model only requires the material properties of the constituents and the laminate geometric parameters, which can be easily determined or just taken as designing variables. The discrepancy in the bridging model predictions with the experiments for a limited number of problems under consideration may be attributed to insufficient experimental data or to the inaccurate incorporation of thermal residual stresses. Further correlation of the bridging model predictions with experimental measurements for the ultimate strengths of fibrous laminates subjected to not only in-plane but also lateral loads would be more beneficial to applications. Such a correlation could be achieved through an additional part of the failure exercise such as Part C, possibly in the future.

1071 References [1] Huang ZM. A bridging model prediction of the ultimate strength of composite laminates subjected to biaxial loads. Compos Sci Technol 2004;64:395–448, see also Chapter 3.16 in this book. [2] Huang ZM. Simulation of the mechanical properties of fibrous composites by the bridging micromechanics model. Composites Part A 2001;32(2):143–72. [3] Soden PD, Hinton MJ, Kaddour AS. Lamina properties, lay-up configurations and loading conditions for a range of fiber-reinforced composite laminates. Compos Sci Technol 1998;58:1011–22. [4] Soden PD, Hinton MJ, Kaddour AS. Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates: Failure exercise benchmark data 2002;62:1489–1514, see also Chapter 2.2 in this book. [5] Huang ZM. Fatigue life prediction of a woven fabric composite subjected to biaxial cyclic loads. Composites Part A 2002;33(2):253–66. [6] Huang ZM. Micromechanical life prediction for composite laminates. Mechanics of Materials 2001;33(4):185–99. [7] Hinton MJ, Kaddour AS, Soden PD. A further assessment of the predictive capabilities of current failure theories for composite laminates: comparison with experimental evidence. Compos Sci Technol 2004;64:449–588, see also Chapter 6 in this book.

Section 6 Predictive Capabilities of Nineteen Failure Theories and Design Methodologies for Polymer Composite Laminates: Part B: Comparison With Experiments

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 Elsevier Ltd. All rights reserved

1073

CHAPTER 6.1

Predictive capabilities of nineteen failure theories and design methodologies for polymer composite laminates. Part B: Comparison with experiments A. S. Kaddoura, M. J. Hintonb and P. D. Sodenc a b

QinetiQ, Farnborough, UK QinetiQ, Fort Halstead, UK c UMIST, Manchester, UK

Abstract This article draws together the results from a co-ordinated study, known as the World-Wide Failure Exercise, whereby nineteen of the leading theories for predicting failure in composite laminates have been tested against experimental evidence. The comparison has been effected through 14 carefully selected Test Cases, which include biaxial strength envelopes for a range of unidirectional and multidirectional laminates, and stress-strain curves for a range of multi-directional laminates, loaded under uniaxial or biaxial conditions. The predictions were provided by the originators of the theories, not by third parties, and were made without prior access to the experimental results. In a later and completely separate stage of the exercise the contributors had an opportunity to modify and refine their theories in the light of the experimental results. The predictions and experimental data have been compared in a systematic and detailed manner, to identify the strengths and weaknesses of each theory and provide a ranking of the overall effectiveness of each theory. Careful reading of this analysis and study of the figures provided will inform the reader when selecting an appropriate failure theory for use in a given design situation. It will also provide qualitative and quantitative assessments of the likely accuracy and reliability of the prediction in a given circumstance.

1. Introduction The editors have organised the World-Wide Failure Exercise (WWFE), with the following aims in mind: • establishing the current level of maturity of theories for predicting the failure response of fibre reinforced plastic (FRP) laminates; • closing the knowledge gap between theoreticians and design practitioners in this field; • stimulating the composites community into providing design engineers with more robust and accurate failure prediction methods, and the confidence to use them.

1074 Details of 14 Test Cases and the associated experimental results that were selected to challenge the current failure theories were given in Section 2 (Chapter 2.1). As described in Section 1 a two-stage approach was adopted and the two stages are referred to as Part A and Part B. In Part A, described fully in Section 3, 19 theories and predictions for the 14 Test Cases were presented by their originators. The organisers (i.e. the authors of this book) have compared all of the theories and their theoretical predictions with one another, and the findings were reported in Section 4. At this stage in the study, the contributors had no knowledge of the experimental results. The purpose of Part B was to compare the theoretical predictions with experimental results and to allow further opportunity for the originators of the theories to provide clarifying remarks. In Section 5 the originators of the theories commented on the correlation between their predictions and the experimental results, and some of them introduced modifications to their theories. This chapter compares the predictions of all 19 theories with the experimental results for the 14 Test Cases. It also offers a qualitative and quantitative assessment of the robustness and accuracy of each of the theories.

2. The experimental results At the heart of the ‘exercise’ is the need for definitive experimental evidence against which the features of any theory could be compared in a rigorous manner. Every effort was made by the present authors (the organisers) to assemble an experimental data set of the highest pedigree during the early stages of the ‘exercise’. Steps taken included: • a thorough search of the research literature; • notices in certain journals and key conferences calling for good test data to be sent to the organisers. After a considerable investigation, suitable data sets were identified, Ref. [17], which were believed (and indeed the ‘exercise’ has proved) to offer a severe challenge to the theoreticians. Out of this emerged the 14 Test Cases (summarised in Table 1), around which the ‘exercise’ has been built. Almost all of the experimental data were obtained from tests on tubular specimens, except that for Test Case 12, where the 0°/90° laminates employed were in the form of coupons. Although the number of Test Cases is relatively small, they serve to highlight a wide range of fundamental issues and physical realities associated with the response of composite materials when subjected to destructive testing. Within each individual Test Case, a number of separate features or characteristics are exhibited. Each feature is listed in Table 4 and will be used later in Appendix III as a point of comparison against which a theory is judged. For the sake of brevity, the features are designated in the form TCi-j where TC stands for Test Case, i ( = 1, 2, 3, . . . , 14) refers to Test Case number and j indicates the feature number. The organisers believe that the experimental data sets that were finally adopted, represented the best available for the purpose. Nevertheless, certain inadequacies were exposed through the ‘exercise’ and these are discussed in detail in Section 6.1, and summarised in Table 3.

1075 3. Revisions in the theories (contained in the Part B papers) Some of the originators of failure theories, listed in Table 2, have made attempts to modify their theories during the second phase (Part B) of the exercise. Part B was initiated by providing the participants in Part A with a data pack, Ref. [17], containing the experimental results for each of the Test Cases. In light of this information and (probably) in the search for a better correlation with the 14 Test Cases listed in Table 1, eleven of the participants provided revised theories in Part B. The key changes are summarised as follows: • Tsai, Ref. [27], introduced a new method of analysing the post-initial failure response. He also showed the influence of the various empirical constants used in the revised theory on the predictions made for certain Test Cases. • Wolfe and Butalia, Ref. [28], submitted revised predictions for the majority of the Test Cases. They studied the effects of relaxing some of the empirically tuned parameters embedded in their theory. • Edge, Ref. [21], made revisions to the failure criteria he used in Part A which included dropping the shear-compression interaction theory and the delamination criterion. • Rotem, Ref. [25], attempted to consider the effects of matrix degradation on the stress– strain curves, whereas he had originally not addressed the issue in Part A. • Sun, Ref. [26], made minor corrections to the stress–strain curves obtained from the linear analysis.

Table 1 Details of the laminates and loading (test) cases. Test Case

Laminate lay-up

Material

Description of Required Prediction

1* 2* 3* 4* 5* 6* 7 8 9* 10 11 12 13 14 15**

0° 0° 0° ( ± 30°/90°) ( ± 30°/90°) (0°/ ± 45°/90°) (0°/ ± 45°/90°) (0°/ ± 45°/90°) ± 55° ± 55° ± 55° (0°/90°) ± 45° ± 45° ± 55°

E-glass/LY556 epoxy T300/BSL914C carbon/epoxy E-glass/MY750 epoxy E-glass/LY556 epoxy E-glass/LY556 epoxy AS4/3501–6 carbon/epoxy AS4/3501–6 carbon/epoxy AS4/3501–6 carbon/epoxy E-glass/MY750 epoxy E-glass/MY750 epoxy E-glass/MY750 epoxy E-glass/MY750 epoxy E-glass/MY750 epoxy E-glass/MY750 epoxy E-glass/MY750 epoxy

y versus xy envelope x versus xy envelope y versus x envelope y versus x envelope x versus xy envelope y versus x envelope Stress-strain curves for y : x = 1 : 0 Stress-strain curves for y : x = 2 : 1 y versus x envelope Stress-strain curves for y : x = 1 : 0 Stress-strain curves for y : x = 2 : 1 Stress-strain curves for y : x = 0 : 1 Stress-strain curves for y : x = 1 : 1 Stress-strain curves for y : x = 1 :–1 Stress variation and development of failure in a tube under pressure

* Biaxial failure stress envelopes under a wide range of biaxial stress ratios. ** This Test Case was provided only in Part B whereas the rest were provided in Part A.

1076

Table 2 A summary of the Test Cases where modifications were introduced in Part B papers. Contributor(s)

Bogetti et al., Refs [32,36] Chamis et al., Refs [3,19] Cuntze, Refs [33,37] Eckold, Refs [4,20] Edge, Refs [5,21] Hart-Smith, Refs [8,22] Hart-Smith, Refs [7,22] Hart-Smith, Refs [18,22] Huang, Refs [35,39] Mayes and Hansen, Refs [34,38] McCartney, Refs [6,23] Puck and Schuermann, Refs [9,24] Rotem, Refs [10,25] Sun et al., Refs [11,26] Sun et al., Refs [11,26] Tsai et al., Refs [12,27] Wolfe and Butalia, Refs [13,28] Zinoviev et al., Refs [14,29]

Approach represented

Maximum strain theory –ICAN and –CODSTRAN Failure modes Concept (FMC) British Standard pressure vessel codes In-house design method Generalised Tresca theory Maximum strain theory Ten-per-cent rule Micromechanics, Generalised max stress Multi-continuum theory Physically based ‘Damage Mechanics’ 3-D phenomenological model Interactive matrix and fibre failure theory Linear analysis Non-linear (FE based) analysis Interactive quadratic failure criterion Maximum strain energy method Development of Maximum stress theory

Theory Designation

Bogetti Chamis Cuntze Eckold Edge Hart-Smith(1) Hart-Smith(2) Hart-Smith(3) Huang Mayes McCartney Puck Rotem Sun (L) Sun (NL) Tsai Wolfe Zinoviev

Test Cases : A: blind submission, *no submission, B: modified or a new submission in Part B. 1

2

3

4

5

6

7

8

9

10 11 12 13 14

A A B A A A A A B A * B B A * B B A

A A A * B A A A A A * A A A * B B A

A A A A B A A A A A * A B A * B B A

A A B A B A A A A A * A A A * B B A

A A B * B A A A A A * A A A * B B A

A A B * B A A A A A * A A A * B B A

A A B * A * * A A A B A A B A B B A

A A B * A * * A A A B A A B A B B A

A A B * B A A A A B * A A A * B B A

A A B B A * * A A A B A A A A B B A

A A B B A * * A A B B B A B A B B A

A A B B A * * A A B B A A B A B B A

A A B A A * * A A B B A A B A B B A

A A B * A * * A A A * A B A A B B A

1077 Table 3 Summary of problems associated with experimental results. Test Case

Description of the problems associated with experimental results

1 2 3

Variation in the uniaxial transverse compressive strength values. Large scatter in the uniaxial shear strength obtained from different laboratories. (a) Data covered mainly one tension-compression quadrant. (b) Results obtained from ± 5° rather than 0° lamina. (c) No results under biaxial compression. (a) No initial failure data provided. (b) Shortage of data in the compression quadrants. No initial failure data provided. (a) Shortage of data under compressive dominated loading, the data shown was influenced by buckling. (b) Large scatter in the uniaxial compressive strength, some results for thin tubes. (c) Shortage of information on initial failure. Not sufficient information on the initial and intermediate failures. Not sufficient information on the initial and intermediate failures. (a) Fibre volume fractions in the compression quadrants are different from those in the tension quadrants. (b) Information missing on the initial failure. No information on the initial failure. (a) No information on the initial failure. (b) Axial strain values at large strains might have some error.

4 5 6

7 8 9

10 11 12 13

14

(a) Final failure points are not obtained. (b) Relatively short specimens used. (c) Strains in x and y directions are not coincident. No information provided on the initial failure.

• Eckold, Ref. [20], terminated some of the stress–strain curves at an arbitrary stress level instead of the arbitrary strain levels used in Part A, Ref. [4]. • In Part A, McCartney, Ref. [6], provided predictions for only two out of the fourteen Test Cases. In Part B he extended the applicability of his damage mechanics approach from cross-plied laminates to include angle ply and quasi-isotropic laminates, Ref. [23]. As a result, solutions were provided for Test Cases requiring stress–strain curves, which had not been possible in Part A. He also introduced the maximum fibre strain limit as a criterion for fibre dominated failure, whereas no final failure criterion had been offered in Part A. • Hart-Smith, Ref. [22], briefly considered the effects of various methods of reducing failed lamina moduli on the behaviour of the stress–strain curves for Test Case No 10 and 11. • Puck, Ref. [24], discussed some of the parameters affecting the stress–strain curves for Test Case No 11 and commented on an error in the results that he had presented in Part A. Also, a revised prediction was provided for Test Case No 1, using a more representative data set. • Cuntze, Ref. [37], made major changes to his modelling procedures, including:

Test Case

Feature No

Description

1*

TC1-1 TC1-2

The shear failure stress is decreased due to application of transverse tensile stress component and vice versa. The shear strength increases above the uniaxial value due to application of moderate compressive transverse stress component but decreases when large compressive stresses are applied. There is a point at which the shear failure stress reaches a maximum value. The envelope passes through the uniaxial shear as well as the transverse tensile and compressive strength values issued by the organisers. The addition of longitudinal tensile stress seems to produce an increase in the shear failure stress, whereas the addition of a shear stress does not lead to an increase in the longitudinal tensile failure stress. The addition of longitudinal compressive stresses appears to reduce the shear strength to lower than the uniaxial strength value. Under combined longitudinal tension and transverse compression, the application of longitudinal tensile stress component tends to reduce the transverse compressive strength and vice versa. There is no recorded enhancement in the biaxial failure stresses. The envelope passes through the four uniaxial strengths issued by the organisers (i.e. the longitudinal and transverse tensile and compressive strengths values). The biaxial strength under combined tension-compression loading is lower than the uniaxial strength values. There is tendency of an enhancement in the biaxial tensile strength, i.e. the maximum biaxial strength is slightly higher than the uniaxial strength values. The application of a shear stress to this laminate reduced the axial strength, i.e the direct strength in the presence of shear loading is always lower than the uniaxial axial strength. When moderate axial tensile stress (up to 0.6 of the uniaxial strength) is applied the shear strength is approximately equal to or slightly higher than the uniaxial shear strength. The biaxial strengths are slightly higher than the uniaxial strength values. In the tension-compression quadrants once stress in the x-direction is applied, the strength in y-direction is reduced below the uniaxial strength and vice versa. The measured stress strain curves exhibited relatively linear behaviour up to final failure (i.e with no pronounced steps). Final failure took place by fibre tension fracture. Both the axial and hoop strain are positive (tensile). The curves exhibited a slight change in the slope at around 400Mpa hoop stress. The hoop curves exhibited a change in the stiffness near the final points.

TC1-3 2*

TC2-2

3*

TC2-3 TC3-3 TC3-4

4*

TC4-2 TC4-3

5*

TC5-1 TC5-2

6*

TC6-3 TC6-4

7

TC7-1 TC7-3 TC8-1 TC8-2 + TC8-3 +

8

1078

Table 4 Features in the experimental results for the Test Cases studied.

Table 4 Continued. Feature No

Description

9

TC9-2

There are two envelopes (a) one representing weeping (leakage of liquid through the wall thickness in thin, unlined tubes) and (b)another representing final failure. The test data, for both the leakage stresses and final failure stresses, in the tension-tension quadrant show that the biaxial strength is larger than the uniaxial strength. The data in the compression-compression quadrant also showed an enhancement in the biaxial compressive strength. The test results are adequate to describe fracture of specimens tested under both biaxial tension and biaxial compression, with no sign of an open envelope. The behaviour is non-linear and the slope decreases gradually from low strains right up to failure. The specimens tested with no liner failed by weepage (at a stress of around 386Mpa) + . The specimens tested with liner failed (at a stress of 600Mpa) + . The failure strains were very high (greater than 8% in the hoop direction and greater than 10% in the axial direction). The behaviour of the hoop stress-hoop strain curve is relatively linear up to final failure. The behaviour of the hoop stress-axial strain curve showed severe non-linearity starting at an earlier stage and there was a decrease in slope. Both the axial and hoop strains are tensile (positive). The axial strain exceeds the hoop strain at hoop stresses above 400 MPa. The specimens tested without a liner failed by weepage (at a stress of around 280 MPa) + while those tested with a liner failed by fibre fracture at a much higher stress (up to 736Mpa) + . Final failure was by fibre fracture. Initial response which was linear. Onset of transverse matrix cracking in the 90° plies (initial failure) (at 117 MPa) + . Reduction in longitudinal modulus from the initial value, accompanied by an increase in transverse crack density. Splitting in the longitudinal plies with an imperceptible change in longitudinal modulus (secondary failure) (at 402 MPa) + . Catastrophic fibre fracture and total loss of integrity (final failure) (at 609 MPa) + . An initial response which was linear-elastic. Onset of transverse matrix cracking (initial failure) starting (at a stress of about 50 MPa) + . Reduction in the slope of the stress strain curves after the cracking started, accompanied by an increase in transverse crack density.

TC9-3 TC9-4 TC9-5 10

11

TC10-1 TC10-2 TC10-3 + TC10-4 TC11-1 TC11-2 TC11-3 TC11-4 TC11-5

12

13

TC11-6 TC12-1 TC12-2 TC12-3 TC12-4 TC12-5 TC13-1 TC13-2 TC13-3

1079

Test Case

1080

Table 4 Continued. Test Case

14

Feature No

Description

TC13-4 TC13-5 + TC13-6 + TC14-1 TC14-2 TC14-3 TC14-4 + TC14-5

Unlined specimens leaked (at a stress of 216 MPa) + . For the lined specimens, the final failure was by fibre fracture at 502 MPa stress. The hoop strain at failure was slightly larger than the axial strain. The stress strain curves are highly non-linear. Above 2% strain, the strain stress curves gradually decrease in slope and become flatter than those at small strain values. No effect of the presence of a liner was observed in the tests. The final failure stress was around 95 MPa. The final failure strains reached 12%. This is much larger than the shear strain at failure given for a unidirectional lamina loaded in pure shear, which was derived from test on a circumferentially wound tube loaded in torsion.

* Biaxial failure stress envelopes under a wide range of biaxial stress ratios. + Information not used in the ranking procedure in Table C-5.

1081 – Improvement of the nonlinear analysis of the computer code. – Simplifications of the failure conditions. – Modification of the lamina stress-strain curves in the ‘softening region’. Some thoughts were also offered on how to account for the effect of radial pressure on failure and on test data evaluation. • Huang, Ref. [39], attempted to consider the effects of thermal stresses on the predicted results. He submitted a revised envelope for Test Case No 1 because he recognised that his Part A prediction was incorrect. Huang also re-analysed Test Case No 3 by taking into account the actual winding angle of the specimens, being ± 5° instead of 0°. • Mayes, Ref. [38], submitted revised curves for Test Case No 9, 11–13. The post failure modelling of Part A was modified so that more representative predictions could be obtained. • Bogetti, Ref. [36], made no detectable changes to the theoretical predictions except that some of the modes of failure reported in Fig. 3 were altered in Part B. As a consequence of these changes, the eleven participants offered revised predictions for a number of the Test Cases – a mapping is provided in Table 2. In the remainder of this chapter, a distinction will be drawn between predictions emanating from the Part A submissions (theories designated as Tsai-A, Wolfe-A, Edge-A, etc.) and those emanating from the Part B submissions (theories designated Tsai-B, Wolfe-B, Edge-B, etc). Graphs are presented for both Part A and Part B predictions, in order to facilitate comparisons. 4. Comparison between theoretical and experimental results The format of the ‘exercise’ provided the organisers with a vast amount of data (19 theories, 11 revisions, 14 Test Cases) generated through Refs [3–14, 18–29, 32–39]. The first stage in the comparison process was to draw this information together into a more tractable form. Graphs have been prepared and presented (Figs 1–14) in which the experimental results (failure envelopes and stress strain curves) for the 14 Test Cases are superimposed on the predictions from all of the theories presented in Part A and from those revised in Part B. Within those graphs, additional bar charts were constructed (Figs 7(b), 8(b), 10(b), 11(b), 12(b), 13(b) and 14(b)) to provide a simple illustration of the degree of correlation for selected Test Cases and selected stress ratios. To obtain a full appreciation of the performance of the theories the reader is encouraged to study each of these figures in detail. Having assembled the raw data, as described above, the next stage was to devise a means of drawing comparisons. The organisers have striven for a concise, traceable and objective methodology, to give a qualitative and quantitative assessment of the correlation between the individual theories and test data, and between the various theories and test data. After much debate, three distinct methods were adopted and each is described below. 4.1. Qualitative assessment of the correlation between each theory and the experimental results, taking each Test Case in turn Using the information in Figs (1–14), each theoretical prediction was compared with the experimental evidence and observations were made on the degree of matching. Weaknesses

1082

Fig. 1. Comparison between the predicted and measured biaxial failure stresses for 0° lamina made of GRP material E-Glass/LY556/HT907/DY063 (Test Case No 1). Prediction based upon original (Part A) and revised (Part B) theories.

1083

Fig. 2. Comparison between the predicted and measured biaxial failure stresses for 0° lamina made of CFRP material T300/914C (Test Case No 2).

were sub-ranked as Minor, Significant and Fundamental. The evidence is grouped by Test Case such that the reader can get a snap shot of the performance of every theory for a given Test Case. Full details and results of this assessment process are contained in Appendix A (note that Tables A-1 to A-14 provide a summary of the salient points for the 14 Test Cases). 4.2. Qualitative assessment of the correlation between the theories and experimental results, taking each theory in turn The results from the assessment described in Appendix A have been transposed and regrouped by theory, such that the reader can get a snap shot of the performance of a given

1084 Fig. 3. Comparison between the predicted and measured biaxial failure envelope of 0° GRP lamina under combined x and y stresses. Material: E-glass/ MY750 epoxy (Test Case No 3). Prediction based upon original (Part A) and revised (Part B) theories.

1085

Fig. 4. Comparison between the predicted and measured biaxial failure envelope for (90°/ ± 30°) laminate under combined x and y stresses. Material: E-glass/LY556 epoxy (Test Case No 4). Prediction based upon original (Part A) and revised (Part B) theories.

theory for every Test Case. Full details are shown in Appendix B where Tables B-1 to B-18 summarise the key features. Table B-19 shows where weaknesses were observed in all of the theories considered. Table B-20 gives the total number of Minor, Significant and Fundamental weaknesses in each theory and the theories are arranged into three groups according to the total number of Fundamental weaknesses. Table 5 (page 1105) summarises the three groupings resulting from this qualitative assessment:

1086

Fig. 5. Comparison between the predicted and measured biaxial failure stresses for (90°/ ± 30°) laminate under xy and x stresses. Material: E-glass/LY556 epoxy (Test Case No 5). Prediction based upon original (Part A) and revised (Part B) theories.

1087

Fig. 6(a). Comparison between the predicted and measured biaxial final failure stresses for (0°/ ± 45°/90°) AS4/3501-6 laminates (Test Case No 6). Prediction based upon original (Part A) and revised (Part B) theories.

Group (1), in which the theories exhibited good predictive capability, none or one fundamental weakness and many relatively minor weaknesses, contains Puck, Zinoviev, Tsai, Sun (L), Cuntze and Bogetti. Within this group Bogetti was perceived to have the largest number of Significant and Minor weaknesses, partly because of his predictions for the strength of a unidirectional lamina (see Test Case 3 and Appendices A and B). Group (2), in which the theories had a few more significant and fundamental weaknesses, includes Edge, Chamis, Wolfe, Rotem, Mayes, Huang, Hart-Smith(3).

1088 Fig. 6(b). Comparison between the predicted and measured ‘initial’ biaxial failure stresses for (0°/ ± 45°/90°) AS4/3501-6 laminates (Test Case No 6).

1089

Fig. 7(a). Comparison between the predicted and measured stress strain curves for (0°/ ± 45°/90°) AS4/3501-6 laminate under uniaxial tension (y/x = 1/0) (Test Case No 7). Prediction based upon original (Part A) and revised (Part B) theories.

1090 Fig. 7(b). A bar chart showing the ratio of predicted to measured values of failure strains, strength and modulus for (0°/ ± 45°/90°) AS4/3501-6 laminate under uniaxial tension (y/x = 1/0) (Test Case No 7). Prediction based upon original (Part A) and revised (Part B) theories.

1091

Fig. 8(a). Comparison between the predicted and measured stress strain curves for (0°/ ± 45°/90°) AS4/3501-6 laminate under biaxial stress y/x = 2/1 (Test Case No 8). Prediction based upon original (Part A) and revised (Part B) theories.

1092

Fig. 8(b). A bar chart showing the ratio of predicted to measured values of final strength for (0°/ ± 45°/90°) AS4/3501-6 laminate under uniaxial tension (y/x = 2/1) (Test Case No 8). Both original (Part A) and revised (Part B) theories are used.

Group (3), where the theories showed clear limitations and many fundamental weaknesses and/or Test Cases not solved, consists of Eckold, McCartney, Hart-Smith, Sun(NL). Some of these theories may have joined one of the other groups had the authors opted to provide solutions for the missing cases.

4.3. Quantitative assessment of the theories In order to assess the predictive capabilities of the theories in a quantitative manner, a third ranking procedure was adopted. The procedure is based upon a systematic comparison of the predicted strengths and deformations against the experimental values. A sufficiently large number of ranking cases (125 cases) was selected which covered five major areas (or five ranking categories): 1. 2. 3. 4. 5.

Biaxial strength of unidirectional laminae. ‘Initial’ biaxial strengths of multi-directional laminates. ‘Final’ strengths of multi-directional laminates. Deformation (stress–strain curves) of multi-directional laminates. Ability to predict the general trends observed in the test data.

These particular categories provided a convenient segmentation of the information, in full expectation that the theoretical base might be more robust at certain classes of prediction than at others (as will be seen later). A scoring system was applied which utilised the following rules: • Grade A (i.e. the prediction lies within ± 10% of the experimental value). • Grade B (i.e. the prediction lies between ± 10% and ± 50% of the experimental value). • Grade C (i.e. the prediction lies below 50% or above 150% of the experimental value). • Grade NA (i.e. no solution offered).

1093

Fig. 9(a). Comparison between the predicted and measured biaxial final failure stresses for ( ± 55°) E-glass/MY750 laminates (Test Case No 9). Prediction based upon original (Part A) and revised (Part B) theories.

1094

Fig. 9(b). Comparison between the predicted and measured biaxial ‘initial’ failure stresses for ( ± 55°) E-glass/MY750 laminates (Test Case No 9). Prediction based upon original (Part A) and revised (Part B) theories.

These rules were chosen arbitrarily by the organisers and are a trade-off between simplicity, needing a sufficiently coarse scale to capture all of the theoretical predictions, and needing a sufficiently fine scale to discriminate between individual theories. The reader should note that an alternate set of rules might change the relative ‘pecking order’ of any given theory relative to the next, but it would not change the overall picture greatly. Complete details of the ranking process are provided in Appendix C. The fit between the predicted and experimental results is shown in a tabular form in Tables C-1 to C-5 and in a graphical form in Figs C-1 to C-5.

1095

Fig. 10(a). Comparison between the predicted and measured stress strain curves for ± 55° E-glass/MY750 laminate under uniaxial tension (y/x = 1/0) (Test Case No 10). Prediction based upon original (Part A) and revised (Part B) theories.

1096 Fig. 10(b). A bar chart showing the ratio of predicted to measured values of failure strains and final strength for ± 55° E-glass/MY750 laminate under uniaxial tension (y/x = 1/0) (Test Case No 10). Both original (Part A) and revised (Part B) theories are used.

1097

Fig. 11(a). Comparison between the predicted and measured stress strain curves for ± 55° E-glass/MY750 laminate under uniaxial tension (y/x = 2/1) (Test Case No 11). Prediction based upon original (Part A) and revised (Part B) theories.

1098

Fig. 11(b). A bar chart showing the ratio of predicted to measured values of initial and final failure strengths for ± 55° E-glass/MY750 laminate under uniaxial tension (y/x = 2/1) (Test Case No 11). Both original (Part A) and revised (Part B) theories are used.

The number of A, B, C and NA grades achieved by the theories in each of the 5 ranking categories is shown in Table C-6 and the number of grades awarded to the theories are compared in Figs C-1 to C-5 (pages 1212–1216). The results for the ‘blind’ (Part A) predictions are grouped together on the left hand side of these figures and the results for the Part B predictions (i.e. revisions by the contributors after they had seen the experimental results) are grouped together on the right hand side of the figures. The highest ranked theories are arranged on the left of each group. The results for each category are summarised in the following sections. 4.3.1. Ranking Category 1 – predicting biaxial strength of unidirectional laminae The theoretical strength predictions are compared with the experimental results for biaxial loading of unidirectional fibre-reinforced laminae in Figs 1–3. Fig. C-1 compares the performance and the percentage grades for each theory. Most theories achieved at least 50% Grade A (i.e. within ± 10%) scores with Tsai, Puck, Cuntze, Wolfe-B and Chamis exceeding 72% Grade A. Hart-Smith(1) achieved a similarly high score but he did not use the curve he presented in Fig. 1a in any subsequent predictions. Some cautionary notes are required whilst considering these results as, by their nature, they do not necessarily tell the whole story. As an example, the Tsai, Wolfe-B, Bogetti and Hart-

1099

Fig. 12(a). Comparison between the predicted and measured stress strain curves for 0°/90° E-glass/MY750 laminate under uniaxial tension (y = 0) (Test Case No 12). Prediction based upon original (Part A) and revised (Part B) theories.

1100 Fig. 12(b). A bar chart showing the ratio of predicted to measured values of failure strains, strengths and modulus for 0°/90° E-glass/MY750 laminate under uniaxial tension (y = 0) (Test Case No 12). Both original (Part A) and revised (Part B) theories are used.

1101

Fig. 13(a). Comparison between the predicted and measured stress strain curves for ± 45° E-glass/MY750 laminate under y/x = 1/1 (Test Case No 13). Prediction based upon original (Part A) and revised (Part B) theories.

1102

Fig. 13(b). A bar chart showing the ratio of predicted to measured values of initial failure strengths for ± 45° E-glass/MY750 laminate under y/x = 1/–1 (Test Case No 11). Both original (Part A) and revised (Part B) theories are used.

Smith(2) theories are seen to perform well in Test Case 3, scoring A Grades against the available experimental data (Table C1). However, these theories predict markedly higher strength levels than the other theories in the compression-compression or tension-tension quadrants, for certain stress ratios, (see Fig. 3). The organisers instincts are that the predictions of these theories are overly optimistic and would have been seen to be so (resulting in a lower grade score) had missing experimental data been available at critical stress ratios. 4.3.2. Ranking Category 2 – predicting ‘initial’ strengths of multi-directional laminates The experimental data available for this category (initial failure strengths) was sparser than the ideal, nevertheless there was sufficient and unambiguous information (particularly in Test Case 12) to raise doubts about the predictions of most of the current theories. Fig. C-2 and Table C-2 show that, in the blind predictions, no theory achieved more than a 27% Grade A score, and that the majority of the theories fell below a 15% Grade A (i.e. for 85% of the test features those theories could not predict to within ± 10% of the experimental values). One has to combine the A and B scores to identify the better performers, who were Bogetti, Zinoviev and Wolfe-B. Those theories achieved predictions to within ± 50% of the experimental values for 80% of the test features. It should be noted (Fig. C-2) that Wolfe only obtained a high ranking after tuning the theory by fitting certain empirical constants to the experimental results. The tuning produced a bulge in the biaxial failure envelope for a unidirectional fibrereinforced lamina (Fig. 3) that the organisers believe is unlikely. The possible presence of residual thermal stresses in the laminates arising from cooling after curing and the wide range of approaches adopted by the contributors for calculating residual stresses in the absence of information on moisture content (discussed in Chapter 4) are complicating factors in assessing the accuracy of predictions. However, this is clearly not the only cause for discrepancy between the predictions and the available experimental results. In all three theories that achieved the highest scores (Bogetti, Zinoviev and Wolfe-B), the

1103

Fig. 14(a). Comparison between the predicted and measured stress strain curves for ± 45° E-glass/MY750 laminate under y/x = 1/–1 (Test Case No 14). Prediction based upon original (Part A) and revised (Part B) theories.

1104

Fig. 14(b). A bar chart showing the ratio of predicted to measured values of failure strain and strength for ± 45° E-glass/MY750 laminate under y/x = 1/–1 (Test Case No 14). Both original (Part A) and revised (Part B) theories are used.

effects of residual thermal stresses were ignored completely within the predictions. On the other hand Tsai, who assumed moisture contents that tended to cancel out the thermal stresses, did less well in predicting initial failure in the multidirectional laminates than he did in predicting the behaviour of unidirectional fibre reinforced laminae under simpler biaxial stress. Another important factor in the correlation is the fact that the events such as leakage and changes in slope of stress-strain curves that were recorded as first obvious failures in the experiments are different from the predicted initial failure events. Chapter 7 offers comments and suggestions on how to deal with this issue. The overall conclusion to be drawn is that a designer wishing to estimate the stress levels at which initial failure might occur in a multi-directional laminate, can only hope to get to within ± 50% of the available experimental results, at best, based on current theories and the limited experimental evidence.

1105 Table 5 Grouping of the theories according to their general characteristics. Group

Theories

General characteristics

1

Puck, Zinoviev, Tsai*, Sun (L), Cuntze, Bogetti**

2

Edge, Chamis, Wolfe, Rotem, Mayes, Huang, Hart-Smith(3) Eckold, McCartney, Hart-Smith, Sun (NL)

Exhibiting good predictive capability, none or one fundamental weakness and many relatively minor weaknesses Having a few more significant and fundamental weaknesses. Having clear limitations and many fundamental weaknesses and/or Test Cases not solved.

3

* See notes about the lack of critical experimental data, in Sections 4.3.1 and 4.3.2 where lack of test data could not confirm a complete benchmarking of the theory. ** The theory predicts two simultaneous modes of failure of a lamina under uniaxial longitudinal compression and un-validated enhancement in the biaxial strength of a lamina under tension-tension and compression-compression stresses in the longitudinal and transverse directions.

4.3.3. Ranking Category 3 – predicting ‘final’ strengths of multi-directional laminates subjected to biaxial loads The results show (see Table C-3 and Fig. C-3) that all of the theories fell below the 40% success rate in achieving Grade A scores for predicting final failure strength. Taking Grade A and B scores together, a number of theories managed to achieve more than a 75% success rate. This category highlights the importance of modelling the post-initial failure response accurately. The theories containing a good method for post-initial failure modelling (Puck, Cuntze, Tsai-B and Zinoviev outperformed those which either did not have any post failure model (Eckold, Hart-Smith(1), Hart-Smith(2)) or had an approach that seemed to lack robustness and maturity (Rotem, Wolfe, Huang, Mayes, Chamis and Edge). The HartSmith(3) ‘modified 10% rule’ performed remarkably well here, which is a great testament to simplicity. McCartney’s theory was a notable casualty, as no attempt was made to predict any final failures in the Part A submission. However, the Part B offering did include strength predictions for a number of Test Cases, with a consequent improvement in scoring. Predictions from the Sun (NL) theory (which also used a damage mechanics approach) were similarly incomplete in the number of final strength predictions made. Bogetti’s use of the Maximum Strain criterion achieved a higher score than Hart-Smith(2) who also employed the Maximum Strain criterion. This is partly because Hart-Smith(2) did not present predictions for the stress-strain curves and final failure strength of the 0°/90° (Test Case 12) and ± 45° laminates (Test Cases 13 and 14), and partly because Bogetti’s progressive failure model is arguably more realistic and avoids the over-optimistic predictions by HartSmith, for final strength of ± 55° tubes (Test Case 9). Once again, the reader is reminded to adopt caution in the interpretation of these results. The general lack of experimental data in the compression-compression quadrant, means that

1106 the theories have been relatively untested in that regime. Particular attention is drawn to Test Cases 3 and 6 (unidirectional lamina, Fig. 3, and quasi-isotropic laminate, Fig. (6) where the Tsai theory is by some way the most optimistic at certain stress ratios. The overall conclusion to be drawn is that a designer wishing to estimate the stress levels at which ultimate failure might occur in a multi-directional laminate, can, at best, hope to: • estimate to within ± 10% in 35% of cases; • estimate to within ± 50% in a further 40–50% of cases; • have an error margin of greater than ± 50% in the remaining cases. 4.3.4. Ranking Category 4 – predicting deformation of multi-directional laminates Most of the theories predicted the stress-strain behaviour in fibre-dominated cases quite well. Many were able to predict non-linear stress-strain curves but most were incapable of predicting the large non-linear deformations observed for angle-ply specimens in Test Cases 10 and 14, where the behaviour was dominated by the properties of the matrix (see Table C-4 and Fig. C-4 for more details of each theory’s performance). Grade B scores were more prevalent than Grade A scores, with the latter being achieved, typically, in only 30% of the Test Cases. Puck achieved the highest number of Grade A scores (39%). Based on ‘blind’ (Part A) predictions, Zinoviev’s approach (which allowed for geometrical non-linearity associated with fibre reorientation but did not allow for material non-linearity), obtained the highest total number of Grade A plus Grade B scores in this category, albeit with a preponderance of Grade B scores that indicates a fairly coarse fit ( ± 50%) to the measured values. Note however that Puck, Cuntze and Bogetti’s approaches gave a better description of non-linear behaviour at moderate levels of strain. Puck predicted the non-linear curves (Figs 10 and 14) very well up to 3% or 4% strain but missed being awarded Grade B for those cases because the failure strain was less than 50% of the measured value. Edge’s theory also did well but the behaviour at large strains seems unrealistic and may be an artefact of the algorithm employed. After making adjustments in allowable strain limits to match the experimental results Cuntze-B achieved a higher all-round ranking. Tsai-B and Mayes-B greatly improved their predictions of deformations by modifying their post initial failure prediction method and hence also achieved high scores. An improved score was awarded to McCartney-B because he offered predictions of the deformations at final failure for certain Test Cases where he had not offered any in his original Part A submission. 4.3.5. Ranking Category 5 – ability to predict a selection of general features There are many qualitative features of a theory (such as correctly identifying the mode of failure) that cannot be represented simply as a ratio of predicted to measured values. In this ranking category the abilities of the theories to predict 41 qualitative features of the experimental results were assessed. The features considered are listed in Table 4. Each feature was considered in turn and Grade A awarded if the theory did predict the feature and Grade C if it did not. Fig. C-5 and Table C-5 show that Cuntze’s theory, closely followed by Puck’s, scored the highest, with 73% of the chosen experimental features being reproduced. This is considered to be a very good performance in view of the wide variety of features under consideration.

1107 Several of the tests provided clear evidence to discriminate between the better and weaker theories. For instance, whilst three distinct (and unambiguous) modes of failure were observed experimentally in the 0°/90° cross ply (Test Case 12), only seven of the theories captured these events. Also only Puck, Tsai and Cuntze showed correctly the increase in shear strength, when transverse compressive stresses were applied to the lamina, in Test Case 1. The overall impression from this category is that a number of the current theories offer the designer reasonable support, should a more detailed probing of failure modes and interactive stress effects be required. 4.3.6. Overall ranking of the theories The total number of A, B, C and NA grades scored by each theory was calculated (see Tables C-6 and 6 in Appendix C) and divided by the total number of cases (125), to give an overall

Table 6 A summary of the grades scored by each theory for all the five ranking categories. The theories are listed according the number of the ‘A’ + ‘B’ grades achieved. Note that the total number of ranking categories is 125 (i.e. a perfect score would be 125 category A‘s) Blind submissions have been ranked separately from those which were modified in the light of the experimental results (the last seven rows). Grade type

A

B

C

NA

A+B

Zinoviev Bogetti Puck Cuntze Tsai Sun(L) Edge Huang Mayes Wolfe Hart-Smith3 Chamis2 Rotem Hart-Smith1 Hart-Smith2 Sun(NL) Eckold McCartney

53 51 64 62 50 52 50 42 52 39 39 44 37 41 36 21 20 10

43 44 30 32 35 31 32 38 27 31 30 24 30 16 19 17 14 4

29 30 31 31 40 42 43 45 46 55 41 53 58 23 25 21 48 4

0 0 0 0 0 0 0 0 0 0 15 4 0 45 45 66 43 107

96 95 94 94 85 83 82 80 79 70 69 68 67 57 55 38 34 14

Cuntze-B Tsai-B Mayes-B Wolfe-B Edge-B Huang-B McCartney-B

59 57 58 53 51 43 26

38 36 33 32 34 37 13

28 32 34 40 40 45 19

0 0 0 0 0 0 67

97 93 91 85 85 80 39

1108

Fig. 15. Overall ranking of the theories according to their capability to fulfil the five ranking categories.

percentage score for each theory. These scores are presented graphically in Fig. 15 where the theories are ranked according to their total Grade A plus Grade B scores. The results (Fig. 15) show that Zinoviev, Bogetti, Puck and Cuntze (and Tsai-B) achieved the highest Grade A + Grade B scores. The low scores achieved by McCartney, Sun(NL), Eckold and Hart-Smith were influenced by the fact that these participants did not attempt to solve all of the Test Cases. It should be noted that in some instances the predictions of these theories were good for cases that they did choose to predict. Note also that in Part B McCartney improved his approach, which enabled him to consider additional loading cases and to predict final failure, and this improved his overall score in Part B (see Fig. 15). Here no attempt has been made to evaluate the various theories by mathematical, logical or intellectual argument or by an assessment of the precepts upon which the theories are based, although we are not blind to such issues. Rather, we have attempted to assess performance of the various theories solely by comparison of theoretical predictions with available experimental observations, and to do so in an objective, reproducible and transparent manner (ie so that the method of assessment is open for inspection). Clearly, the result of such a comparison depend as much upon the availability and accuracy of the experimental results as it does upon the accuracy of theoretical predictions. One important outcome of the exercise is that we have identified weaknesses in experimental results and areas where further experimental work is required (see Section 6). Relatively strict rules have been applied to rank the theories. However, it is inevitable that some of the strengths and weaknesses of each of the theories could be masked by the nature of the ranking procedures adopted. Some features not highlighted in the ranking procedure will be touched upon in Section 5. It is, of course, to be expected that had different categories

1109 or criteria been applied, then a slightly different result would be achieved. It has already been noted that some critical experimental results are not available and that also has some influence on the results. The ranges, assigned to Grades A, B and C, are a further influencing factor. Nonetheless, the results of all five ranking categories assembled together in Fig. 15 do reflect our overall impressions of the theories and do agree with the more simplified qualitative procedures used in Section 4.2 above (and Appendix B). 5. Comments on individual theories Nineteen different theoretical approaches are represented in this study. The following is a brief summary of our overall impressions of the performance of each of the theories. The theories are considered in alphabetical order with comments reflecting the ranking of the theories depicted in the bar charts (Fig. 15 and Figs C-1 to C-5 in Appendix C). 5.1. Bogetti (with Hoppel, Harik, Newill and Burns), Refs [32,36] This theory employs a Maximum Strain failure criterion and takes into account lamina nonlinear shear behaviour and progressive failure. The theory was formulated in its three-dimensional form. The predicted failure envelopes for unidirectional laminae subjected to combined shear and direct loading were of the same form as those for other non-interactive criteria. The envelope for an isolated lamina under combined longitudinal and transverse loading did not pass through the value of transverse compressive strength provided in the data pack, and this raises a clear question mark. A step was predicted near the uniaxial longitudinal compression loading condition, and this theory gave much higher predictions of strength in the biaxial tension, and biaxial compression quadrants than other theories. Unfortunately the apparently un-conservative predicted peaks could not be validated with the limited available experimental data. The prediction of initial failure, due to through-thickness failure even in the absence of through-thickness stress, is a feature in this theory, but the available test data could not confirm this behaviour. Intuitively the organisers have reservations about this feature. The theory gave a good prediction for the initial portion of the observed non-linear stressstrain curves of the multi-directional laminates and also for the final strengths where fibre failure was the dominant mode of failure. The post failure analysis was simple and effective but, perhaps, imperfect in some respects. The overall performance was good, predicting a wide range of features of the available experimental data. Some of the borderline (but favourable) scores and the lack of critical experimental data to really challenge certain features of the theory (e.g. through-thickness failure) has contributed to the high overall ranking. 5.2. Chamis (with Gotsis and Minnetyan), Refs [3,19] This micro-mechanics based theory gives a good description of measured unidirectional lamina failure envelopes although it did not predict the maximum shear stresses observed in experiments under combined direct and shear loading.

1110 The predicted initial failure stresses for multi-directional laminates were lower than the limited available experimental data and lower than the predictions of many other theories, largely due to the full allowance made for residual thermal stresses. There appears to be a fundamental weakness in the prediction of laminate behaviour in certain instances where, experimentally, initial failure occurred at modest stress but final failure occurred at much larger stress levels. A problem in the post-initial failure modelling assumptions is indicated and this often resulted in predictions of final failure stresses that were much lower than the experimental results. Additionally, differences were observed between predicted and measured laminate stiffness (Fig. 12), the predictions being noticeably stiffer. Furthermore, this theory had no means of predicting the observed non-linear stress-strain curves. Its overall performance was not quite as good as some other theories. 5.3. Cuntze (with Freund), Refs [32,37] This theory gains many of its strong features from that of Puck. It considers five failure mechanisms and the predictions of the three failure envelopes for the unidirectional laminae bear some resemblance to Puck’s. Unlike Puck’s theory, interaction between the failure mechanisms is assumed, due to probabilistic effects and Cuntze’s theory does not require the determination of the angle of the fracture plane. The Cuntze theoretical failure envelopes for the unidirectional laminae were in very good agreement with the experimental results. The predicted final failure envelopes and stress– strain curves for the multi-directional laminates were also generally in good agreement with the measured ones The theory performed very well overall and seems to have provided a good fit with large number of test data. Some of the improvements in performance offered in Part B were partly as a result of altering the lamina stress-strain input curves, assuming convenient degradation profiles of the lamina stress strain curves and (to a lesser extent) by a suitable choice of the curve fitting parameters embedded in the theory. Certain other modifications, made in Part B were offered as a means of simplifying the computation and appear to have no physical basis. The reader should form his/her own view as to the fidelity of these revisions. The theory is intended to handle multiple non-linearity but sometimes appears to suffer some difficulty in achieving convergence in the numerical algorithm, especially for large deformations and for cases involving combined non-linearity. It is intended to be capable of considering failure under three-dimensional applied loads but those aspects have not been tested here. In summary, the theory performed very well but further work is required to verify and improve some of the theoretical modelling and computational aspects involved. 5.4. Eckold, Refs [4,20] The theory is an extension of the philosophy used in the British Standards for commercial GRP pressure vessel design. The method is simple in concept and aims to hold strains to a very low level, below the threshold for initial failure in a uniaxial test. Although this ‘exercise’ is concerned with failure, the approach offered by Eckold actually provides predictions of safe working stresses and is thus expected to provide conservative predictions in all cases. The

1111 theory was applied only to GRP laminates and no predictions were offered for carbon fibre reinforced composites or for cases involving shear loading. When the longitudinal and shear strains were limited to 0.4% and the transverse strains to 0.1% the design method was always conservative, sometimes very conservative, compared with the available experimental results. For two Test Cases (Cases 4 and 9) Eckold used a different method that employed ultimate values of lamina transverse, longitudinal and shear strains in order to predict ultimate (final) failure envelopes. In applying the theory, an assumption was made that the lamina tensile and compressive strength in the fibre direction was identical, even though information was supplied to the participants to the contrary, Ref. [3]. Large discrepancies resulted between the predicted and measured ultimate strengths of laminae and laminates, leading to unconservative predictions in certain instances. No allowance was made for non-linear behaviour and that sometime led to an underestimate of strains and an overestimate of stiffness. The theory represents a brave attempt at stretching a very simple design philosophy for a niche component into a more generally applicable method. Unfortunately the ‘exercise’ has confirmed that this theory is not sufficiently robust, flexible or accurate for wide spread use. 5.5. Edge, Refs [5,21] The performance of this theory was something of a mixed bag. It gave moderate agreement with the measured shape of failure envelopes for unidirectional laminae. It did not predict the interactions observed between strengths in some quadrants of the experimental failure envelopes but it did in others. In general the laminate initial failure stress predictions were low due, in part, to full allowance being made for residual thermal stresses. Laminate final failure strength predictions were sometimes conservative and sometimes unconservative, with no clear pattern to each. This was one of the few theories with the capability to predict non-linear forms of stress– strain curve up to the large strains observed in some Test Cases. However, the predicted shapes were very different from those observed in the experiments. Convergence problems were encountered in the post initial failure algorithms and it is not clear if this was a contributory factor to the mismatch. The revisions provided in the Part B submission produced some small improvements but did not impact greatly on the general picture. Although this theory contains some interesting features, overall it is not in the leading rank. 5.6. Hart-Smith(1), Refs [8,22] Hart-Smith’s generalised maximum-shear-stress theory employed detailed argument to construct a failure envelope from individual curves that describe each different failure mode. The failure envelopes for unidirectional laminae were sometimes a little unconservative. Interestingly, Hart-Smith introduced an additional equation into the theory that gave a curved shape of failure envelope for combined shear and transverse tension (Test Case 1) closely

1112 resembling the experimental data. However, that equation was dropped from the subsequent analysis used to predict the strength of multi-directional laminates. The predicted shapes of the final failure envelopes for laminates were often of broadly similar shape to the experimental data but were sometimes unconservative. The predictions offered did not include any initial or intermediate modes of failure, nor any stress–strain curves. Part of the reason for this is believed to be because the originator has yet to translate the theory into a suitable computer model. Thus all calculations have been carried out by hand in the first instance. The Hart-Smith(1) theory, in its current formulations, has to be categorised as one requiring further development before it could explain all aspects of the experimental results presented here. It was never intended to predict initial failures. It gave a good description of the failure envelope for the quasi-isotropic carbon fibre reinforced laminate but did not predict all of the other measured final failure stresses as well as some other theories (see Table C-4 and Fig. C-4). 5.7. Hart-Smith(2), Refs [7,22] Hart-Smith’s presentation of the maximum strain and truncated maximum strain criteria was similar in style to that described above. The predicted failure envelopes for unidirectional laminae subjected to combined shear and direct loading were of the same form as those for other non-interactive criteria. The envelope for ‘an isolated lamina’ under combined longitudinal and transverse loading did not reproduce the value of longitudinal compressive strength provided in the datapack and gave much higher predictions of strength in the biaxial tension and biaxial compression quadrants than other theories. Unfortunately these apparently unconservative peaks could not be validated with the limited available experimental data. Many of the final failure envelopes for laminates were similar to those for Hart-Smith(1) and were sometimes unconservative (i.e. higher than the experimental results). Once again the predictions offered did not include any initial or intermediate modes of failure, nor any stress– strain curves. However, the theory seems to work relatively well in some areas pertaining to fibre failure predictions, as for instance in Test Case 6. The failure to supply stress–strain curves meant that it was impossible to judge the reliability of the theory for several Test Cases, such as No 7, 8 and 13. A limited amount of evidence has been provided through the ‘exercise’ in support of this theory. Furthermore, there are concerns about the predicted peaks in certain of the lamina failure envelopes which, although experimental data is sparse, may prove to be unconservative. It is worth noting here that, in a more recent article prepared for publication, Hart-Smith has made attempts to include initial failure prediction in his theories, see Ref. [44]. 5.8. Hart-Smith (3), Refs [18,22] This is an extension of the widely used ‘Hart-Smith 10% rule’, which was originally intended as a simple and approximate method for sizing of components (made from a combination of 0°, ± 45° and 90° plies) in design, before embarking on a detailed prediction of laminate behaviour.

1113 The method did not do well in predicting failure envelopes for unidirectional laminae. The shapes of these envelopes were rectangular (similar in form to those for other noninteractive (e.g. Maximum Stress) criteria), but the method overestimated the transverse tensile strength and underestimated the transverse compressive strength, values for which were provided to all participants as basic data, Ref. [2]. The method predicted final failure envelopes for all of the laminates that were in good general agreement with the available experimental results. However, the theory has no means of predicting initial or intermediate failure modes or stresses. It gave linear approximations that were reasonably close to the experimental stress–strain curves except in cases where significant non-linear behaviour came in to play (e.g. Test Cases 10 and 14). In those instances the theory gave linear curves that were non-representative of the experimental curves. Although having less ability than other theories to predict the detail, the Hart-Smith(3) theory proved to be quite effective in predicting the upper bound performance of most laminates. With further refinement and a ‘good users guide’, the method could continue to provide the basis for ‘quick’ sizing calculations for the design community, where ultimate strength is sought and where initial failure is not of concern. 5.9. Huang, Refs [35,39] This micro-mechanics based theory combines a number of ideas, (micro-mechanics, anisotropic plasticity and general Maximum Stress theories) into a single theoretical framework. It is ambitious but appears to be somewhat immature and lacks, for instance, physically based concepts of fracture. Whilst the theory provides a basis for obtaining the lamina and laminate elastic constants and strengths from the constituent fibre and matrix properties, the predictions show a large number of discrepancies in comparison with the experimental results. The theory predicted large values for the uniaxial transverse tensile and shear strengths of a unidirectional fibre reinforced lamina that were significantly at variance with the experimental observations. This discrepancy was carried forwards into the analysis of the Test Cases and resulted in some very over optimistic predicted values of initial failure strength. The theory lacks good progressive modelling capabilities and gave very low predictions of final strength and deformation of multi-directional laminates for some loading conditions. Although it appears to capture some of the large deformations observed in the tests, it did so because of the falsely high, ‘tuned’, large strain limits imposed on the unidirectional lamina, rather than any other inherent capability to predict such large deformation. The inclusion of thermal residual stresses on micro- and macro-levels is a feature of the theory and is unique amongst the WWFE contributions. However, the theory requires significant development work to bring this to the maturity level of some other theories. 5.10. Mayes (with Hansen), Refs [34,38] This is a finite element micro-mechanics based theory that takes the lamina nonlinear shear behaviour into account. It performed moderately well in predicting unidirectional lamina failure envelopes. It did not predict the maximum shear stresses observed in experiments

1114 under combined direct and shear loading and for biaxial tension and compression predicted an envelope rather similar to that of the non-interactive theories The predicted initial failure stresses for the multi-directional laminates were in better agreement with experiment than many of the other theories but, as for all theories, confidence in this aspect of the correlation is not high. The ‘blind’ predictions of final failure stresses for the multi-directional laminates were sometimes very low in comparison with the experimental results, especially for angle ply laminates (e.g. Test Cases 9, 11 and 13) but in Part B further attention to the post initial failure analysis greatly improved the final failure stress predictions in these cases. The stress–strain curves in Part A were also often truncated at much lower strains than the final strains observed in the experiments and the post failure modelling gave some other trends that were not observed in the experiments (e.g. the stress-strain curves in Test Case 11 in Part B). The modifications introduced in Part B have improved the correlation between the theory and experimental results for some cases. Overall, the performance of this theory was not ranked as quite as highly as that of the very best theories. 5.11. McCartney, Refs [6,23] McCartney’s mathematical models describe the initiation and multiplication of cracks in a unidirectional fibre reinforced layer, caused by tensile stresses perpendicular to the fibres, and the effects of the cracks on the laminate stress-strain behaviour. It was recognised from the outset of the ‘exercise’ that McCartney’s theory was unlikely to be capable of tackling all of the Test Cases. In Part A of the ‘exercise’ just two Test Cases were analysed. The theory predicted the shape of the stress–strain curve for the 0°/90° laminate loaded under uniaxial tension (Test Case 12) very well, but overestimated the stiffness of the ± 45° laminate under biaxial tension (Test Case 13). The theory presented in Part A had no means of predicting final failure. In Part B the application of the theory was extended, including the introduction of a maximum fibre strain criterion to assist with final failure prediction, and was used to predict stress-strain curves for 0°/90°/ ± 45° quasi-isotropic and ± 55° angle-ply laminates under uniaxial and biaxial tension (Test Cases 7, 8, 10, 11). The predictions for the quasi-isotropic laminate were good. The theory did not accommodate non-linear lamina properties, nor the consequences of failure due to shear, nor cracking in more than one layer. Consequently the theoretical stiffness of the angle-ply laminate was too high and the non-linear deformations were not captured. The agreement between the theoretical and measured strengths for the quasi-isotropic and 0°/90° laminates was good. The number of Test Cases analysed was small and the theory has many limitations and is still too immature to be used as a general design aid. However, the approach offers the future promise (after development) of providing a rigorous and flexible method for modelling the initial and post initial failure response of laminates, which remains as a clear problem area for many of the competing theories. 5.12. Puck (with Schürmann), Refs [9,24] Puck’s theory has evolved from extensive experimental studies of the mechanisms by which failure occurs in a lamina when subjected to a biaxial stress state. Most of the failure theories

1115 represented in this ‘exercise’ have some means of addressing ‘matrix cracking’. However, Puck’s theory contains a very sophisticated treatment of inter fibre failure, through the introduction of the several competing ply cracking mechanisms and consideration of the orientation (angle) of the fracture plane. Given the pedigree of the theory it is no surprise to find that the Puck theoretical failure envelopes for the unidirectional laminae were in very good agreement with the experimental results. The predicted final failure envelopes and stress–strain curves for the multidirectional laminates were also generally in good agreement with the measured ones. Discrepancies between theory and experiment emerged in the Test Cases where large nonlinear deformations were present. In some instances, predictions of final failure strain were much smaller than the observed values and in others the failure envelope was not closed. It is also worth recording that the need (or otherwise) for designing structures to operate in the large displacement/non-linear region is worthy of a philosophical debate, in an arena separate from this ‘exercise’. Overall, the results show that the Puck theory captures most features of the experimental results and appears to be one of the best available currently.

5.13. Rotem, Refs [10,25] The Rotem theoretical failure envelopes for the unidirectional laminae were in reasonable agreement with the experimental results although giving slightly conservative predictions for some loading combinations. The predicted initial failure stresses for the multi-directional laminates were in better agreement with experiment than were the predictions for many of the other theories. However, the predicted final failure envelopes for the multi-directional laminates were frequently extremely conservative in comparison with experiment. The stress–strain curves were also generally truncated at much lower strains than the final strains observed in the experiments. The indications are that the theory does not discriminate adequately between initial and final failure, and that there may be fundamental weaknesses in the algorithms for dealing with the post-initial failure laminate response. Some attempt was made in Part B to improve one of the algorithms and this gave a noticeable improvement over the Part A results, in one of the Test Cases. In its current state, this theory requires fundamental modifications, especially in the post failure modelling, in order for it to be in the leading rank.

5.14. Sun(L) (with Tao), Refs [11,26] This Sun’s theory, employing linear stress analysis, contains an interesting amalgam of methods. It gave reasonable predictions of unidirectional lamina failure stress envelopes although, for some combinations of loading, the non-interactive features resulted in slightly unconservative predictions. The behaviour under combined shear and transverse tension was predicted using a criterion first proposed by Rotem.

1116 The predicted initial failure envelopes for the multi-directional laminates were in moderate agreement with the measured ones. However, the predicted final failure envelopes for the multi-directional laminates were better. The predicted stress–strain curves were rather mixed. Where the response was heavily fibre dominated (Test Cases 7, 8, 12 and 13), the strength predictions were in good agreement with experiment. However, where shearing of the laminae was a dominant feature, this linearelastic based theory greatly underestimated the large non-linear deformations observed in the experiments (Test Cases 10 and 14). Interestingly, the Sun(L) theory performed better that the Rotem theory in modelling the post-initial failure response, even though a common (the Rotem) failure criterion was used by both under conditions of combined shear/transverse tension. Overall the Sun(L) theory had few fundamental weaknesses and performed sufficiently well to be in the leading group of theories. 5.15. Sun(NL) (with Tao), Refs [11,26] At the outset of the ‘exercise’ Sun proffered a mixture of two theories to address the range of Test Cases. The Sun(NL) theory, capable of utilising Non-Linear lamina input data, used Finite Element based fracture mechanics to predict the shape of all of the stress–strain curves. However, the theory was not employed to predict final failure levels nor any of the failure envelopes. In practise, predictions from the Sun(NL) theory were not significantly better than those from the Sun(L) theory. The one marginal improvement was in Test Case No 14, a shear dominated situation where the use of the non-linear lamina stress–strain curves gave slightly better performance than the linear theory at low strains. Like many others the theory proved incapable of predicting the large deformations observed in those experiments. Given the limited number of Test Cases attempted and the lack of improvement over the alternative Sun(L) theory, the added complexity within the Sun(NL) theory of employing FE and fracture mechanics does not seem justified. Thus, in its current state, the Sun(NL) is not recommended for use as a design aid involving ultimate failure prediction, although continuous development may aid in the study of the effects of crack multiplication. 5.16. Tsai (with Liu and Kuraishi), Refs [12,27] Tsai’s contribution to this ‘exercise’ utilises his well known theory which has been made widely available as inexpensive software with an accompanying text book. It employs the interactive Tsai-Wu failure criterion, which is arguably one of the best known and mathematically satisfying theories available. Tsai has advocated the theory for its simplicity and as a tool that can be applied under a wide range of conditions. It is not intended to capture the detailed physics associated with the various failure mechanisms. Starting with the failure envelopes for unidirectional laminae, in this exercise the Tsai theory has described the available experimental results better than any other theory. However it has been noted that the theory predicts enhancement of strength under compressioncompression biaxial loading (Fig. 3) which, though not tested by the experimental data presented here, should be viewed with some concern.

1117 The predicted initial failure stresses for the multi- directional laminates in the Part A submission were in poor agreement with the available experiments. The predicted shapes of the final failure envelopes for the multi- directional laminates agreed quite well with the available experimental results. In many cases the predicted final failure stresses were conservative (i.e. lower than the experimental results and the predictions of some other theories). Once again there is a large predicted enhancement in compressioncompression biaxial strength for the quasi-isotopic laminate (Fig. 6) under conditions which have not been tested, and that should be viewed with caution. Like many of the theories featured in the ‘exercise’, the Tsai theory is linear-elastic and it could not predict the large non-linear strains observed in those Test Cases where high lamina shear was involved. Revisions presented in Part B significantly improved the agreement with the experimental initial and final failure stresses, and with the deformations in several cases. Also the introduction of a new post-initial failure model is seen to be a useful step. Overall, with the reservations indicated above, the Tsai theory proved to be in the leading group of those tested in the ‘exercise’.

5.17. Wolfe (with Butalia), Refs [13,28] This theory gave reasonably good descriptions of the unidirectional lamina failure envelopes except for predicting two concurrent values for each of the longitudinal uniaxial tension and compression strengths, probably due to a problem in the logic of the failure algorithms. Predictions of initial failure strengths for the multi-directional laminates were in quite good agreement with the experiments, relative to the other theories (though overall none performed well). Some of the predicted final failure envelopes for the multi-directional laminates were extremely conservative compared with the measured strengths. This is believed to be a result of a fundamental difficulty within the theory in distinguishing effectively between initial and final failure, and a subsequent problem associated with the post-initial failure algorithm (i.e. a method adopted for reducing both the transverse and shear moduli simultaneously, immediately after initial failure). This also resulted in stress-strain curves being truncated at unrealistically low strains. Revised predictions in Part B resulted in much improved agreement for final failure with experimental results but also produced an unlikely bulge in the biaxial stress failure envelope for a unidirectional lamina (see Fig. 3). Overall the Wolfe theory performed creditably and is considered to be in the second group of theories, as some questions linger over its robustness and particularly of the post-initial failure algorithms.

5.18. Zinoviev (with Grigoriev, Labedeva and Tairova), Refs [14,29] Zinoviev used a development of the Maximum Stress failure theory, which embodies a very simple, but carefully structured, set of non-interactive criteria to identify failure mechanisms

1118 and to take appropriate post-initial failure action. There is less of an attempt (compared to Puck, for example) to provide physically based failure models. The theory gave reasonably good descriptions of the unidirectional lamina failure envelopes though, as expected for a non-interactive failure theory, it overestimated the measured lamina strengths at certain loading combinations. The theory was one of the best at predicting initial failure events for multi-directional laminates, albeit none of the theories were particularly robust in this area and he ignored residual thermal stresses. The theory gave a reasonably good fit to the experimental final failure envelopes for all of the multi-directional laminates, by giving careful attention to effective lamina property degradation and unloading after initial failure, and by allowing for fibre re-orientation during loading, Although the theory assumed linear-elastic material properties, it also gave reasonably good descriptions of nearly all of the stress–strain curves and only failed to predict the observed large deformations in Test Case No 10. Some problems were encountered in terminating predictions involving large deformations and this resulted in a (perhaps) unrealistic open failure envelope in Test Case No 9. The Zinoviev theory performed very well overall and it appears to be one of the best available currently It is worth noting here that space does not allow detailed discussion of pros and cons of all features of each theory and, in order to highlight other aspects not discussed in this paper, readers should consult the individual contributions in Refs [3–14, 18–29, 32–39].

6. Discussion A relatively large amount of information has been generated in this comparison study where the theoretical predictions of nineteen failure theories, and their modified versions, have been compared with experimental results for 14 Test Cases. Much of the correlation between the theoretical predictions and the experimental results could be visualised from the graphs and bar charts in Figs 1–14 and from the tables listed in Appendices I, II and III. However, to increase awareness of some of the potential factors affecting the validity of the comparison and to highlight the extreme cases of discrepancies between theories and experiments, it was felt appropriate to make the following points.

6.1. Comments on the experimental results The reader will recall that the experimental data selection issues were discussed in Section 2 of this paper. Not entirely unexpectedly, as the ‘exercise’ proceeded, it became clear that the experimental data was judged to be inadequate in a number of critical areas (see Table 3). The inadequacies need to be put in context. Rather like a set of Russian dolls, there was more than enough experimental evidence to satisfy the major aims of the ‘exercise’ (i.e. the largest doll). However, as one probed the theories more deeply (the inner dolls), the experimental evidence became insufficient, or unreliable to draw further conclusions. Fuller descriptions of the experimental weak points are provided below. The aim in providing this information being to

1119 remind the readers that definitive experimental data is hard to come by in the field of composite materials, and that a degree of greyness, and scepticism, is always present when using such in any comparisons with theories. The primary weak points were: 6.1.1. Incomplete lamina biaxial data The shortage of failure envelope data for the unidirectional E-glass/MY750 epoxy lamina subjected to biaxial loading (Test Case (3)) made it impossible to confirm the accuracy of the unusual behaviour displayed by Tsai, Bogetti and Hart-Smith(1) in the biaxial compression quadrant, and by Wolfe-B, Bogetti and Hart-Smith(1) in the biaxial tension quadrant. 6.1.2. Some unfamiliar trend in lamina biaxial data Fig. 2, showed the failure envelope for the unidirectional (T300/914C) carbon fibre reinforced epoxy lamina under combined shear and longitudinal loading. Those experiments were conducted at several different laboratories and there was a degree of scatter in some of the results (e.g. in the uniaxial shear strength). One important feature of the experimental results was that the shear strength appeared to increase significantly in the presence of longitudinal tension. As this was contradicted by nearly all of the theoretical predictions, further experimental work is clearly needed in this area. 6.1.3. Pre-mature and structural failure In Test Case 6 (the quasi-isotropic [0°/ ± 45°/90°]s AS4/3501–6 carbon/epoxy laminate), there is a belief that shell buckling had influenced the test data in the compression – compression quadrant, leading to lower ultimate strengths than anticipated. It was therefore not possible to confirm the accuracy of the theories in that quadrant, where Tsai, in particular, had predicted significantly higher levels of strength than all others. 6.1.4. Accuracy of measuring deformation In determining experimental stress-strain curves, small inaccuracies are always to be expected. In presenting the experimental stress-strain curves results for the ± 55° and ± 45° tubes in Test Cases 10 to 14, it was reported [17] that there was some variation in the axial strain measurements from one strain gauge to the next. That variation was small compared with the wide variation in theoretical predictions shown, for example, in Fig. 11 for the ± 55° tubes or the large discrepancies between the theoretical and non-linear experimental results in Figs 10 and 14. Further experiments would be advisable in those cases before very detailed comparisons could be justified (e.g. if or at exactly what strain did the axial strain curve intersect the circumferential stress–strain curve in Fig. 11). The accuracy of using strain gauge readings to identify failure through change in stiffness (such as those shown in Figs 7 and 8) is also questionable, without there being convincing independent evidence of the nature of the failure (such as in Test Case 12). In general, however, the accuracy of the strain gauge results has been more than adequate for the purposes of this ‘exercise’.

1120 6.1.5. Consideration of bulging and scissoring When tubular specimens are used as test pieces to determine the uniaxial and biaxial behaviour of laminates, the change in geometry during loading may affect the stress ratio and the stress-strain curves. Bulging (or necking) and scissoring (change in fibre direction) are two phenomena that may take place when pressurising tubular specimens. While none of the contributors was expected to take into account the bulging effects in Part A of the ‘exercise’ some theories, notably that of Zinoviev (see, for example, Test Case 14)), did consider the scissoring effects. In Part B Test Case 15 did provide an opportunity for contributors with access to advanced analysis tools to allow for the effects of change in specimen geometry through a re-examination of the results from Test Case 10 but no one took up that challenge. 6.1.6. Lack of data on initial failure In almost all of the Test Cases selected for the ‘exercise’, information was rather sparse on the nature and location of initial failure, see Ref. [43]. Clear visual evidence of cracking was available for Test Cases 12 and 13 (the 0°/90°and ± 45° glass-fibre reinforced epoxy composites). However the data presented for cracking in carbon-fibre reinforced epoxy laminates was inferred mainly from some limited fractography and from a slight changes in slope of stress-strain curves for Test Cases 7 and 8 (the 0°/ ± 45°/90° laminates), see Ref. [17] for more details. Future studies would certainly benefit from better attempts to capture these features. 6.1.7. Lack of data on crack propagation Data on the post-initial failure damage evolution process were also rather sparse. Test Case 12 and 13 (the cross ply 0°/90° and ± 45° glass-fibre reinforced epoxy laminates) were the only examples where data on the development of cracks between initial and final failure might have been utilised, but were not. As a result, the crack density prediction of the McCartney and Sun(NL) theories were not tested in this exercise. 6.2. Unresolved Issues During the ‘exercise’, several disparate aspects of laminate analysis have emerged where there is no consensus as to their treatment. Whilst it was to be hoped that the ‘exercise’ might have provided definitive answers, in practise the issues remain open due to lack of evidence. They include: 6.2.1. Residual thermal stress Multi-directional laminates that are cured at elevated temperature will develop built-in residual stresses due to the curing process. The significance or otherwise of these stresses remains a debating issue, with a confusing picture emerging from the ‘exercise’ so far. 6.2.2. In-situ strength A number of the participants (Sun, Rotem and Hart-Smith) chose to differentiate between the strength behaviour of an isolated lamina and that of a lamina embedded within a laminate. Insufficient evidence is available here to resolve this issue.

1121 6.2.3. Leakage This mode of failure was encountered in testing tubes under internal pressure loading, where ‘initial’ failure is associated with escape of the pressurisation liquid through the damaged wall. The initial failure predictions greatly underestimated the stresses at which leakage occurred, presumably because considerable damage and possibly some delamination must accumulate to create a sufficiently large path for the pressurising liquid to weep through the wall. Initial indications point to the need for some form of damage evolution within the theories to handle this problem. However, the problem is unresolved at present. 7. Concluding remarks 1. In the present paper the theoretical predictions of nineteen failure theories have been compared with experimental results. 2. The theories have been assessed and ranked according to their abilities to predict the experimental results for failure of a unidirectional fibre reinforced lamina, initial and final failure of multi-directional laminates and large deformation of laminates under biaxial loads. 3. Some theories have been shown to give good agreement with experimental results and others to have limited capabilities. In particular, the following has been observed: (a) On a lamina level, Tsai was seen to be the highest scorer but Cuntze and Puck also did well and further experiments are required to confirm Tsai’s predictions of increased lamina strength under biaxial compression. (b) Doubts arose from the shortage of good experimental data and from the variation in the theoretical treatment of residual thermal stresses and effective in-situ ply strengths, that made it difficult to arrive at straightforward conclusions regarding the best theory to use for predicting initial failure. (c) On final strength prediction for multi-directional laminates under biaxial loads, Puck, Cuntze, Tsai and Zinoviev achieved the highest scores. Again Tsai predicted higher strengths than the others for some biaxial compression loading cases and that is not confirmed by the experiments. The importance of a good post initial failure analysis method in the prediction of final failure has been demonstrated. (d) Zinoviev was ranked highest overall for blind predictions of deformations, although Puck was awarded the highest number of Grade A scores and gave better predictions of non-linear stress-strain curves at moderate strains. The results have shown that it is necessary to handle multiple non-linearities arising from damage, non-linear shear behaviour and change in fibre orientation in order to accurately predict the stress-strain behaviour up to failure of laminates under loading conditions that result in matrix dominated behaviour. (e) Puck and Cuntze captured more of the general features of the experimental results than the other theories. (f) Puck, Cuntze Zinoviev and Bogetti did best in the overall ranking together with Tsai (after he had made some improvements to his post failure analysis). (g) All of the participants made valuable contributions to the exercise, including those whose theories that have not scored highly in the overall ranking. In some instances the

1122 low score was because the contributor did not offer solutions to many of the wide range of problems posed. 4. The authors believe that the three aims of the ‘exercise’ have been met in full: • Aim 1: establishing the current level of maturity of theories for predicting the failure response of fibre reinforced plastic (FRP) laminates. Through Parts A and B, an extremely detailed examination has been carried out of the current level of maturity of theories for predicting the failure response of fibre reinforced plastic (FRP) laminates. Within the summarising papers, the organisers believe that a comprehensive critique of the current situation is now available to the composites community. • Aim 2: closing the knowledge gap between theoreticians and design practitioners in this field. When the organisers first initiated this ‘exercise’, it was acknowledged that a major philosophical question, would have to be addressed before the activity was concluded. That question is,‘what is failure?’ In the various instructions issued to participants, the organisers sought answers to this question. Whilst most, if not all, of the participants have some form of personal working definition, it has proved to be extremely difficult for them to articulate their thoughts through the ‘exercise’. In many respects, this goes to the heart of why there is a perceived credibility gap between the community of composite theoreticians and the engineers who are responsible for designing the next generation of composite structures. Through this ‘exercise’ it is believed that some theoreticians, in particular, now recognise the size of gulf between an interesting theory and a practical predictive tool. Recognising the gap and being reminded of the issues that are critical to designers is a major step forward in encouraging the next wave of improved theories. • Aim 3: stimulating the composites community into providing design engineers with more robust and accurate failure prediction methods, and the confidence to use them. Through Parts A and B, the leading failure theories have been showcased and tested together for the first time. This very process, and the publicity associated with it, has been extremely valuable in providing an impetus to the theoreticians and the designers alike, to take a renewed interest in an area perceived as being stagnant for some time. Three major advances illustrate the point: – The ‘exercise’ has provided a convenient framework by which several of the theoreticians have chosen to examine ‘the other’ theories in more depth than perhaps they had done previously. A number of ad-hoc gatherings have taken place between the participants and it is pleasing to see the degree of cross fertilisation that has already taken place. – The two part process adopted for the ‘exercise’ has already resulted in revisions to eleven of the nineteen original theories. It is arguable if this rate of improvement would have been achieved without the motivation (and competitive environment) created by the ‘exercise’. Unarguably, many of the revisions have resulted in significant improvements in the predictive capabilities of the theories.

1123 – The ‘exercise’ provided an immense source of data from which it was possible to identify the more robust theories from the weaker ones and to identify those areas in the general design space where reliable predictions might be made. The organisers have distilled the information into ‘Recommendations for Designers’ [42], which are presented in the final chapter of this book (Chapter 7). We believe this is a significant step forward, which should help steer designers towards improved ‘best practice’ and one that can be updated, as new theories emerge and are validated. 5. Finally, and through debate with the composites community, the authors intend to explore in more detail the lessons learned from the ‘exercise’ and to identify the next steps required towards improved and validated failure prediction models. Two points for early contemplation and debate are worth citing here: • Of the nineteen theories considered so far, the strengths and weaknesses of each are now more readily apparent, as a result of the ‘exercise’. Rather than the proliferation of yet further theories, it is timely that the community begins to focus on those that have proven to be the most robust, currently. The analysis presented in this paper provides an ample source of information from which the better performing algorithms could be selected and grafted together, to accelerate the emergence of improved models. We strongly urge the theoreticians of the leading models to pursue this approach. • A common complaint from the design community is that the theoreticians do not make available their theories in a form that can be readily utilised. One cannot expect that an engineer faced with a challenging component design issue will have the time to read the literature, identify a suitable paper, then translate the (usually) complex theory into a suitable design tool (usually software). It is incumbent on the theoreticians to cross this divide if their theory is ever to be anything more than a useful diversion. We strongly urge the theoreticians take that step and it is pleasing to see that several of the participants in the ‘exercise’ are developing, or have already developed, suitable software aids.

References 1 M J Hinton, A S Kaddour and P D Soden, ‘The World-Wide Failure Exercise: Its Origin, Concept And Content’, see Chapter 1 in this book. 2 P D Soden, M J Hinton and A S Kaddour, ‘Lamina properties and lay-up configurations and loading conditions of a range fibre reinforced composite laminates’, Compos Sci Techno, V58, pp 1011–1022, 1998, see also Chapter 2.1 in this book. 3 P K Gotsis, C C Chamis and L Minnetyan, ‘Prediction of composite laminate fracture: micromechanics and progressive fracture’, Compos Sci Techno, V58, pp 1137–1150, 1998, see also Chapter 3.1 in this book. 4 G C Eckold, ‘Failure criteria for use in the design environment’, Compos Sci Techno, V58, pp 1095–1106, 1998, see also Chapter 3.2 in this book. 5 E C Edge, ‘Stress based Grant-Sanders method for predicting failure of composite laminates’, Compos Sci Techno, V58, pp 1043–1044, 1998, see also Chapter 3.3 in this book. 6 L N McCartney, ‘Predicting transverse crack formation in cross-ply laminate’, Compos Sci Techno, V58, pp 1069–1082, 1998, see also Chapter 3.4 in this book.

1124 7 L J Hart-Smith, ‘Predictions of the original and truncated maximum strain failure models for certain fibrous composite laminates’, Compos Sci Techno, V58, pp 1151–1178, 1998, see also Chapter 3.5 in this book. 8 L J Hart-Smith, ‘Predictions of a generalised maximum-shear-stress failure criterion for certain fibrous composite laminates’, Compos Sci Techno, V58, pp 1179–1208, 1998, see also Chapter 3.6 in this book. 9 A Puck and H Schürmann, ‘Failure analysis of FRP laminates by means of physically based phenomenological models’, Compos Sci Techno, V58, pp 1045–1068, 1998, see also Chapter 3.7 in this book. 10 A Rotem, ‘Prediction of laminate failure with Rotem failure criterion’, Compos Sci Techno, V58, pp 1083–1094, 1998, see also Chapter 3.8 in this book. 11 C T Sun and J X Tao, ‘Prediction of failure envelopes and stress strain behaviours of composite laminates’, Compos Sci Techno, V58, pp 1125–1136, 1998, see also Chapter 3.9 in this book. 12 K-S Liu and S W Tsai, ‘A progressive quadratic failure criterion of a laminate’, Compos Sci Techno, V58, pp 1023–1032, 1998, see also Chapter 3.10 in this book. 13 W E Wolfe and T S Butalia, ‘A strain energy based failure criterion for nonlinear analysis of composite laminates subjected to biaxial loading’, Compos Sci Techno, V58, pp 1107–1124, 1998, see also Chapter 3.11 in this book. 14 P Zinoviev, S V Grigoriev, O V Labedeva and L R Tairova, ‘Strength of multilayered composites under plane stress state’, Compos Sci Techno, V58, pp 1209–1224, 1998, see also Chapter 3.13 in this book. 15 P D Soden, M J Hinton, and A S Kaddour, ‘Comparison of the predictive capabilities of current failure theories for composite laminates’, Compos Sci Techno, V58, pp 1225–1254, 1998, see also the up-date in Chapter 4 in this book. 16 M J Hinton, P D Soden and A S Kaddour, ‘Evaluation of failure prediction in composite laminates: background to ‘Part B’ of the exercise’, Compos Sci Techno, V62, pp 1481–1488, 2002, see also the up-date in Chapter 1 in this book. 17 P D Soden, M J Hinton and A S Kaddour, ‘Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates: Failure exercise benchmark data’, Compos Sci Techno, V62, pp 1489–1514, 2002, see also Chapter 2.2 in this book. 18 L J Hart-Smith, ‘Expanding the capabilities of the ten-percent rule for predicting the strength of fibre-polymer composites’, Compos Sci Techno, V62, pp 1515–1544, 2002, see also Chapter 3.17 in this book. 19 P K Gotsis, C C Chamis and L Minnetyan, ‘Application of progressive fracture analysis for predicting failure envelopes and stress-strain behaviors of composite laminates: A comparison with experimental results’, Compos Sci Techno, V62, pp 1545–1560, 2002, see also Chapter 5.1 in this book. 20 G C Eckold, ‘Failure criteria for use in the design environment – Part B’, Compos Sci Techno, V62, pp 1561–1570, 2002, see also Chapter 5.2 in this book. 21 E C Edge, ‘Theory v. experiment comparison for stress based Grant-Sanders method’, Compos Sci Techno, V62, pp 1571–1590, 2002, see also Chapter 5.3 in this book. 22 L J Hart-Smith, ‘Comparison between theories and test data concerning the strength of various fibre-polymer composites’, Compos Sci Techno, V62, pp 1591–1618, 2002, see also Chapter 5.4 in this book. 23 L N McCartney, ‘Predicting ply crack formation and failure in laminates’, Compos Sci Techno, V62, pp 1619–1632, 2002, see also Chapter 5.5 in this book. 24 A Puck and H Schürmann, ‘Failure analysis of FRP laminates by means of physically based phenomenological models- Part B –’, Compos Sci Techno, V62, pp 11633–1672, 2002, see also Chapter 5.6 in this book. 25 A Rotem, ‘The Rotem failure criterion theory and practice’ Compos Sci Techno, V62, pp 1663–1672, 2002, see also Chapter 5.7 in this book.

1125 26 C T Sun, J. Tao and A S Kaddour, ‘Prediction of failure envelopes and stress-strain behavior of composite laminates: Comparison with experimental results’, Compos Sci Techno, V62, pp 1672–1682, 2002, see also Chapter 5.8 in this book. 27 A Kuraishi, S W Tsai and K Liu, ‘A progressive quadratic failure criterion, Part B’, Compos Sci Techno, V62, pp 1682–1696, 2002, see also Chapter 5.9 in this book. 28 T S Butalia and W E Wolfe, ‘Strain energy based non-linear failure criterion: comparison of numerical predictions and experimental observations for symmetric composite laminates’, Compos Sci Techno, V62, pp 1695–1710, 2002, see also Chapter 5.10 in this book. 29 P A Zinoviev, O V Lebedeva and L P Tairova, ‘Coupled analysis of experimental and theoretical results on the deformation and failure of laminated composites under a plane state of stress’, Compos Sci Techno, V62, pp 11711–1724, 2002, see also Chapter 5.11 in this book. 30 M J Hinton, A S Kaddour and P D Soden, ‘A comparison of the predictive capabilities of current failure theories for composite laminates, judged against experimental evidence’, Compos Sci Techno, V62, pp 1725–1797, 2002, see the up-date this Chapter. 31 M J Hinton, P D Soden and A S Kaddour, ‘Evaluation of failure prediction in composite laminates: background to Part C of the exercise’, Compos Sci Techno, 2004, see also the up-date in Chapter 1 in this book. 32 T A Bogetti, C P R Hoppel, V M Harik, J F Newill and B P Burns, ‘Predicting the nonlinear response and progressive failure of composite laminates’, Compos Sci Techno, V64, pp 329–342, 2004, see also Chapter 3.15 in this book. 33 R G Cuntze and A Freund, ‘The predictive capability of failure mode concept – based strength criteria for multidirectional laminates’, Compos Sci Techno, V64, pp 343–377, 2004, see also Chapter 3.14 in this book. 34 S Mayes and A C Hansen, ‘Composite laminate failure analysis using multicontinuum theory’, Compos Sci Techno, V64, pp 379–394, 2004, see also Chapter 3.15 in this book. 35 Zheng-ming Huang, ‘A bridging model prediction of the tensile strength of composite laminates subjected to biaxial loads’, Compos Sci Techno, V64, pp 395–448, 2004, see also Chapter 3.16 in this book. 36 T A Bogetti, C P R Hoppel, V M Harik, J F Newill and B P Burns, ‘Predicting the nonlinear response and failure of composite laminates: correlation with experimental results’, Compos Sci Techno, V64, pp 477–485, 2004, see also Chapter 5.12 in this book. 37 R G Cuntze, ‘The predictive capability of failure mode concept – based strength criteria for multidirectional laminates: Part B’, Compos Sci Techno, V64, pp 487–516, 2004, see also Chapter 5.13 in this book. 38 S J Mayes and A C Hansen, ‘A comparison of multicontinuum theory based failure simulation with experimental results‘, Compos Sci Techno, V64, pp 517–527, 2004, see also Chapter 5.14 in this book. 39 Zheng ming Huang, ‘Correlation of the bridging model predictions of the biaxial failure strengths of fibrous laminates with experiments’, Compos Sci Techno, V64, pp 529–548, 2004, see also Chapter 5.15 in this book. 40 A S Kaddour, M J Hinton and P D Soden, ‘A comparison of the predictive capabilities of current failure theories for composite laminates: Additional contributions’, Compos Sci Techno, V64, pp 449–476, 2004, see also the up-date Chapter 4 in this book. 41 M J Hinton, P D Soden And A S Kaddour, ‘A further assessment of the predictive capabilities of current failure theories for composite laminates: comparison with experimental evidence’, Compos Sci Techno, V64, pp 549–588, 2004, see also the up-date in this Chapter of this book. 42 P D Soden, M J Hinton and A S Kaddour, ‘Recommendations for Designers and Researchers Resulting from the World-Wide Failure Exercise’, Compos Sci Techno, V64, pp 589–604, 2004, see also Chapter 7 in this book. 43 M J Hinton, A S Kaddour and P D Soden, ‘Predicting failure in fibre composites: The final chapter of the world-wide failure exercise’, Presented at ICCM-14, July 2003.

1126 44 L J Hart-Smith, ‘How to fake solutions for matrix-dominated failures in fibre-polymer composites – additional predictions for the world-wide failure exercise on predicting the strength of composite laminates’, in preparation.

8. Appendix A: Further details of the Qualitative assessment of the correlation between theoretical approaches and experimental results arranged on a Test Case-byTest Case basis The procedure adopted for comparing the theoretical and experimental results on a Test Case by Test Case basis was as follows: The predictions of all of the theories are shown superimposed on the experimental results for the 14 Test Cases in Figs 1–14. These figures were examined one at a time and a table was compiled for each Test Case showing the strengths and weaknesses of all the theoretical predictions. The severity of each weakness was assessed and classified as Fundamental, Significant or Minor. The tables, one for each Test Case, are labelled Tables A1–14. Each table contains the following information: 1 – Description of the Test Case, (see Table 1 for more details of the Test Cases). 2 – Features observed in the experiment for this Test Case, (see Table 4 for the various features observed in all of the Test Cases). 3 – Column 1 showing the theory used, see Table 2 for a description of the approaches represented. 4 – Column 2 showing Strengths: These points describe areas where good correlation is achieved between the theory and test data. 5 – Column 3 showing Weaknesses: These points describe areas where the correlation between theory and test data was seen to be imperfect and attempt to indicate (classify) the degree of weaknesses in the theory. 6 – Column 4 showing the figure number from which the information in Columns 2 and 3 are derived.

1127 Table A-1 Remarks on the correlation between theoretical prediction of all theories and experimental results for Test Case 1 Test Case No.: 1 Description: E-glass/LY556 epoxy unidirectional lamina tested under a combination of in-plane shear and transverse to fibre direction biaxial stresses (i.e. y versus xy envelope) Major Features The shape of the biaxial envelope, and the degree of fit for ultimate strength between theory and experiment TC1-1

The shear failure stress is decreased due to application of transverse tensile stress component and vice versa

TC1-2

The shear strength increases due to application of moderate compressive transverse stress component but decreases when large compressive stresses are applied. There is a point at which the shear failure stress reaches a maximum value.

Theory designator

Plus Points

Bogetti

Minus Points

Fig No.

Non-interactive failure theory does not fit the general shape of the experiments & is a little unconservative at some stress ratios. Minor Weakness.

1

Chamis

Conservative in one quadrant, good fit in the other quadrant.

Does not predict max shear stress. Minor Weakness.

1

Cuntze

Fits experiments reasonably well.

The curve predicted in Part A does not have a pronounced peak in the (–y, xy) quadrant. Minor Weakness. Revised envelope gives a better correlation by curve fitting. Minor Weakness.

1

In Part B Various fitted curves were tried and the best fitted parameters gave a good fit to test data. Eckold

Two envelopes, the inner using BS4994 is conservative.

Outer envelope very conservative in one quadrant and unconservative in another, Uniaxial compressive strength does not fit given data. Significant Weakness.

1

Edge

Theory fits the shape of envelope in one quadrant.

Non-interactive form, does not fit general shape in the second quadrant. Minor Weakness.

1

Conservative in one quadrant and un-conservative in the second. Minor Weakness.

1

Hart-Smith 2

1128 Theory designator

Plus Points

Hart-Smith 3

Minus Points

Fig No.

Does not fit the general shape of the experiments, extremely unconservative in one quadrant. Does not fit uniaxial strengths provided. Significant Weakness.

1

Hart-Smith 1

Theory fits the shape of the experimental envelope.

Predicted envelope does not reflect the basic equations governing the theory. Envelope not used in the laminate prediction. Minor Weakness in the non interactive theory actually used.

1

Huang

Predicts interaction and enhancement in shear strength.

Does not fit the general shape of the experiments, extremely unconservative in places. Does not fit uniaxial strengths provided. Significant Weakness.

1

Mayes

Fits experimental trend in the ( + y, xy) quadrant.

Does not fit general shape in the (–y, xy) quadrant. Gives conservative prediction. Minor Weakness.

1

No prediction offered. Fundamental Weakness.

N/A

McCartney Puck

Fits shape of test data well. Excellent revised Pt B predictions.

Rotem

Fits experiment very well in the ( + y, xy) quadrant.

Does not fit general shape in the (–y, xy) quadrant. Gives conservative prediction. Revised Pt B predictions indicate some Minor Weakness remains

1

Sun(L)

Fits experiment well in the ( + y, xy) quadrant.

Does not predict maximum shear in the (–y, xy) quadrant. Minor Weakness.

1

Case not attempted. Fundamental Weakness.

NA

Sun(NL)

1 1

Tsai

Fits shape well in both the ( + y, xy) & (–y, xy) quadrants. Excellent revised Pt B predictions.

Wolfe

Fits shape reasonably well in the ( + y, xy) quadrant.

Predictions slightly unconservative in parts. Max shear strength not reproduced.

Part B revisions fitted revised compression and shear strengths.

Pt B predictions did not greatly improve the fit. Minor Weakness.

1

1

1129 Theory designator Zinoviev

Plus Points

Minus Points

Fig No.

Non-interactive failure theory does not fit the general shape of the experiments & is a little unconservative at some stress ratios. Minor Weakness?

1

1130 Table A-2 Remarks on the correlation between theoretical prediction of all theories and experimental results for Test Case 2. Test Case No.: 2 Description: T300/914C carbon epoxy unidirectional lamina tested under a combination of in-plane shear and longitudinal to fibre direction biaxial stresses (i.e. xy versus x envelope) Major Features The shape of the biaxial envelope, and the degree of fit for ultimate strength between theory and experiment TC2-2

The addition of longitudinal tensile stress produces an increase in the shear failure stress, whereas the addition of a shear stress does not lead to an increase in the longitudinal tensile failure stress

TC2-3

The addition of longitudinal compressive stresses appears to reduce the shear strength although scatter makes interpretation of the results difficult.

Theory designator

Plus Points

Bogetti

Minus Points

Fig No.

Does not fit general shape very well. A little conservative in one quadrant, and un-conservative in the other. Minor Weakness.

2

Chamis

Generally conservative.

Does not predict max shear stress Minor Weakness.

2

Cuntze

Theory generally fits the test data reasonably well.

Did not capture possible increase in shear strength in the tension quadrant. Minor Weakness.

2

No prediction offered. Fundamental Weakness.

N/A

Does not fit general shape in the second quadrant very well. Minor Weakness.

2

Hart-Smith 2

Does not fit general shape very well, un-conservative in one quadrant, conservative in the other. Minor Weakness.

2

Hart-Smith 3

Does not fit general shape very well, slightly conservative in one quadrant, un-conservative in the other. Minor Weakness.

2

Hart-Smith 1

Does not fit general shape in the second quadrant, un-conservative in places. Minor Weakness.

2

Eckold Edge

Theory fits the shape of envelope in one quadrant reasonably well.

1131 Theory designator

Plus Points

Minus Points

Fig No.

Does not fit general shape very well. A little conservative in one quadrant, and un-conservative in the other. Minor Weakness.

2

Does not predict max shear stress. Minor Weakness.

2

No prediction offered. Fundamental Weakness.

N/A

Did not capture possible increase in shear strength in the tension quadrant. Minor Weakness.

2

Rotem

Does not fit general shape of envelope very well, Possibly slightly un-conservative in places. Minor Weakness.

2

Sun(L)

Does not fit general shape very well, conservative in one quadrant and unconservative in the other. Minor Weakness.

2

Sun(NL)

No prediction offered. Fundamental Weakness.

N/A

Huang

Mayes

Predicts interaction, generally conservative.

McCartney Puck

Theory generally fits the test data well in both the + x and –x quadrants.

Tsai

Good fit of the shape in both the + x & –x quadrants.

Did not predict the high measured maximum shear strength. Minor Weakness.

2

Wolfe

Good general fit of the shape in both the + x and –x quadrants.

Max shear strength not predicted. Minor Weakness.

2

Zinoviev

Does not fit general shape. A little conservative in one quadrant, and un-conservative in the other. Minor Weakness.

1132 Table A-3 Remarks on the correlation between theoretical prediction of all theories and experimental results for Test Case 3. Test Case No.: 3 Description: E-glass/MY750 epoxy unidirectional lamina tested under a combination of in-plane biaxial stress acting along and transverse to the fibre direction (i.e. x versus y envelope) Major Features The shape of the biaxial envelope, and the degree of fit for ultimate strength between theory and experiment TC3-3

Under combined longitudinal tension and transverse compression, the application of longitudinal stress component tends to reduce the transverse strength and vice versa. There is no enhancement in the strength over the uniaxial values when loaded under biaxial stress.

Theory designator

Plus Points

Minus Points

Fig No.

Bogetti

Good prediction in one quadrant.

Predicted uniaxial transverse compressive strength different from the one supplied and inconsistent with Fig 1. Step in envelope near uniaxial compression (-x). Biaxial tension and biaxial compression envelopes seem un-conservative but not proven. Three Significant Weaknesses.

3

Chamis

Good fit.

Maybe slightly un-conservative in one small region Minor Weakness.

3

Cuntze

Fits shape and magnitude of the experimental data reasonably well.

A little un-conservative in the + x & –y quadrant. Minor Weakness.

3

Eckold

Two envelopes, inner envelope conservative.

Inner envelope very conservative in some cases, outer envelope conservative in one quadrant and un-conservative in another. Significant Weakness.

3

Edge

Does not fit general shape of envelope, un-conservative in places. Minor Weakness.

3

Hart-Smith 2

Predicted uniaxial compressive strength different from the one supplied. Biaxial tension and biaxial compression envelopes seem un-conservative but not proven. Significant Weakness.

3

1133 Theory designator

Plus Points

Minus Points

Fig No.

Hart-Smith 3

Extremely un-conservative in transverse tension, predicted curve does not pass through measured uniaxial data. Significant Weakness.

3

Hart-Smith 1

Does not fit general shape of envelope very well, unconservative in places. Minor Weakness.

3

Huang

Extremely un-conservative in transverse tension, predicted curve does not pass through measured uniaxial data. Significant Weakness.

3

Mayes

Almost non-interactive failure prediction is a little unconservative (particularly in the ( + x, –y) quadrant). Minor Weakness.

3

McCartney

No prediction offered. Fundamental Weakness.

N/A

Puck

Fits shape and magnitude of the experimental data reasonably well.

A little unconservative in the + x & –y quadrant. Minor Weakness.

3

Rotem

Fits the shape and magnitude of the experimental data quite well.

Possibly a little unconservative. MinorWeakness. Revised Pt B gives more unconservative prediction and slightly overestimates transverse compressive strength. Minor Weakness.

3

Sun(L)

Non-interactive failure theory does not fit the experiments very well, a little un-conservative in the + x & –y quadrant. Minor Weakness.

3

Sun(NL)

Case not attempted. Fundamental Weakness.

N/A

Theory predicts 90% increase strength at SR = –14 : –1 in comparison with uniaxial strength values. Seems doubtful, but lack of experimental data in (–x, –y) quadrant to prove or disprove this possible Significant Weakness under biaxial compression

3

Tsai

Excellent fit to the shape and magnitude of the experimental data.

1134 Theory designator

Plus Points

Minus Points

Fig No.

Wolfe

Conservative prediction.

Theory produced discontinuity in the envelope near uniaxial longitudinal loading. Significant weakness. Revised Pt B predictions introduced a radically different shape of failure envelope with an unlikely bulge in the ( + x, + y) quadrant. Lack of experimental data in the( + x, + y) quadrant to prove or disprove this – Significant Weakness.

3

Non-interactive failure theory is a little unconservative (particularly in the ( + x, –y) quadrant). Minor Weakness.

3

Pt B revisions provide a better fit with tests data Also remove the discontinuity in the envelope.

Zinoviev

1135 Table A-4 Remarks on the correlation between theoretical prediction of all theories and experimental results for Test Case 4. Test Case No.: 4 Description: E-glass/LY556 epoxy ( ± 30°/90°) laminate tested under a combination of in-plane biaxial stress (x versus y envelope) Major Features The shape of the biaxial envelope, and the degree of fit for ultimate strength between theory and experiment TC4-2

The biaxial strength under combined tension-compression loading is lower than the uniaxial strength values

TC4-3

There is tendency of an enhancement in the biaxial tensile strength, i.e. the biaxial strength is slightly higher than the uniaxial strength values.

Theory designator

Plus Points

Minus Points

Fig No.

Bogetti

Fits shape and magnitude of the experimental data reasonably well.

Slightly un-conservative in the (x, –y) and (–x, –y) quadrants (though experimental data is sparse in the latter). Minor Weakness.

4

Chamis

Always conservative.

Predictions much too low in places. Significant Weakness.

4

Cuntze

Fits shape and magnitude of the experimental data reasonably well.

Slightly un-conservative in the (x, –y) and (–x, –y) quadrants (though experimental data is sparse in the latter). Minor Weakness. Part B prediction is a poorer fit to the experimental results.

4

Eckold

Outer envelope similar shape (but not magnitude) to experiments in 3 quadrants. Inner envelope conservative.

Inner envelope very conservative in some cases. Outer envelope unconservative in 3 quadrants, compression strengths overestimated. Significant Weakness.

4

Edge

Shape of envelope broadly fits the experimental data in shape and magnitude.

Theory is overly conservative in portions of one quadrant, unconservative in a portion of another quadrant. Minor weakness.

4

Pt B revised theory provided better fit in one quadrant.

Pt B revised theory did not improve the fit in one quadrant. No evidence that Pt B revised theory has solved the implied Minor Weakness in the original theory.

1136 Theory designator

Plus Points

Minus Points

Fig No.

Hart-Smith 2

Broadly fits shape of the experimental data. Truncation gives good fit in one quadrant.

Theory moderately un-conservative in places. Minor Weakness.

4

Hart-Smith 3

Broadly fits shape and magnitude of the experimental data.

Un-conservative in places. Minor Weakness.

4

Hart-Smith 1

Broadly fits shape of the experimental data.

Theory is generally moderately unconservative. Minor Weakness.

4

Prediction does not fit the experimental data in shape or magnitude. Very conservative in one quadrant ( + x, + y) and unconservative in another. Possible problems with post failure analysis. Significant Weakness.

4

Theory is slightly over conservative in portions of one quadrant, un-conservative in a portion of another quadrant. Minor Weakness.

4

No prediction offered. Fundamental Weakness.

N/A

Huang

Mayes

Shape of envelope broadly fits the experimental data in shape and magnitude.

McCartney Puck

Fits shape and magnitude of the experimental data reasonably well.

Slightly unconservative in the (x, –y) and (–x, –y) quadrants (though experimental data is sparse in the latter). Minor Weakness.

4

Rotem

Very conservative predictions in both the ( + x, + y) and the (–x, + y) quadrants.

Prediction does not fit the experimental data in shape or magnitude. Very conservative in two quadrants (reaching a factor of 5 at SR = 1/–1) and unconservative in a third, Possible problems with post failure analysis. Significant Weakness.

4

Sun(L)

Good fit to the shape and magnitude of the experimental data, in all four quadrants.

A little unconservative in the (–x, –y) quadrant (though experimental data is sparse). Minor Weakness.

4

Case not attempted. Fundamental Weakness.

N/A

Sun(NL)

1137 Theory designator

Plus Points

Minus Points

Fig No.

Tsai

Broadly fits the experimental data. Slightly conservative predictions in the ( + x, + y) and the ( + x, –y) quadrants. – Pt B revised predictions fit the shape and magnitude of the experimental envelope very well.

A little unconservative in the (–x, –y) quadrant (though experimental data is sparse).

4

Conservative predictions in the ( + x, + y) , (–x, + y) & (–x, + y) quadrants.

Overly conservative (reaching a factor of 5 under SR = 1/1) but generally within 50% of the measured data. Significant Weakness in post initial failure theory or its application.

Wolfe

Pt B revised theory gives a good fit to experiments in three quadrants.

– Slightly more unconservative in the (–x, –y) quadrant (though experimental data is sparse). Minor Weakness. 4

Un-conservative in the (-x, –y) quadrant (though experimental data is sparse). Significant Weakness. Zinoviev

Broadly fits shape and magnitude of the experimental data.

A little unconservative in (x, –y) and (–x, –y) quadrants (though experimental data is sparse in the latter). Minor Weakness.

4

1138 Table A-5 Remarks on the correlation between theoretical prediction of all theories and experimental results for Test Case 5. Test Case No.: 5 Description: E-glass/LY556 epoxy ( ± 30°/90°) laminate tested under a combination of in-plane biaxial stress (x versus xy envelope) Major Features The shape of the biaxial envelope, and the degree of fit for ultimate strength between theory and experiment TC5-1

The application of a shear stress to this laminate reduced the axial strength, i.e the direct strength in the presence of shear loading is always lower than the uniaxial axial strength

TC5-2

The shear strength exhibited a slight enhancement due to application of moderate values of axial tension.

Theory designator

Plus Points

Minus Points

Fig No.

Bogetti

Fits shape and magnitude of the experimental data very well.

Does not exactly capture the shape of the envelope under combined compression and shear. Minor Weakness.

5

Chamis

Always conservative.

Prediction too low. Significant Weakness.

5

Cuntze

Fits shape and magnitude of the experimental data very well.

Slightly un-conservative in one small region. Minor Weakness.

5

No prediction offered. Fundamental Weakness.

N/A

Doesn’t fit shape and magnitude of the experimental data in the second quadrant very well, gives some unconservative predictions. Minor Weakness.

5

Eckold Edge

Broadly fits the shape of the experimental data and is conservative in one quadrant. Pt B revised theory offers improved fit in one quadrant.

Hart-Smith 2

Fits shape and magnitude of the experimental data in part.

Un-conservative in some portions. Minor Weakness.

5

Hart-Smith 3

Fits shape and magnitude of the experimental data in part.

Slightly un-conservative in one portion. Minor Weakness.

5

Hart-Smith 1

Fits shape and magnitude of the experimental data in part.

Un-conservative in some portions. Minor Weakness.

5

Huang

Fits shape of envelope well, generally conservative.

Considerably conservative under uniaxial tension. Minor Weakness.

5

1139 Theory designator

Plus Points

Minus Points

Fig No.

Mayes

Broadly fits the shape of the experimental data and is conservative in both quadrants.

Predicted uniaxial tensile strength considerably lower than measured data. Minor Weakness.

5

No prediction offered. Fundamental Weakness.

N/A

McCartney Puck

Fits shape and magnitude of the experimental data very well.

Slightly unconservative in one small region. Minor Weakness.

5

Rotem

Very conservative predictions for all quadrants.

Prediction does not match the experimental data. Very small envelope, Possible problems with post failure analysis. Significant Weakness.

5

Sun(L)

Reasonably good fit of experimental data in some parts.

Overall, the fit is moderate and suggests that the theory is not capturing all of the correct features. Minor Weakness.

5

No prediction offered. Fundamental Weakness.

N/A

Broadly fits shape of the experimental data. Predictions are very conservative (i.e. safe) for both quadrants.

Overall the predictions are approx 50% too conservative suggesting that there is some Significant Weakness in the theory or its application.

5

Pt B revised theory produces a significantly different predicted envelope which is in slightly better agreement with experiment.

Pt B revision gives a little better accuracy in the ( + x, xy) quadrant but there is still Minor Weakness.

Very conservative predictions for both quadrants.

Extremely small envelope that does not match the experimental data (by a factor reaching 5 at SR = 1/1 but generally more than 100% deviation in many places). Overly conservative predictions seem to stem from Significant Weakness in post initial failure theory.

Pt B revised theory produces a radically different predicted envelope which is in better agreement with experiment.

With Pt B revision only Minor Weaknesses remains.

Sun(NL) Tsai

Wolfe

5

1140 Theory designator

Plus Points

Minus Points

Fig No.

Zinoviev

Best of theories overall (with Puck) in general shape & fit to experiment. Absence of zones where predictions are unconservative makes this the most appropriate theory for Test Case 5.

A little too conservative (30%) in the (–x, xy) quadrant?

5

Minor Weakness in theory

1141 Table A-6 Remarks on the correlation between theoretical prediction of all theories and experimental results for Test Case 6. Test Case No.: 6 Description: AS4/3501–6 carbon fibre/epoxy (0°/ ± 45°/90°) laminate tested under a combination of inplane biaxial stress (x versus y envelope) Major Features The shape of the biaxial envelope, and the degree of fit for initial and ultimate strength between theory and experiment TC6-3

The biaxial strengths are slightly higher than the uniaxial strength values.

Theory designator

Plus Points

Minus Points

Fig No.

Bogetti

Good agreement in one quadrant.

Slightly un-conservative under tension-compression and unconservative in compressioncompression quadrant where experimental results may have been influenced by buckling. Minor Weakness.

6

No prediction offered for final failure. Fundamental Weakness.

6

Chamis

Very low initial failure strength. Cuntze

Good agreement in one quadrant.

Eckold

Slightly un-conservative under tension-compression and unconservative in compressioncompression quadrant where experimental results may have been influenced by buckling. Minor Weakness.

6

No prediction offered. Fundamental Weakness.

N/A

Edge

Good agreement in one quadrant.

Slightly un-conservative under tension-compression and doubt about compression-compression quadrant. Minor weakness Very low initial strength prediction. Minor Weakness.

6

Hart-Smith 2

Good prediction of shape of experiments, truncation appears good. Fibre failure prediction good.

Possibly un-conservative and doubt about the compressioncompression quadrant. Minor Weakness.

6

1142 Theory designator

Plus Points

Minus Points

Fig No.

Hart-Smith 3

Good prediction of shape of experiments, truncation appears good. Fibre failure prediction good.

Possibly un-conservative and doubt about the compressioncompression quadrant. Minor Weakness.

6

Hart-Smith 1

Good agreement in two quadrants. Truncation in tension-compression seems good.

Possibly un-conservative, doubt about compression-compression quadrant. Minor Weakness.

6

Does not fit the shape of test data, very conservative in one quadrant, un-conservative in the others. Significant Weakness.

6

Unconservative in compressioncompression quadrant where experimental results may have been influenced by buckling. Sharp steps in predicted envelope. Minor Weakness.

6

No prediction offered. Fundamental Weakness.

N/A

Slightly un-conservative under tension-compression and doubt about compression-compression quadrant. Minor Weakness.

6

Unconservative in the (–x, –y) quadrant (though experimental data is possibly unreliable). Envelope does not match the experimental data at all in the tension-tension quadrant. Possible problems with post failure analysis. Significant Weakness.

6

Unconservative in the (–x, –y) quadrant (though experimental data is possibly unreliable). Minor Weakness.

6

No prediction offered. Fundamental Weakness.

N/A

Huang

Mayes

Good fit in one quadrant and part of another.

McCartney Puck

Good agreement in one quadrant.

Rotem

Sun(L)

Sun(NL)

Good fit in two quadrants. This theory introduces some truncations in the envelope in the region of (x : y = –1 : 1).

1143 Theory designator

Plus Points

Minus Points

Fig No.

Tsai

Unusual and distinctive envelope broadly fits some of the experimental data. Matches experimental data well in the (–x, + y) quadrant.

Theory predicts 300% increase in strength at SR = –1 : –1 in comparison with uniaxial strength values. Doubts over the validity of the experimental data in (–x, –y) quadrant mean that definitive proof or disproof of this theory is not possible. May be a Significant Weakness here.

6

Pt B revision slightly improves the fit to experiment in tensiondominated quadrants.

Pt B revision reproduces perceived weakness in the Pt A theory. Thus there remains a possible Significant Weakness in the theory for biaxial compressive modes of failure.

Predicted envelope fits the experimental data quite well in parts.

In-accurate prediction of fibre failure, Unconservative in the (–x, –y) quadrant (though experimental data is possibly unreliable, due to premature structural buckling rather than material failure).

– Pt B revision slightly improves the fit to experiment in the (–x, + y) quadrant.

– After Part B revision Minor Weakness still remains.

Shape and magnitude of the predictions are near identical to those of Edge, Puck, Hart-Smith (2). Broad agreement with the experimental envelope in shape and magnitude for the (x, y) & (–x, y) quadrants.

Un-conservative in the (–x, –y) quadrant (though experimental data is possibly unreliable, due to premature structural buckling rather than material failure). Minor Weakness in theory.

Wolfe

Zinoviev

6

6

1144 Table A-7 Remarks on the correlation between theoretical prediction of all theories and experimental results for Test Case 7. Test Case No.: 7 Description: Stress-strain curves for AS4/3501–6 carbon fibre/epoxy (0°/ ± 45°/90°) laminate tested under an in-plane y : x = 1 : 0 stress state Major Features The degree of fit between theory and experiment, for the shape of the stress-strain curve up to failure, and the values of strains and stress at failure TC7-1

The measured stress strain curves exhibited relatively linear behaviour up to final failure

TC7-2

There was a sign of a slight change in the slope at around 400 MPa stress

TC7-3

Final failure took place by catastrophic fibre fracture in tension.

Theory designator

Plus Points

Minus Points

Fig No.

Bogetti

Shape of the predicted stress-strain curve in quite good agreement with experiment, (failure strength and strain within 8%). Prediction of various modes of failure

Predicted curves softer than those measured. Predicted small steps in the curves not seen in the experiments. Minor Weakness.

7

Chamis

Theory fits the experiment at small strain.

Predicts low stiffness and strength. Significant Weakness.

7

Cuntze

Shape and magnitude of the predicted stress-strain curve in quite good agreement with experiment.

Predicted curve showed un-usual nonlinearity near the end. Problems were reported in stiffness reduction after initial failure. Significant Weakness.

7

Revised prediction gave larger failure stresses.

Revised curves still softer than measured data. Minor Weakness.

Eckold Edge

Theory fits well the shape and magnitude of stress strain curves.

Hart-Smith 2 Hart-Smith 3

Shape of the predicted curves in fair agreement with experiment. Good for predicting fibre failure.

No prediction offered. Fundamental Weakness.

N/A

No prediction offered. Fundamental Weakness.

7

No prediction offered. Fundamental Weakness.

N/A

Slightly softer Poisson’s strain response than observed. Minor Weakness.

7

1145 Theory designator

Plus Points

Huang

Mayes

Predicted final strength close to measured.

McCartney Prediction now offered in the Pt B revision. General shape of the predicted stress-strain curve predicted in Pt B is in good agreement with experiment.

Minus Points

Fig No.

Major departure from experiment at approx. 300 MPa, large axial strain predicted, significant effects of thermal stresses on final fibre tension failure. Significant Weakness.

7

Departure from experiment at approx. 400 MPa, large axial and hoop strains predicted. Significant Weakness.

7

No prediction offered in Pt A.

N/A

Pt B revision gave a reasonable strength prediction but the theory does handle hear dominated failure. Significant Weakness.

7

7

Puck

Shape and magnitude of the predicted stress-strain curve in very good agreement with experiment.

Rotem

Conservative strength prediction (75% of observed value).

Major departure from experiment at approx 350 MPa when initial failure was detected, resulting in an increment in strain (particularly in the minor direction) followed by bi-modular response to failure. Theory seems to have a problem in predicting beyond onset of initial failure. Significant Weakness.

7

Sun(L)

Shape of the predicted stress-strain curve in quite good agreement with experiment, (failure strength and strain within 7%). Prediction of various modes of failure.

Stress/ major strain response is identical to Sun (NL) prediction (slightly softer than experiment). Stress/ minor strain response differs from Sun (NL) prediction and is in excellent agreement with experiment.

7

Pt B revision provided a better representation of the theory.

Pt B revision did not improve the fit with test data and produced steps in the curves not seen in the experiments. Minor Weakness.

Shape and magnitude in good agreement with experiment (failure strength and strain within 10%).

Minor strain is 50% larger than measured strain. Strength prediction similar to that using linear analysis. Minor Weakness.

Sun(NL)

7

1146 Theory designator

Plus Points

Minus Points

Fig No.

Tsai

Shape of the predicted stress-strain curve in fair agreement with experiment (failure strength within 15% and failure strain within 10%).

Softer major and minor strain response than experiment.

7

Pt B revised prediction gives improved accuracy (failure strength within 5%). Wolfe

Zinoviev

Pt B revision does not change the oversoft strain prediction. Minor Weakness.

Shape of the predicted stress-strain curve in good agreement with experiment up to approx 450 MPa (60% of failure point).

Steps in curves and premature final failure. Significant Weakness.

Pt B revised theory produces a radical improvement – now in quite good agreement with experiment.

Pt B revision has greatly improved the initial to final failure prediction technique for this case. Significant Weakness.

Shape of the predicted stress-strain curve in good agreement with experiment (failure strength within 5% and strain within 10%).

7

7

1147 Table A-8 Remarks on the correlation between theoretical prediction of all theories and experimental results for Test Case 8. Test Case No.: 8 Description: Stress-strain curves for AS4/3501–6 carbon fibre/epoxy (0°/ ± 45°/90°) laminate tested under an in-plane y : x = 2 : 1 stress state Major Features The degree of fit between theory and experiment, for the shape of the stress-strain curve up to failure, and the values of strains and stress at failure TC8-1

Both the axial and hoop strain are positive (tensile)

TC8-2

The curves exhibited a slight change in the slope at approx y = 400MPa

TC8–3

The major strain response exhibited some non-linearity as failure is approached.

Theory designator

Plus Points

Minus Points

Fig No.

Bogetti

Shape of the predicted stress-major strain curve in good agreement with experiment (failure strength within 5%.

Predicted curves softer than those measured. Predicted steps in the curves not seen in the experiments. Minor Weakness.

8

Chamis Cuntze

Theory fits the stress strain curves Shape and magnitude of the predicted majorstress-strain curve in very good agreement with experiment.

Minor strain over estimated slightly. Minor Weakness.

Eckold Edge

No prediction offered. Fundamental Weakness. Theory fits well the shape and magnitude of stress strain curves.

Hart-Smith 2 Hart-Smith 3

Hart-Smith 1

Predicted stress-strain curve in good agreement with experiment.

8 8

N/A 8

No prediction offered. Fundamental Weakness.

N/A

Neglect of matrix failure, slightly overstiff strain response, change is slope in the experimental curves not captured. Minor Weakness.

8

No prediction offered. Fundamental Weakness.

N/A

1148 Theory designator

Plus Points

Huang

Mayes

Good correlation with the minor strain curve, final strength predicted well.

McCartney General shape of the predicted stress-strain curve predicted in Pt B is in good agreement with experiment.

Puck

Fig No.

Major departure of axial strain from experiment at approx 270 MPa Initial failure detected, resulting in a large increment in strain and immediate final failure. Problems in predicting beyond onset of initial failure and not capturing fibre tension failure. Substantial effects of thermal stresses Fundamental Weakness.

8

Departure from experiment at approx. 350 MPa. Minor Weakness.

8

No prediction offered in Pt A.

N/A

Pt B revision gave a reasonable strength prediction but the theory does handle hear dominated failure. Curve extended beyond fibre failure. Significant Weakness.

8

8

Shape and magnitude of the predicted stress-strain curve in very good agreement with experiment. Major departure from experiment at approx 300 MPa. Initial failure detected, resulting in a large increment in strain and immediate final failure. Problems in predicting beyond onset of initial failure and not capturing fibre tension failure. Significant Weakness.

8

Shape of the predicted stress-major strain curve in good agreement with experiment (failure strength within 2%.

Minor strain response 50% larger than that obtained from experiments, after initial failure at approx 200 MPa, probably stems from imprecise assumptions made about the residual lamina properties after initial failure.

8

Pt B revision provided a better representation of the theory.

Pt B revision did not improve the fit and produced steps in the curves. Minor Weakness.

Rotem

Sun(L)

Minus Points

1149 Theory designator

Plus Points

Sun(NL)

Shape of the predicted stress-strain curve in quite good agreement with experiment (failure strength and strain within 10%).

Tsai

Shape of the predicted stress-strain curve in quite good agreement with experiment (failure strength and strain within 10%).

Major strain response is slightly softer than experiment.

Pt B revised prediction gives slightly improved accuracy.

Pt B revision does not change the over-soft strain prediction. Minor Weakness.

Shape of the predicted stress-strain curve in fair agreement with experiment (failure strength and strains within 15%).

Steps in stress-strain curves, small changes in slope before final failure. Minor weakness in the theory or its application?

Pt B revised prediction gives improved accuracy (failure strength and strain within 10%).

The Pt B revisions moderate the Minor Weakness.

Wolfe

Zinoviev

Almost exact fit with experiment (failure strength and strain well within 5%). The best fit (together with Puck) of all Pt A theories for this Test Case.

Minus Points

Fig No. 8

8

8

8

1150 Table A-9 Remarks on the correlation between theoretical prediction of all theories and experimental results for Test Case 9. Test Case No.: 9 Description: E-glass/MY750 epoxy ( ± 55°) laminate tested under a combination of in-plane biaxial stress (x versus y envelope) Major Features The degree of fit between theory and experiment in terms of overall shape of the biaxial envelope, and the values of ultimate failure strength TC9-2

There are two envelopes (a) one representing weeping (leakage of liquid through the wall thickness in thin, unlined tubes) and (b)- another representing final failure

TC9-3

The test data, for both the leakage stresses and final failure stresses, in the tensiontension quadrant show that the biaxial strength is larger than the uniaxial strength.

Theory designator

Plus Points

Minus Points

Fig No.

Bogetti

Very good fit to experimental envelope in shape and magnitude for 80% of the region. Good fit in three quadrants.

The theory under-predicts the strength in one portion of the biaxial tension quadrant. Minor Weakness.

9a

Chamis

Theory broadly fits the shape of envelope. Fibre tension prediction. Conservative in places.

Un-conservative in places, Overly conservative in places. Significant Weakness.

9a

Cuntze

Very good fit to experimental envelope in shape and magnitude for 80% of the region. Good fit in three quadrants.

Underpredicts strength in part of biaxial tension quadrant. Minor Weakness.

9a

The revised envelope still slightly conservative in part of the tensiontension envelope an in the compression-compression quadrant. Un-conservative in a small area. Minor Weakness. Eckold

Outer envelope broadly fits the experimental data in shape and magnitude in two quadrants. Inner envelope conservative.

Outer envelope does not fit the experimental data in two quadrants, un-conservative. Significant Weakness.

9a

1151 Theory designator

Plus Points

Minus Points

Fig No.

Edge

Theory fits the shape and magnitude of half the envelope.

Theory does not fit all experimental results. Minor weakness.

9a

Pt B revisions are insignificant.

Pt B revision resulted in unconservative prediction in certain areas. Minor Weakness remains.

Hart-Smith 2

Theory fits the general trend.

Un-conservative in places, conservative in others. Minor Weakness.

9a

Hart-Smith 3

Envelope fits the experimental data in shape and magnitude quite well.

Slightly un-conservative in some places, conservative in others. Minor Weakness.

9a

Hart-Smith 1

Theory fits the general trend.

Un-conservative in places, conservative in others. Minor Weakness.

9a

Huang

Predicted envelope fits the leakage stresses well.

No distinction between fibre and matrix failure, very unconservative under biaxial tension. Fundamental Weakness.

9

Mayes

Fits measured data in certain areas.

Part A prediction very conservative. Fundamental Weakness.

9a

Revised curve produces better correlation in the tension-tension quadrant. Significant Weakness. McCartney

No prediction offered. Fundamental Weakness.

N/A

Puck

Very good fit to experimental envelope in shape and magnitude for 80% of the region. Good fit in three quadrants.

The theory has a unique open envelope which needs to be reconciled with experimental evidence. Significant Weakness.

9

Rotem

Strength predictions are conservative in all quadrants. Works best in the (–x, –y) quadrant.

Very conservative (a factor of 6 at SR = 3.3/1) under biaxial tension. Apparent lack of effective post failure matrix degradation mechanism when combined shear, transverse and longitudinal tensile stresses act at the lamina level. Fundamental Weakness.

9

1152 Theory designator

Plus Points

Minus Points

Fig No.

Sun(L)

Envelope fits the experimental data quite well. Good fit in the (–x, y) quadrant. Fair fit in the (–x, –y) quadrant.

Mixed fit to experiment in (x, y) quadrant – unconservative in places, very conservative elsewhere. Minor Weakness.

9

Case not attempted

N/A

Envelope broadly fits the experimental data in shape (80%) and magnitude (40%). Strength predictions are conservative in all quadrants.

No leakage prediction. Theory is quite conservative in the (–x, –y) and ( + x, + y) quadrants, particularly at the –2 : 1 ‘netting analysis’ condition. A Significant Weakness.

9

Pt B revised theory produces some improved accuracy in the (–x, –y) and ( + x, + y) quadrants. Good improvement.

Pt B revisions still conservative in places. Minor Weakness.

Moderate fit to experimental envelope overall. Best fit in the (–x, –y) quadrant.

Very conservative by a factor of 6.5 at SR = 2/1 in the ( + x, + y) quadrant, possibly due to inadequate treatment of post failure process and lack of discrimination between initial and final failure when there is high combined shear and transverse tensile stress acting at the lamina level.

Pt B revised theory produces some improved accuracy in the (–x, –y) quadrant and very minor improvement in the ( + x, + y) quadrant.

Pt B revisions do not overcome the Pt A shortfall. Significant Weakness remains.

Envelope broadly fits the experimental data. Strength predictions are conservative in all quadrants.

Upper line in the ( + x, + y) quadrant is not real as the envelope is open in this area. Significant Weakness remains here.

Sun(NL) Tsai

Wolfe

Zinoviev

9

9

1153 Table A-10 Remarks on the correlation between theoretical prediction of all theories and experimental results for Test Case 10. Test Case No.: 10 Description: E-glass/MY750 epoxy ( ± 55°) laminate tested under an in-plane stress state (y : x = 1 : 0) Major Features The degree of fit between theory and experiment, for the shape of the stress-strain curve up to failure, and the values of strains and stress at failure TC10-1

The behaviour is highly non-linear from low strains right up to failure

TC10-2

The specimens tested with no liner failed by weepage at a stress of approx 386 MPa

TC10-3

The specimens tested with a liner failed at a stress of 600 MPa

TC10-4

The failure strains were 9% in the hoop direction and approx –11% in the axial direction.

Theory designator

Plus Points

Minus Points

Fig No.

Bogetti

Theory captures part of the nonlinear behaviour, predicts initial modulus and initial part of the nonlinear curves very well.

The curves terminate at approx 2% strain (one fifth of the measured hoop strain and one seventh of the measured axial strain). Significant Weakness.

10

Theory does not fit the shape of stress strain curves except at low strain. Large horizontal plateau. Fundamental Weakness.

10

The shape and magnitude of the stress-strain curve is in very good agreement with experiment up to approx 3% strain (50% of measured final failure strain).

No leakage prediction, further work needed to capture final predicted strength. Unusual nonlinearity before failure. Significant Weakness.

10

The revised prediction gave a larger stress at failure.

The revised curves still show an unusual nonlinerity at approx. 400MPa. Significant Weakness.

Chamis

Cuntze

Eckold

Does not capture the shape of stress strain curves or modes of failure, Design allowables very low compared with leakage and fracture values. Fundamental Weakness.

10

1154 Theory designator

Plus Points

Minus Points

Fig No.

Edge

The shape and magnitude of stressstrain curve is in good agreement with experiment up to approx 3% strain.

The theory does not fit the experimental curves at large strains. Fundamental problem in post failure analysis. Fundamental Weakness.

10

No prediction offered. Fundamental Weakness.

N/A

Theory predicts very soft stress strain curves; predicted curves do not match the test data. Minor Weakness.

10

No prediction offered. Fundamental Weakness.

N/A

Hart-Smith 2 Hart-Smith 3

Gives conservative predictions of failure stress and strain.

Hart-Smith 1 Huang

Captured part of the nonlinear behaviour.

Predicted curves deviates considerably from test data at approx. 200MPa. Predicted stresses 50% lower than measured. Significant Weakness.

10

Mayes

Captured part of the nonlinear behaviour.

Failure strains more than 5 times lower than those measured. Significant Weakness.

10

No prediction offered in Pt A.

N/A

Prediction now offered in the Pt B revision.

Prediction diverges significantly from experiment by being overstiff (cannot model non-linear materials response). Significant Weakness.

10

Puck

The shape and magnitude of the stress-strain curve is in very good agreement with experiment up to approx 4% strain (50% of measured final failure strain).

No leakage prediction, further work needed to capture final predicted strength. Significant Weakness.

10

Rotem

Non-linear curves predicted.

Predicted curves terminate at approx 1.5% strain (less than 20% of the measured strain at final failure). Significant Weakness.

10

Linear prediction, predicted final strains lower than test data by a factor of more than 5. Significant Weakness.

10

Failure strains more than 5 times lower than those measured. Significant Weakness.

10

McCartney

Sun(L)

Sun(NL)

Captured part of the nonlinear behaviour.

1155 Theory designator

Plus Points

Minus Points

Fig No.

Tsai

Prediction captures initial modulus.

Linear response, replicates approximately 5% of the actual stress-strain response. Significant Weakness in the theory or its application.

10

Pt B revised prediction gives improved accuracy. The magnitude of the stress at ‘final failure’ equates well (within 5%) to the leakage stress observed experimentally.

The Pt B revised curves produced a horizontal plateau not seen in experiment. The revised theory now replicates approximately 60% of the observed stress-strain response. Clearly, Pt B revisions have corrected some of the Significant Weaknesses observed in the Pt A theory. Minor Weakness remains.

Theory captures part of the nonlinear behaviour, predicts Initial modulus and initial part of the nonlinear curves very well.

The curves terminate at approx 1.5% strain (one fifth of the measured hoop strain and one seventh of the measured axial strain). Significant Weakness

Pt B revised theory predicts a very small increase in strength.

Pt B revisions did not correct the Pt A shortfall. Significant Weakness remains.

The magnitude of the stress at ‘final failure’ equates well (within 5%) to the leakage stress observed experimentally.

Prediction is a bi-modular, linear approximation to the non-linear experimental curve – a reasonable fit with experiment up to approx 380 MPa. Curves terminate at this point, as final failure is detected. Thus the theory only replicates approximately 55% of the actual stress-strain response. Significant Weakness in the theory or its application?

Wolfe

Zinoviev

10

10

1156 Table A-11 Remarks on the correlation between theoretical prediction of all theories and experimental results for Test Case 11. Test Case No.: 11 Description: E-glass/MY750 epoxy ( ± 55°) laminate tested as a tube under an in-plane stress state (y : x = 2 : 1) (i.e. hoop : axial = 2 : 1) Major Features The degree of fit between theory and experiment, for the shape of the stress-strain curve up to failure, and the values of strains and stress at failure TC11-1

The behaviour of the hoop stress-hoop strain curve is relatively linear up to failure

TC11-2

The behaviour of the hoop stress-axial strain curve showed severe non-linearity starting at an earlier stage

TC11-3

Both the axial and hoop strains are tensile (positive)

TC11-4

The axial strain exceeds the hoop strain at hoop stresses above 400MPa

TC11-5

The specimens tested without a liner failed by weepage at a stress of around 280MPa while those tested with a liner failed by fibre fracture at a much higher stresses, up to 736MPa

TC11-6

Final failure was by fibre fracture.

Theory designator

Plus Points

Minus Points

Fig No.

Bogetti

Predicts strength only a little less than the measured strength. The axial strain crosses the hoop strain.

Theory predicts early part of stress (y) vs strain (x) curve well, up to approx 140 MPa. Thereafter the theoretical curve deviates from the observed trend. Steps in predicted curve not observed in tests. The other curve does not capture the nonlinear response observed in the test data. Significant Weakness.

11

Chamis

Theory fits one branch of stress strain curves.

Theory does not fit the second branch of stress strain curves. Significant Weakness.

11

Cuntze

Prediction fits well with one of the experimental curves. Strength is over predicted by 20%.

Theory does not predict the second branch of the experimental curves well. Significant Weakness.

11

The revised curves showed two stages of failure.

The revised curve still does not capture the shape of the stress versus axial strain curve well. Significant Weakness.

1157 Theory designator

Plus Points

Minus Points

Fig No.

Eckold

Design allowable conservative compared with observed leakage stress.

Does not capture the shape of stress strain curves or modes of failure. No strength prediction. Significant Weakness.

11

Edge

Theory does not fit the shape of one branch of the stress strain curves and gives large failure strain in the other branch of the curves. Fundamental Weakness.

11

Hart-Smith 2

No prediction offered. Fundamental Weakness.

N/A

Prediction does not fit the shape of the other branch of stress train curves, linear prediction, does not capture matrix failure. Minor Weakness.

11

No prediction offered. Fundamental Weakness.

N/A

Hart-Smith 3

Prediction fits the shape of one branch of stress strain curves.

Hart-Smith 1 Huang

Theory replicates the early part of the curves moderately well.

The curves terminate at very low strain. Prediction does not capture matrix failure progression, fibre tension failure and leakage failure. Theory has a problem in predicting beyond onset of initial failure. Fundamental Weakness.

11

Mayes

Predicted curves good at the initial stages.

Part A prediction captured only 10% of the measured curve. Fundamental Weakness.

11

Revised curves strength prediction close to measured data.

The revised curves deviate largely from measured data. Final strain for the fibre dominated curve is very conservative. The shape of axial strain does not resemble the measured curve, problems with post failure analysis. Fundamental Weakness.

McCartney Prediction now offered in the Pt B revision. One predicted branch of curves fits test data well.

No prediction offered in Pt A.

N/A

The theory diverges significantly from experimental ex by being over-stiff (cannot model non-linear materials response). Significant Weakness.

11

1158 Theory designator

Plus Points

Minus Points

Fig No.

Puck

Prediction fits well with one of the experimental curves. Strength is over predicted by 20%.

Theory does not predict the second branch of the experimental curves well (Possibly, partly due to ignoring fibre rotation in this case). Significant Weakness.

11

Rotem

Theory replicates the early part of the curves moderately well.

The curves terminate at very low strain. Prediction does not capture matrix failure progression, fibre tension failure and leakage failure. The theory only replicates approximately 17% of the actual stress-strain response. Theory has a problem in predicting beyond onset of initial failure. Fundamental Weakness.

11

Sun(L)

predicts strength only less than 10% higher than the measured strength. The axial strain crosses the hoop strain.

Theory predicts early part of stress () vs strain (x) curve well, up to approx 140 MPa. Thereafter the theoretical curve is too stiff. The other curve does not capture the nonlinear response observed in the test data. Significant Weakness.

11

Pt B revision gave a better representation of the theory.

Pt B revision shows steps in the curves and no improvement. Significant Weakness.

Sun(NL)

Major stress (y) vs major strain (y) curve fits the experiment reasonably well for most part of the curve.

The theory does not actually predict a final failure. Beyond 200Mpa the theoretical curve (y  x) is very unrepresentative of the experiment (hardening behaviour as opposed to softening behaviour). Significant Weakness.

11

Tsai

Major stress (y) vs major strain (y) curve fits shape of experiment reasonably well for most of the curve.

Theory predicts the initial part of major stress (y) vs minor strain (x) curve well, up to approx 250 MPa. Thereafter the theoretical curve is very unrepresentative of the experiment (too stiff).

11

Pt B revision provides an improvement in the strength prediction.

Pt B revision does not improve the (x) strain curve. There is still a Significant Weakness.

1159 Theory designator

Plus Points

Minus Points

Fig No.

Wolfe

The theory replicates the early part of the curve well.

The curves terminate at very low strain.. Theory appears to have difficulty in predicting beyond onset of initial failure. Fundamental Weakness in the theory or its application.

11

Pt B revision predicts a very small increase in strength.

Pt B revisions do not address the Pt A shortfall. Fundamental Weakness remains.

Zinoviev

The theory predict a bilinear behaviour of major stress (y) vs minor strain (x) curve. Both branches of the predicted curves are stiffer than the experimental curves. Minor Weakness.

11

1160 Table A-12 Remarks on the correlation between theoretical prediction of all theories and experimental results for Test Case 12. Test Case No.: 12 Description: E-glass/MY750 epoxy (0°/90°) laminate tested under a uniaxial in-plane stress state (y : x = 0 : 1) Major Features The degree of fit between theory and experiment, for the shape of the stress-strain curve up to failure, and the values of strains and stress at failure TC12-1

Initial response was linear

TC12-2

Onset of transverse matrix cracking in the 90° plies (initial failure) at 117.5 MPa

TC12-3

Reduction in longitudinal modulus from the initial value, accompanied by an increase in transverse crack density

TC12-4

Splitting in the longitudinal plies with an imperceptible change in longitudinal modulus (secondary failure) at 402 MPa

TC12-5

Catastrophic fibre fracture and total loss of integrity (final failure) at 609 MPa.

Theory designator

Plus Points

Minus Points

Fig No.

Bogetti

The predicted and experimental curves are very similar. Final strength close to measured. Two modes of failure captured.

Post failure analysis does not capture the smooth transition, Intermediate failure not predicted. Initial failure some 50% lower than those recorded in experiments. Minor Weakness.

12

Chamis

Theory predicts all stages of observed failure. Good fibre strength prediction.

Problems with predicting slope after initial and intermediate failure stages. Significant Weakness.

12

Cuntze

The predicted and experimental curves are similar. Final strength close to measured one (13% higher than the measured). Three modes of failure captured.

The initial and intermediate failure strengths are some 50% lower than those recorded in experiments. Minor Weakness.

12

The revised curves did not solve the above problems. Minor Weakness. Eckold

Design allowable transverse strain (0.1%) conservative compared with observed matrix cracking.

Does not predict the shape of stress strain curves. No strength prediction. Design allowable very conservative compared with observed final failure. Fundamental Weakness.

12

1161 Theory designator

Plus Points

Minus Points

Fig No.

Edge

The predicted and measured curves are similar in shape and magnitude.

Did not predict the second, intermediate, mode of failure. Initial failure stress low. Minor Weakness.

12

No prediction offered. Fundamental Weakness.

N/A

No predictions of modes of failure. Significant Weakness.

12

No prediction offered. Fundamental Weakness.

N/A

Hart-Smith 2 Hart-Smith 3

Predicted stress strain curves and failure strains fit experiments.

Hart-Smith 1 Huang

Good prediction of the shape of one branch of stress strain curves.

Theory did not capture all stages (modes) of failure. Predicted transverse strain larger than experimental. Final failure strength low. Significant Weakness.

12

Mayes

Three modes of failure captured, final strength close to measured.

Initial stress strain curves showed peculiar and unexpected behaviour indicating difficulties in dealing with thermal stresses. Initial failure stresses well below the measured values. The predicted stress strain curve is softer than measured and showed steps. Minor Weakness.

12

McCartney

Theory predicts crack density.

Theory did not capture the various modes of failure.

12

Pt B revision gave improved stresses above those predicted in Part A.

Pt B revision does not capture intermediate mode of failure and the stress/strain curves were extended beyond fibre failure strain. Significant Weakness.

Puck

The predicted and experimental curves are similar. Final strength close to measured one (13% higher than the measured). Three modes of failure captured.

The initial and intermediate failure strengths are some 50% lower than those recorded in experiments. Minor Weakness.

12

Rotem

Final strength close to measured one (6% higher than the measured). Good prediction of one branch of stress strain curves.

Theory did not capture the intermediate mode of failure. Predicted transverse strain larger than experimental (by a factor of 6). Significant Weakness.

12

1162 Theory designator

Plus Points

Sun(L)

Final strength close to measured one (6% higher than the measured). Captured three modes of failure.

Minus Points

12

Pt B revision reflects what the theory actually predicts.

Pt B revision did not significantly alter the fit between prediction and experiments. Steps in curves. Minor Weakness.

Sun(NL)

Final predicted point conservative (86% of measured value). Major strain curve in good agreement with test data.

No final strength prediction. Theory did not show the various modes of failure. The behaviour of the transverse strain curve deviates slightly from the experimental curve. Significant Weakness.

Tsai

Predicted curves resemble the observed ones.

Only two stages (modes) of failure predicted. Final strength reached only two third of the measured value. Significant Weakness.

Pt B revision was able to capture the three observed modes of failure. Improvement in strength prediction (within 5% of measured strength). Very good revised prediction. Wolfe

Zinoviev

Captures the initial behaviour of the experimental curves.

The theory did not predict the intermediate mode of failure. Final strength less than 50% of the measured value. Significant Weakness in the theory or its application?

Pt revision provides a better strength prediction and reaches 85% of the measured value.

Pt B revision is still unable to predict the second mode of failure. Significant Weakness.

All three modes of failure were captured. Final Strength prediction is 8% higher than measured value. Very good agreement.

Fig No.

12

12

12

1163 Table A-13 Remarks on the correlation between theoretical prediction of all theories and experimental results for Test Case 13. Test Case No.: 13 Description: E-glass/MY750 epoxy ( ± 45°) laminate tested as a tube under an in-plane stress state (y : x = 1 : 1) (i.e. hoop : axial = 1 : 1) Major Features The degree of fit between theory and experiment, for the shape of the stress-strain curve up to failure, and the values of strains and stress at failure TC13-1

The initial stress-strain response was linear-elastic

TC13-2

Onset of transverse matrix cracking (initial failure) at a stress ranging from 50 to 70 MPa

TC13-3

Reduction in the slope of the stress-strain curves after the cracking started, accompanied by an increase in transverse crack density

TC13-4

For the unlined specimens, leakage at a stress of 216 MPa

TC13-5

For the lined specimens, the final failure was by fibre fracture (at 502 MPa stress).

TC13-6

The hoop strain at failure was slightly larger than the axial strain.

Theory designator

Plus Points

Minus Points

Fig No.

Bogetti

Shape of predicted curve resembles that observed.

Theory predicts the strains y and x to be coincident, predicted strength was some 20% larger than the measured strength. The curve shows steps in the predicted curve not observed in tests. Significant Weakness.

13

Chamis

Theory predicts two stages of failure. Good fibre strength prediction.

Theory does not fit the shape of stress strain curves. Leakage failure not predicted. Significant Weakness.

13

Cuntze

The predicted and measured curves are similar in shape and magnitude.

Step in curve. Predicted strength was some 20% larger than the measured strength. Significant Weakness.

13

Revised curve gave a smoother transition beyond initial failure.

The revised prediction still gives un-conservative strength prediction, not capturing strain deviations. Minor Weakness.

1164 Theory designator

Plus Points

Minus Points

Fig No.

Eckold

Design allowable transverse strain (0.1%) conservative compared with observed matrix cracking.

Does not predict the shape of stress strain curves. No strength prediction. Design allowable very low compared with observed leakage and fracture. Significant Weakness.

13

Edge

The predicted and measured curves are similar in shape and magnitude.

Un-conservative strength prediction, not capturing strains deviations, Minor Weakness.

13

Hart-Smith 2

No prediction offered. Fundamental Weakness.

N/A

Hart-Smith 3

Linear behaviour. Not capturing matrix cracking and change in slope. Minor Weakness.

13

Hart-Smith 1

No prediction offered. Fundamental Weakness.

N/A

Huang

Theory replicates the early part of the curves moderately well.

Truncated curve predicted with a strength of less than 40% of the measured value and with a predicted failure strain of 20% of measured value. No leakage prediction. Theory unable to predict final fibre tension failure. Fundamental Weakness.

13

Mayes

Shape of predicted curve resembles that observed.

Part A submission produced truncated curves covering only 10% of the measured curve. Fundamental Weakness.

13

Pt B revision provides a better correlation with test data.

The revised curve Theory predicts the strains y and x to be coincident, predicted strength was some 20% larger than the measured strength. The revision shows steps in the stress strain curve not observed in tests. Significant Weakness.

1165 Theory designator

Plus Points

Minus Points

Fig No.

McCartney

Ability to predict crack density.

Predicted curve stiffer than the measured one. No final strength prediction. Significant Weakness.

13

Pt B revision resulted in an improved (extension) of the stress strain curves.

Predicted curves stiffer than those observed, prediction extended beyond fibre failure strain, initial failure stress twofold above cracking level. Significant Weakness.

Puck

The predicted and measured curves are similar in shape and magnitude.

Un-conservative strength prediction, not capturing strain deviations. Minor Weakness.

13

Rotem

Theory replicates the early part of the curves moderately well.

Truncated curve predicted with a strength of less than 20% of the measured value and with a predicted failure strain of 10% of measured value. No leakage prediction. Theory unable to predict final fibre tension failure. Fundamental Weakness.

13

Sun(L)

Shape of predicted curve resembles that observed.

Theory predicts the strains y and x to be coincident, predicted strength was some 20% larger than the measured strength.

13

Pt B revision provides an accurate representation of the theory.

Pt B revision shows steps in the stress strain curve not observed in tests. Significant Weakness.

Sun(NL)

Shape and magnitude of predicted curve in reasonable agreement with experiments.

No final strength prediction. Significant Weakness.

13

Tsai

Conservative prediction of final strength.

Theory predicts both strains to be identical.

13

Pt B revision provided an improvement in strength prediction.

Minor Weakness in terms of capturing final failure and leakage.

1166 Theory designator

Plus Points

Minus Points

Fig No.

Wolfe

Initial part of curve predicted.

Curves truncated at very low strains. Does not capture the softening behaviour of stress strain curves. Final strength prediction is more than 7 times lower than measured strength. Fundamental Weakness.

13

Pt B revision resulted in a very small improvement in the strength.

Pt B revision still under-predicts the strength (by a factor of approx 5). There is still a Fundamental Weakness in the Pt B prediction.

The predicted and measured curves are broadly similar.

Predicted curves stiffer than the measured curves and predicted final strength is 20% higher than average measured strength. Minor Weakness.

Zinoviev

13

1167 Table A-14 Remarks on the correlation between theoretical prediction of all theories and experimental results for Test Case 14. Test Case No.: 14 Description: E-glass/MY750 epoxy ( ± 45°) laminate tested as a tube under an in-plane stress state (y : x = 1 : –1) (i.e. hoop : axial = 1 : –1) Major Features The degree of fit between theory and experiment, for the shape of the stress-strain curve up to failure, and the values of strains and stress at failure TC14-1

The stress-strain curves are highly non-linear

TC14-2

The stress-strain curves above 2% strain become flatter than those at small strain values

TC14-3

No effect of the presence of a liner was observed in the tests

TC14-4

The final failure stress was around 95 MPa

TC14-5

The final failure strains reached 12%. This is much larger than the shear strain at failure given for a unidirectional lamina loaded in pure shear, which was derived from test on a circumferentially wound tube loaded in torsion.

Theory designator

Plus Points

Minus Points

Fig No.

Bogetti

Predicted curves are in agreement with trend of measured data up to 2% strain.

Theory unable to capture the large final measured strains (a factor of more than 5). Fundamental Weakness.

14

Theory does not fit the shape of stress strain curves. Maximum predicted strain very low. Fundamental Weakness.

14

Part A submission predicted the nonlinear shape of the stress strain curve.

Theory unable to capture the large final measured strains (a factor of more than 5). Fundamental Weakness.

14

The revised curves gave improved predicted failure strains.

The revised curves were stiffer than measured below 4% strain and softer than measured above 4% strain. Minor Weakness.

Chamis

Cuntze

1168 Theory designator

Plus Points

Minus Points

Fig No.

Does not predict the shape of stress strain curves. No strength prediction. Design allowable for this shear dominated case (implied direct strain 0.2%) very conservative compared with observed final failure strain. Fundamental Weakness.

14

Curves at large strain unconservative and un-representative. Fundamental Weakness in post failure analysis.

14

Hart-Smith 2

No prediction offered. Fundamental Weakness.

N/A

Hart-Smith 3

Not capturing initial stiffness nor the nonlinear shape of experimental curves. Fundamental Weakness.

14

Hart-Smith 1

No prediction offered. Fundamental Weakness.

N/A

Eckold

Edge

Theory fits the shape of curves and captures non-linear behaviour at low strains.

Huang

Predicted curves showing good non-linear behaviour, final strains captured well.

Predicted curves are above the measured ones, predicted final stresses 30% larger than measured. Significant Weakness.

14

Mayes

Predicted curves are in agreement with measured data up to 2% strain.

Theory unable to capture the large final measured strains (a factor of more than 5). Fundamental Weakness.

14

No prediction offered, theory unable to predict failure under shear. Fundamental Weakness.

N/A

McCartney

Puck

Predicted curves are in agreement with measured data up to 2% strain.

Theory unable to predict large deformation. Fundamental Weakness.

14

Rotem

Non-linear curves up to 2% strain.

Stiffness over-predicted. Theory unable to predict large deformation. Predicted strains cover only 0.16% of measured value. Fundamental weakness.

14

Pt B revision provided a closer fit with measured final strain values.

Pt B still under-predicts the failure strain by a factor of 2. Significant Weakness.

1169 Theory designator

Plus Points

Sun(L)

Sun(NL)

Fig No.

Theory unable to capture nonlinear behaviour and the large final measured strains. Predicted strains are some 20 times lower than experimental value. Fundamental Weakness. predicted curves are in agreement with measured data up to 2% strain.

Tsai

Wolfe

Minus Points

Theory unable to capture the large final measured strains (a factor of more than 5). Fundamental Weakness.

14

Theory unable to predict nonlinear behaviour and capture the large final measured strains.. Fundamental Weakness in of the theory for large deformation?

14

Pt B revision offered.

Pt B revision produced a horizontal plateau dissimilar to the observed trend. Fundamental Weakness remains.

Predicted curves are in agreement with measured data up to 2% strain.

Theory unable to capture the large final measured strains by a factor of 5 to 6.

14

Fundamental Weakness in the theory for large deformation? Zinoviev

Theory gave a good prediction of the stress strain curve up to final failure. Good for large deformation.

Theory could be improved by taking the nonlinear behaviour into account. Minor Weakness.

14

1170 Appendix B: Further details of the Qualitative assessment of the correlation between theoretical approaches and experimental results, arranged on theory-by-theory basis In this appendix details are presented for assessing qualitatively the fit between the individual theories and the 14 Test Cases used in the ‘exercise’ and the degree of weaknesses in these theories. Basically, the information in Tables A1–A14, Appendix A, was transposed to produce the Tables B1–B18 contained in Appendix B. Each of the Tables B1–B18 (shown below) contains the following information for one theory: • • • • •

Theory number and Name Column 1: Test Case number Column 2: Strengths Column 3 Weaknesses Column 4: Figure Number from which the comments in Columns 2 and 3 were derived.

Careful reading of Tables B1–B18, together with study of Figs 1–14 should give a picture of the overall performance of each theory. To convey our overall impressions of the theories’ performance the results are summarised in Tables B19 and B20. After the weaknesses had been classified as Fundamental, Significant or Minor, Table B19 was constructed to summarise the places (Test Cases) where these weaknesses were observed. The total number of weaknesses in each category and the number of cases for which no solution was offered were then counted and recorded for each theory in Table B20. The allocation of the categories Fundamental Significant and Minor is somewhat subjective and an alternative procedure for assessing and ranking the theories is presented in Appendix C.

1171 Table B-1 Remarks on the correlation between Bogetti’s theory and experimental data for the 14 Test Cases. Theory No: 1 Bogetti Test Case

Plus Points

Minus Points

Fig No.

1

Non-interactive failure theory does not fit the general shape of the experiments & is a little un-conservative at some stress ratios. Minor Weakness.

1

2

Does not fit general shape very well. A little conservative in one quadrant, and un-conservative in the other. Minor Weakness.

2

3

Good prediction in one quadrant.

Predicted uniaxial transverse compressive strength different from the one supplied and inconsistent with Fig 1. Step in envelope near uniaxial compression (–x). Biaxial tension and biaxial compression envelopes seem unconservative but not proven. Three Significant Weaknesses.

3

4

Fits shape and magnitude of the experimental data reasonably well.

Slightly un-conservative in the (x, –y) and (–x, –y) quadrants (though experimental data is sparse in the latter). Minor Weakness.

4

5

Fits shape and magnitude of the experimental data very well.

Does not exactly capture the shape of the envelope under combined compression and shear. Minor Weakness.

5

6

Good agreement in one quadrant.

Slightly un-conservative under tensioncompression and unconservative in compression-compression quadrant where experimental results may have been influenced by buckling. Minor Weakness.

6

7

Shape of the predicted stress-strain curve in quite good agreement with experiment, (failure strength and strain within 8%). Prediction of various modes of failure.

Predicted curves softer than those measured. Predicted small steps in the curves not seen in the experiments. Minor Weakness.

7

8

Shape of the predicted stress-major strain curve in good agreement with experiment (failure strength within 5%.

Predicted curves softer than those measured. Predicted steps in the curves not seen in the experiments. Minor Weakness.

8

1172 Test Case

Plus Points

Minus Points

Fig No.

9

Very good fit to experimental envelope in shape and magnitude for 80% of the region. Good fit in three quadrants.

The theory under-predicts the strength in one portion of the biaxial tension quadrant. Minor Weakness.

9

10

Theory captures part of the nonlinear behaviour, predicts initial modulus and initial part of the nonlinear curves very well.

The curves terminate at approx 2% strain (one fifth of the measured hoop strain and one seventh of the measured axial strain). Significant Weakness.

10

11

Predicts strength only a little less than the measured strength. The axial strain crosses the hoop strain.

Theory predicts early part of stress (y) vs strain (x) curve well, up to approx 140 MPa. Thereafter the theoretical curve deviates from the observed trend. Steps in predicted curve not observed in tests The other curve does not capture the nonlinear response observed in the test data. Significant Weakness.

11

12

The predicted and experimental curves are very similar. Final strength close to measured. Two modes of failure captured.

Post failure analysis does not capture the smooth transition, Intermediate failure not predicted. Initial failure some 50% lower than those recorded in experiments. Minor Weakness.

12

13

Shape of predicted curve resembles that observed.

Theory predicts the strains y and x to be coincident, predicted strength was some 20% larger than the measured strength. The curve shows steps in the predicted curve not observed in tests. Significant Weakness.

13

14

Predicted curves are in agreement with trend of measured data up to 2% strain.

Theory unable to capture the large final measured strains (a factor of more than 5). Fundamental Weakness.

14

1173 Table B-2 Remarks on the correlation between Chamis’s theory and experimental data for the 14 Test Cases. Theory No: 2 Chamis Test Case

Plus Points

Minus Points

Fig No.

1

Conservative in one quadrant, good fit in the other quadrant.

Does not predict max shear stress. Minor Weakness.

1

2

Generally conservative.

Does not predict max shear stress. Minor Weakness.

2

3

Good fit.

Maybe slightly un-conservative in one small region. Minor Weakness.

3

4

Always conservative.

Predictions much too low in places. Significant Weakness.

4

5

Always conservative.

Prediction too low. Significant Weakness.

5

No prediction offered for final failure. Fundamental Weakness.

N/A

Very low initial failure strength.

6

Predicts low stiffness and strength. Significant Weakness.

7

Theory fits the stress strain curves.

8

Un-conservative in places, Overly conservative in places. Significant weakness.

9

Theory does not fit the shape of stress strain curves except at low strain. Large horizontal plateau. Fundamental Weakness.

10

6

7

Theory fits the experiment at small strain.

8 9

Theory broadly fits the shape of envelope. Fibre tension prediction. Conservative in places.

10

11

Theory fits one branch of stress strain curves.

Theory does not fit the second branch of stress strain curves. Significant Weakness.

11

12

Theory predicts all stages of observed failure. Good fibre strength prediction.

Problems with predicting slope after initial and intermediate failure stages. Significant Weakness.

12

13

Theory predicts two stages of failure. Good fibre strength prediction.

Theory does not fit the shape of stress strain curves. Leakage failure not predicted. Significant Weakness.

13

Theory does not fit the shape of stress strain curves. Maximum predicted strain very low. Fundamental Weakness.

14

14

1174 Table B-3 Remarks on the correlation between Cuntze’s theory and experimental data for the 14 Test Cases. Theory No: 3 Cuntze Test Case

Plus Points

Minus Points

Fig No.

1

Fits experiments reasonably well. In Part B Various fitted curves were tried and the best fitted parameters gave a good fit to test data.

The curve predicted in Part A does not have a pronounced peak in the (–y xy) quadrant. Minor Weakness. Revised envelope gives a better correlation by curve fitting. Minor Weakness.

1

2

Theory generally fits the test data reasonably well.

Did not capture possible increase in shear strength in the tension quadrant. Minor Weakness.

2

3

Fits shape and magnitude of the experimental data reasonably well.

A little un-conservative in the + x & –y quadrant. Minor Weakness.

3

4

Fits shape and magnitude of the experimental data reasonably well.

Slightly un-conservative in the (x, –y) and (–x, –y) quadrants (though experimental data is sparse in the latter). Minor Weakness. Part B prediction is a poorer fit to the experimental results.

4

5

Fits shape and magnitude of the experimental data very well.

Slightly un-conservative in one small region. Minor Weakness.

5

6

Good agreement in one quadrant.

Slightly un-conservative under tensioncompression and unconservative in compression-compression quadrant where experimental results may have been influenced by buckling. Minor Weakness.

6

7

Shape and magnitude of the predicted stress-strain curve in quite good agreement with experiment.

Predicted curve showed un-usual nonlinearity near the end. Problems were reported in stiffness reduction after initial failure. Significant Weakness.

7

Revised prediction gave larger failure stresses.

Revised curves still softer than measured data. Minor Weakness.

Shape and magnitude of the predicted major stress-strain curve in very good agreement with experiment.

Minor strain over estimated slightly. Minor Weakness.

8

8

1175 Test Case

Plus Points

Minus Points

Fig No.

9

Very good fit to experimental envelope in shape and magnitude for 80% of the region. Good fit in three quadrants.

Underpredicts strength in part of biaxial tension quadrant. Minor Weakness.

9

The revised envelope still slightly conservative in part of the tensiontension envelope an in the compressioncompression quadrant. Un-conservative in a small area. Minor Weakness. 10

11

12

The shape and magnitude of the stressstrain curve is in very good agreement with experiment up to approx 3% strain (50% of measured final failure strain).

No leakage prediction, further work needed to capture final predicted strength. Unusual nonlinearity before failure.

The revised prediction gave a larger stress at failure.

The revised curves still show an unusual nonlinerity at approx. 400MPa. Significant Weakness.

Prediction fits well with one of the experimental curves. Strength is over predicted by 20%.

Theory does not predict the second branch of the experimental curves well. Significant Weakness.

The revised curves showed two stages of failure.

The revised curve still does not capture the shape of the stress versus axial strain curve well. Significant Weakness.

The predicted and experimental curves are similar. Final strength close to measured one (13% higher than the measured). Three modes of failure captured.

The initial and intermediate failure strengths are some 50% lower than those recorded in experiments. Minor Weakness.

10

11

12

The revised curves did not solve the above problems. Minor Weakness. 13

14

The predicted and measured curves are similar in shape and magnitude.

Step in curve. Predicted strength was some 20 % larger than the measured strength. Significant Weakness.

Revised curve gave a smoother transition beyond initial failure.

The revised prediction still gives unconservative strength prediction, not capturing strain deviations. Minor Weakness.

Part A submission predicted the nonlinear shape of the stress strain curve.

Theory unable to capture the large final measured strains (a factor of more than 5). Fundamental Weakness.

The revised curves gave improved predicted failure strains.

The revised curves were stiffer than measured below 4% strain and softer than measured above 4% strain. Minor Weakness.

13

14

1176 Table B-4 Remarks on the correlation between Eckold’s theory and experimental data for the 14 Test Cases. Theory No: 4 Eckold Test Case

Plus Points

Minus Points

Fig No.

1

Two envelopes, the inner using BS4994 is conservative.

Outer envelope very conservative in one quadrant and un-conservative in another, Uniaxial compressive strength does not fit given data. Significant Weakness.

1

No prediction offered*.

N/A

2 3

Two envelopes, inner envelope conservative.

Inner envelope very conservative in some cases, outer envelope conservative in one quadrant and un-conservative in another. Significant Weakness.

3

4

Outer envelope similar shape (but not magnitude) to experiments in 3 quadrants. Inner envelope conservative.

Inner envelope very conservative in some cases. Outer envelope unconservative in 3 quadrants, compression strengths over-estimated. Significant Weakness.

4

5

No prediction offered*.

N/A

6

No prediction offered*.

N/A

7

No prediction offered*.

N/A

8

No prediction offered*.

N/A

Outer envelope does not fit the experimental data in two quadrants, unconservative. Significant Weakness.

9

Does not capture the shape of stress strain curves or modes of failure, Design allowables very low compared with leakage and fracture values. Fundamental Weakness.

10

9

Outer envelope broadly fits the experimental data in shape and magnitude in two quadrants. Inner envelope conservative.

10

11

Design allowable conservative compared with observed leakage stress.

Does not capture the shape of stress strain curves or modes of failure. No strength prediction. Significant Weakness.

11

12

Design allowable transverse strain (0.1%) conservative compared with observed matrix cracking.

Does not predict the shape of stress strain curves. No strength prediction. Design allowable very conservative compared with observed final failure. Fundamental Weakness.

12

1177 Test Case

Plus Points

Minus Points

Fig No.

13

Design allowable transverse strain (0.1%) conservative compared with observed matrix cracking.

Does not predict the shape of stress strain curves. No strength prediction. Design allowable very low compared with observed leakage and fracture. Significant Weakness.

13

Does not predict the shape of stress strain curves. No strength prediction. Design allowable for this shear dominated case (implied direct strain 0.2%) very conservative compared with observed final failure strain. Fundamental Weakness.

14

14

* Fundamental weakness.

1178 Table B-5 Remarks on the correlation between Edge’s theory and experimental data for the 14 Test Cases. Theory No: 5 Edge Test Case

Plus Points

Minus Points

Fig No.

1

Theory fits the shape of envelope in one quadrant.

Non-interactive form, does not fit general shape in the second quadrant. Minor Weakness.

1

2

Theory fits the shape of envelope in one quadrant reasonably well.

Does not fit general shape in the second quadrant very well. Minor Weakness.

2

Does not fit general shape of envelope, un-conservative in places. Minor Weakness.

3

Shape of envelope broadly fits the experimental data in shape and magnitude.

Theory is overly conservative in portions of one quadrant, unconservative in a portion of another quadrant. Minor Weakness.

4

Pt B revised theory provided better fit in one quadrant.

Pt B revised theory did not improve the fit in one quadrant. No evidence that Pt B revised theory has solved the implied Minor Weakness in the original theory.

Broadly fits the shape of the experimental data and is conservative in one quadrant.

Doesn’t fit shape and magnitude of the experimental data in the second quadrant very well, gives some unconservative predictions. Minor Weakness.

5

Slightly un-conservative under tensioncompression and doubt about compression-compression quadrant. Minor Weakness. Very low initial strength prediction. Minor Weakness.

6

3

4

5

Pt B revised theory offers improved fit in one quadrant. 6

Good agreement in one quadrant.

7

Theory fits well the shape and magnitude of stress strain curves.

7

8

Theory fits well the shape and magnitude of stress strain curves.

8

9

Theory fits the shape and magnitude of half the envelope.

Theory does not fit all experimental results. Minor Weakness.

Pt B revisions are insignificant.

Pt B revision resulted in unconservative prediction in certain areas. Minor Weakness remains.

9

1179 Test Case

Plus Points

Minus Points

Fig No.

10

The shape and magnitude of stressstrain curve is in good agreement with experiment up to approx 3% strain.

The theory does not fit the experimental curves at large strains. Fundamental problem in post failure analysis. Fundamental Weakness.

10

Theory does not fit the shape of one branch of the stress strain curves and gives large failure strain in the other branch of the curves. Fundamental Weakness

11

11

12

The predicted and measured curves are similar in shape and magnitude.

Did not predict the second, intermediate, mode of failure. Initial failure stress low. Minor Weakness.

13

The predicted and measured curves are similar in shape and magnitude.

Un-conservative strength prediction, not capturing strains deviations, Minor Weakness.

14

Theory fits the shape of curves and captures non-linear behaviour at low strains.

Curves at large strain un-conservative and un-representative. Fundamental Weakness in post failure analysis.

14

1180 Table B-6 Remarks on the correlation between Hart-Smith-1’s theory and experimental data for the 14 Test Cases. Theory No: 6 Hart-Smith 1 Test Case

Plus Points

Minus Points

Fig No.

1

Theory fits the shape of the experimental envelope.

Predicted envelope does not reflect the basic equations governing the theory. Envelope not used in the laminate prediction. Minor Weakness in the non interactive theory actually used.

1

2

Does not fit general shape in the second quadrant, un-conservative in places. Minor Weakness.

2

3

Does not fit general shape of envelope very well, un-conservative in places. Minor Weakness.

3

4

Broadly fits shape of the experimental data.

Theory is generally moderately unconservative. Minor Weakness.*

4

5

Fits shape and magnitude of the experimental data in part.

Un-conservative in some portions. Minor Weakness.*

5

6

Good agreement in two quadrants. Truncation in tension-compression seems good.

Possibly un-conservative, doubt about compression-compression quadrant. Minor Weakness.*

6

7

No prediction offered.**

N/A

8

No prediction offered.**

N/A

Un-conservative in places, conservative in others. No initial failure (only one envelope predicted). Fundamental Weakness.

9

10

No prediction offered**.

N/A

11

No prediction offered**.

N/A

12

No prediction offered**.

N/A

13

No prediction offered**.

N/A

14

No prediction offered**.

N/A

9

Theory fits the general trend.

* No initial failure prediction. Fundamental weakness. ** Fundamental weakness.

1181 Table B-7 Remarks on the correlation between Hart-Smith-2’s theory and experimental data for the 14 Test Cases. Theory No: 7 Hart-Smith 2 Test Case

Plus Points

Minus Points

Fig No.

1

Conservative in one quadrant and unconservative in the second. Minor Weakness.

1

2

Does not fit general shape very well, un-conservative in one quadrant, conservative in the other. Minor Weakness.

2

3

Predicted uniaxial compressive strength different from the one supplied. Biaxial tension and biaxial compression envelopes seem un-conservative but not proven. Significant weakness.

3

4

Broadly fits shape of the experimental data. Truncation gives good fit in one quadrant.

Theory moderately un-conservative in places. Minor Weakness.*

4

5

Fits shape and magnitude of the experimental data in part.

Un-conservative in some portions. Minor Weakness.*

5

6

Good prediction of shape of experiments, truncation appears good. Fibre failure prediction good

Possibly un-conservative and doubt about the compression-compression quadrant Minor Weakness.*

6

7

No prediction offered**.

N/A

8

No prediction offered**.

N/A

Un-conservative in places, conservative in others. Minor Weakness.*

9

10

No prediction offered**.

N/A

11

No prediction offered**.

N/A

12

No prediction offered**.

N/A

13

No prediction offered**.

N/A

14

No prediction offered**.

N/A

9

Theory fits the general trend.

* No initial failure prediction Fundamental weakness. ** Fundamental Weakness.

1182 Table B-8 Remarks on the correlation between Hart-Smith-3’s theory and experimental data for the 14 Test Cases. Theory No: 8 Hart-Smith 3 Test Case

Plus Points

Minus Points

Fig No.

1

Does not fit the general shape of the experiments, extremely un-conservative in one quadrant. Does not fit uniaxial strengths provided. Significant Weakness.

1

2

Does not fit general shape very well, slightly conservative in one quadrant, un-conservative in the other. Minor Weakness.

2

3

Extremely un-conservative in transverse tension, predicted curve does not pass through measured uniaxial data. Significant Weakness.

3

4

Broadly fits shape and magnitude of the experimental data.

Un-conservative in places. Minor Weakness.*

4

5

Fits shape and magnitude of the experimental data in part.

Slightly un-conservative in one portion. Minor Weakness.*

5

6

Good prediction of shape of experiments, truncation appears good. Fibre failure prediction good.

Possibly un-conservative and doubt about the compression-compression quadrant Minor Weakness.*

6

7

Shape of the predicted curves in fair agreement with experiment. Good for predicting fibre failure.

Slightly softer Poisson’s strain response than observed. Minor Weakness.

7

8

Predicted stress-strain curve in good agreement with experiment.

Neglect of matrix failure, slightly overstiff strain response, change is slope in the experimental curves not captured. Minor Weakness.

8

9

Envelope fits the experimental data in shape and magnitude quite well.

Slightly un-conservative in some places, conservative in others. Minor Weakness.*

9

10

Gives conservative predictions of failure stress and strain.

Theory predicts very soft stress strain curves, predicted curves do not match the test data. Minor Weakness.

10

11

Prediction fits the shape of one branch of stress strain curves.

Prediction does not fit the shape of the other branch of stress train curves, linear prediction, does not capture matrix failure. Minor Weakness.

11

1183 Test Case

Plus Points

Minus Points

Fig No.

12

Predicted stress strain curves and failure strains fit experiments.

No predictions of modes of failure. Significant Weakness.

12

13

Linear behaviour. Not capturing matrix cracking and change in slope. Minor Weakness.

13

14

Not capturing initial stiffness nor the nonlinear shape of experimental curves. Fundamental Weakness.

14

* Initial failure not predicted Fundamental Weakness.

1184 Table B-9 Remarks on the correlation between Huang’s theory and experimental data for the 14 Test Cases. Theory No: 9 Huang Test Case

Plus Points

Minus Points

Fig No.

1

Predicts interaction and enhancement in shear strength.

Does not fit the general shape of the experiments, extremely un-conservative in places. Does not fit uniaxial strengths provided. Significant Weakness.

1

2

Does not fit general shape very well. A little conservative in one quadrant, and un-conservative in the other. Minor Weakness.

2

3

Extremely un-conservative in transverse tension, predicted curve does not pass through measured uniaxial data. Significant Weakness.

3

4

Prediction does not fit the experimental data in shape or magnitude. Very conservative in one quadrant ( + x, + y) and un-conservative in another. Possible problems with post failure analysis. Significant Weakness.

4

Considerably conservative under uniaxial tension. Minor Weakness.

5

6

Does not fit the shape of test data, very conservative in one quadrant, unconservative in the others. Significant Weakness.

6

7

Major departure from experiment at approx. 300MPa, large axial strain predicted, significant effects of thermal stresses on final fibre tension failure. Significant Weakness.

7

8

Major departure of axial strain from experiment at approx 270MPa Initial failure detected, resulting in a large increment in strain and immediate final failure. Problems in predicting beyond onset of initial failure and not capturing fibre tension failure. Substantial effects of thermal stresses. Fundamental Weakness.

8

5

Fits shape of envelope well, generally conservative.

1185 Test Case

Plus Points

Minus Points

Fig No.

9

Predicted envelope fits the leakage stresses well.

No distinction between fibre and matrix failure, very un-conservative under biaxial tension. Fundamental Weakness.

9

10

Captured part of the nonlinear behaviour.

Predicted curves deviates considerably from test data at approx. 200MPa. Predicted stresses 50% lower than measured. Significant Weakness.

10

11

Theory replicates the early part of the curves moderately well.

The curves terminate at very low strain. Prediction does not capture matrix failure progression, fibre tension failure and leakage failure. Theory has a problem in predicting beyond onset of initial failure. Fundamental Weakness.

11

12

Good prediction of the shape of one branch of stress strain curves.

Theory did not capture all stages (modes) of failure. Predicted transverse strain larger than experimental. Final failure strength low. Significant Weakness.

12

13

Theory replicates the early part of the curves moderately well.

Truncated curve predicted with a strength of less than 40% of the measured value and with a predicted failure strain of 20% of measured value. No leakage prediction. Theory unable to predict final fibre tension failure. Fundamental Weakness.

13

14

Predicted curves showing good nonlinear behaviour, final strains captured well.

Predicted curves are above the measured ones, predicted final stresses 30% larger than measured. Significant Weakness.

1186 Table B-10 Remarks on the correlation between Mayes’ theory and experimental data for the 14 Test Cases. Theory No: 10 Mayes Test Case

Plus Points

Minus Points

Fig No.

1

Fits experimental trend in the ( + y, xy) quadrant.

Does not fit general shape in the (–y, xy) quadrant. Gives conservative prediction. Minor Weakness.

1

2

Predicts interaction, generally conservative.

Does not predict max shear stress. Minor Weakness.

2

Almost non-interactive failure prediction is a little un-conservative (particularly in the ( + x, –y) quadrant). Minor Weakness.

3

3

4

Shape of envelope broadly fits the experimental data in shape and magnitude.

Theory is slightly over conservative in portions of one quadrant, unconservative in a portion of another quadrant. Minor Weakness.

4

5

Broadly fits the shape of the experimental data and is conservative in both quadrants.

Predicted uniaxial tensile strength considerably lower than measured data. Minor Weakness.

5

6

Good fit in one quadrant and part of another.

Unconservative in compressioncompression quadrant where experimental results may have been influenced by buckling. Sharp steps in predicted envelope. Minor Weakness.

6

7

Predicted final strength close to measured.

Departure from experiment at approx. 400 MPa, large axial and hoop strains predicted. Significant Weakness.

7

8

Good correlation with the minor strain curve, final strength predicted well.

Departure from experiment at approx. 350 MPa. Minor Weakness.

8

9

Fits measured data in certain areas.

Part A prediction very conservative. Fundamental Weakness.

9

Revised curve produces better correlation in the tension-tension quadrant. Significant Weakness. 10

Captured part of the nonlinear behaviour.

Failure strains more than 5 times lower than those measured. Significant Weakness.

10

1187 Test Case

Plus Points

Minus Points

Fig No.

11

Predicted curves good at the initial stages.

Part A prediction captured only 10% of the measured curve. Fundamental Weakness.

11

Revised curves strength prediction close to measured data.

The revised curves deviate largely from measured data. Final strain for the fibre dominated curve is very conservative. The shape of axial strain does not resemble the measured curve, problems with post failure analysis. Fundamental Weakness.

Three modes of failure captured, final strength close to measured.

Initial stress strain curves showed peculiar and unexpected behaviour indicating difficulties in dealing with thermal stresses.

12

12

Initial failure stresses well below the measured values. The predicted stress strain curve is softer than measured and showed steps. Minor Weakness. 13

14

Shape of predicted curve resembles that observed.

Part A submission produced truncated curves covering only 10% of the measured curve. Fundamental Weakness.

Pt B revision provides a better correlation with test data.

In the revised curve, theory predicts the strains y and x to be coincident, predicted strength was some 20% larger than the measured strength. The revision shows steps in the stress strain curve not observed in tests. Significant Weakness.

Predicted curves are in agreement with measured data up to 2% strain.

Theory unable to capture the large final measured strains (a factor of more than 5). Fundamental Weakness.

13

14

1188 Table B-11 Remarks on the correlation between McCartney’s theory and experimental data for the 14 Test Cases. Theory No: 11 McCartney Test Case

Plus Points

Minus Points

Fig No.

1

No prediction offered**.

N/A

2

No prediction offered**.

N/A

3

No prediction offered**.

N/A

4

No prediction offered**.

N/A

5

No prediction offered**.

N/A

6

No prediction offered**.

N/A

7

No prediction offered in Pt A.

N/A

Pt B revision does not contain any criterion to handle fibre failure nor shear dominated failure. Fundamental Weakness.

7

No prediction offered in Pt A.

N/A

Pt B revision does not contain any criterion to handle fibre and shear dominated failure. End point terminated but not predicted. Significant Weakness.

8

9

No prediction offered**.

N/A

10

No prediction offered in Pt A.

N/A

Prediction diverges significantly from experiment by being over-stiff (cannot model non-linear materials response). Significant Weakness.

10

No prediction offered in Pt A.

N/A

The theory diverges significantly from experimental ex by being over-stiff (cannot model non-linear materials response). Significant Weakness.

11

Prediction now offered in the Pt B revision. General shape of the predicted stress-strain curve predicted in Pt B is in good agreement with experiment. 8 General shape of the predicted stressstrain curve predicted in Pt B is in good agreement with experiment.

Prediction now offered in the Pt B revision.

11 Prediction now offered in the Pt B revision. One predicted branch of curves fits test data well.

1189 Test Case

Plus Points

Minus Points

Fig No.

12

Theory predicts crack density.

Theory did not capture the various modes of failure.

12

Pt B revision gave improved stresses above those predicted in Part A.

Pt B revision does not capture intermediate mode of failure and the stress/strain curves were extended beyond fibre failure strain. Significant Weakness.

Ability to predict crack density.

Predicted curve stiffer than the measured one. No final strength prediction. Significant Weakness.

13

Pt B revision resulted in an improved (extension) of the stress strain curves.

Predicted curves stiffer than those observed, prediction extended beyond fibre failure strain, initial failure stress twofold above cracking level. Significant Weakness.

13

No prediction offered, theory unable to predict failure under shear**.

N/A

13

14

** Fundamental Weakness.

1190 Table B-12 Remarks on the correlation between Puck’s theory and experimental data for the 14 Test Cases. Theory No: 12 Puck Test Case

Plus Points

Minus Points

Fig No.

1

Fits shape of test data well. Excellent revised Pt B predictions.

2

Theory generally fits the test data well in both the + x and –x quadrants.

Did not capture possible increase in shear strength in the tension quadrant. Minor Weakness.

2

3

Fits shape and magnitude of the experimental data reasonably well.

A little unconservative in the + x & –y quadrant. Minor Weakness.

3

4

Fits shape and magnitude of the experimental data reasonably well.

Slightly unconservative in the (x, –y) and (–x, –y) quadrants (though experimental data is sparse in the latter). Minor Weakness.

4

5

Fits shape and magnitude of the experimental data very well.

Slightly unconservative in one small region. Minor Weakness.

5

6

Good agreement in one quadrant

Slightly un-conservative under tensioncompression and doubt about compression-compression quadrant. Minor Weakness.

6

7

Shape and magnitude of the predicted stress-strain curve in very good agreement with experiment.

7

8

Shape and magnitude of the predicted stress-strain curve in very good agreement with experiment.

8

9

Very good fit to experimental envelope in shape and magnitude for 80% of the region. Good fit in three quadrants.

The theory has a unique open envelope Which needs to be reconciled with experimental evidence. Significant Weakness.

9

10

The shape and magnitude of the stressstrain curve is in very good agreement with experiment up to approx 4% strain (50% of measured final failure strain).

No leakage prediction, further work needed to capture final predicted strength. Significant Weakness.

10

11

Prediction fits well with one of the experimental curves. Strength is over predicted by 20%.

Theory does not predict the second branch of the experimental curves well (Possibly, partly due to ignoring fibre rotation in this case). Significant weakness.

11

1

1191 Test Case

Plus Points

Minus Points

Fig No.

12

The predicted and experimental curves are similar. Final strength close to measured one (13% higher than the measured). Three modes of failure captured.

The initial and intermediate failure strengths are some 50% lower than those recorded in experiments. Minor Weakness.

12

13

The predicted and measured curves are similar in shape and magnitude.

Un-conservative strength prediction, not capturing strain deviations, Minor Weakness.

13

14

Predicted curves are in agreement with measured data up to 2% strain.

Theory unable to predict large deformation. Fundamental Weakness.

14

1192 Table B-13 Remarks on the correlation between Rotem’s theory and experimental data for the 14 Test Cases. Theory No: 13 Rotem Test Case

Plus Points

Minus Points

Fig No.

1

Fits experiment very well in the ( + y, xy) quadrant.

Does not fit general shape in the (–y, xy) quadrant. Gives conservative prediction. Revised Pt B predictions indicate some Minor Weakness remains.

1

Does not fit general shape of envelope very well, Possibly slightly unconservative in places. Minor Weakness.

2

Possibly a little unconservative. Minor Weakness. Revised Pt B gives more unconservative prediction and slightly overestimates transverse compressive strength. Minor Weakness.

3

2

3

Fits the shape and magnitude of the experimental data quite well.

3

4

Very conservative predictions in both the ( + x, + y) and the (–x, + y) quadrants.

Prediction does not fit the experimental data in shape or magnitude. Very conservative in two quadrants (reaching a factor of 5 at SR = 1/–1) and unconservative in a third, Possible problems with post failure analysis. Significant Weakness.

4

5

Very conservative predictions for all quadrants.

Prediction does not match the experimental data. Very small envelope, Possible problems with post failure analysis. Significant Weakness.

5

Unconservative in the (–x, –y) quadrant (though experimental data is possibly unreliable). Envelope does not match the experimental data at all in the tension-tension quadrant. Possible problems with post failure analysis. Significant Weakness.

6

6

1193 Test Case

Plus Points

Minus Points

Fig No.

7

Conservative strength prediction (75% of observed value).

Major departure from experiment at approx 350 MPa when initial failure was detected, resulting in an increment in strain (particularly in the minor direction) followed by bi-modular response to failure. Theory seems to have a problem in predicting beyond onset of initial failure. Significant Weakness.

7

Major departure from experiment at approx 300 MPa. Initial failure detected, resulting in a large increment in strain and immediate final failure. Problems in predicting beyond onset of initial failure and not capturing fibre tension failure. Significant Weakness.

8

8

9

Strength predictions are conservative in all quadrants. Works best in the (–x, –y) quadrant.

Very conservative (a factor of 6 at SR = 3.3/1) under biaxial tension. Apparent lack of effective post failure matrix degradation mechanism when combined shear, transverse and longitudinal tensile stresses act at the lamina level. Fundamental Weakness.

9

10

Non-linear curves predicted.

Predicted curves terminate at approx 1.5% strain (less than 20% of the measured strain at final failure). Significant Weakness.

10

11

Theory replicates the early part of the curves moderately well.

The curves terminate at very low strain. Prediction does not capture matrix failure progression, fibre tension failure and leakage failure. The theory only replicates approximately 17% of the actual stress-strain response. Theory has a problem in predicting beyond onset of initial failure. Fundamental Weakness.

11

12

Final strength close to measured one (6% higher than the measured). Good prediction of one branch of stress strain curves.

Theory did not capture the intermediate mode of failure. Predicted transverse strain larger than experimental (by a factor of 6). Significant Weakness.

12

1194 Test Case

Plus Points

Minus Points

Fig No.

13

Theory replicates the early part of the curves moderately well.

Truncated curve predicted with a strength of less than 20% of the measured value and with a predicted failure strain of 10% of measured value. No leakage prediction. Theory unable to predict final fibre tension failure. Fundamental Weakness.

13

14

Non-linear curves up to 2% strain.

Stiffness over-predicted. Theory unable to predict large deformation. Predicted strains cover only 0.16% of measured value. Fundamental weakness.

14

Pt B revision provided a closer fit with measured final strain values.

Pt B still under-predicts the failure strain by a factor of 2. Significant Weakness.

1195 Table B-14 Remarks on the correlation between Sun(L)’s theory and experimental data for the 14 Test Cases. Theory No: 14 Sun (L) Test Case

Plus Points

Minus Points

Fig No.

1

Fits experiment well in the ( + y, xy) quadrant.

Does not predict maximum shear in the (–y, xy) quadrant. Minor Weakness.

1

2

Does not fit general shape very well, conservative in one quadrant and unconservative in the other. Minor Weakness.

2

3

Non-interactive failure theory does not fit the experiments very well, a little unconservative in the + x & –y quadrant. Minor Weakness.

3

4

Good fit to the shape and magnitude of the experimental data, in all four quadrants.

A little unconservative in the (–x, –y) quadrant (though experimental data is sparse). Minor Weakness.

4

5

Reasonably good fit of experimental data in some parts.

Overall, the fit is moderate and suggests that the theory is not capturing all of the correct features. Minor Weakness.

5

6

Good fit in two quadrants. This theory introduces some truncations in the envelope in the region of (x : y = –1 : 1).

Unconservative in the (–x, –y) quadrant (though experimental data is possibly unreliable). Minor Weakness.

6

7

Shape of the predicted stress-strain curve in quite good agreement with experiment, (failure strength and strain within 7%). Prediction of various modes of failure.

Stress/major strain response is identical to Sun (NL) prediction (slightly softer than experiment). Stress/minor strain response differs from Sun (NL) prediction and is in excellent agreement with experiment.

7

Pt B revision provided a better representation of the theory.

Pt B revision did not improve the fit with test data and produced steps in the curves not seen in the experiments. Minor Weakness.

Shape of the predicted stress-major strain curve in good agreement with experiment (failure strength within 2%.

Minor strain response 50% larger than that obtained from experiments, after initial failure at approx 200 MPa, probably stems from imprecise assumptions made about the residual lamina properties after initial failure.

Pt B revision provided a better representation of the theory.

Pt B revision did not improve the fit and produced steps in the curves. Minor Weakness.

8

8

1196 Test Case

Plus Points

Minus Points

Fig No.

9

Envelope fits the experimental data quite well. Good fit in the (–x, y) quadrant. Fair fit in the (–x, -y) quadrant.

Mixed fit to experiment in (x, y) quadrant – unconservative in places, very conservative elsewhere. Minor Weakness.

9

Linear prediction, predicted final strains lower than test data by a factor of more than 5. Significant Weakness.

10

predicts strength only less than 10% higher than the measured strength. The axial strain crosses the hoop strain.

Theory predicts early part of stress (y) vs strain (x) curve well, up to approx 140 MPa. Thereafter the theoretical curve is too stiff. The other curve does not capture the nonlinear response observed in the test data. Significant Weakness.

11

Pt B revision gave a better representation of the theory.

Pt B revision shows steps in the curves and no improvement. Significant Weakness.

10

11

12

13

14

12

Final strength close to measured one (6% higher than the measured). Captured three modes of failure. Pt B revision reflects what the theory actually predicts.

Pt B revision did not significantly alter the fit between prediction and experiments. Steps in curves. Minor Weakness.

Shape of predicted curve resembles that observed.

Theory predicts the strains y and x to be coincident, predicted strength was some 20% larger than the measured strength.

Pt B revision provides an accurate representation of the theory.

Pt B revision shows steps in the stress strain curve not observed in tests. Significant Weakness. Theory unable to capture nonlinear behaviour and the large final measured strains. Predicted strains are some 20 times lower than experimental value. Fundamental Weakness.

13

14

1197 Table B-15 Remarks on the correlation between Sun(NL)’s theory and experimental data for the 14 Test Cases. Theory No: 15 Sun (NL) Test Case

Plus Points

Minus Points

Fig No.

1

Case not attempted*.

NA

2

Case not attempted*.

N/A

3

Case not attempted*.

N/A

4

Case not attempted*.

N/A

5

Case not attempted*.

N/A

6

Case not attempted*.

N/A

Minor strain is 50% larger than measured strain. Strength prediction similar to that using linear analysis. Minor Weakness.

7

7

Shape and magnitude in good agreement with experiment (failure strength and strain within 10%).

8

Shape of the predicted stress-strain curve in quite good agreement with experiment (failure strength and strain within 10%).

9

8

Case not attempted*.

N/A

10

Captured part of the nonlinear behaviour.

Failure strains more than 5 times lower than those measured. Significant Weakness.

10

11

Major stress (y) vs major strain (y) curve fits the experiment reasonably well for most part of the curve.

The theory does not actually predict a final failure. Beyond 200Mpa the theoretical curve (y  x) is very unrepresentative of the experiment (hardening behaviour as opposed to softening behaviour). Significant Weakness.

11

12

Final predicted point conservative (86% of measured value). Major strain curve in good agreement with test data.

No final strength prediction. Theory did not show the various modes of failure. The behaviour of the transverse strain curve deviates slightly from the experimental curve. Significant Weakness.

12

13

Shape and magnitude of predicted curve in reasonable agreement with experiments.

No final strength prediction. Significant Weakness.

13

1198 Test Case

Plus Points

Minus Points

Fig No.

14

Predicted curves are in agreement with measured data up to 2% strain.

Theory unable to capture the large final measured strains (a factor of more than 5). Fundamental Weakness.

14

* Fundamental Weakness.

1199 Table B-16 Remarks on the correlation between Tsai’s theory and experimental data for the 14 Test Cases. Theory No: 16 Tsai Test Case

Plus Points

1

Fits shape well in both the ( + y, xy) & (–y, xy) quadrants. Excellent revised Pt B predictions.

2

Good fit of the shape in both the + x & –x quadrants.

Did not predict the high measured maximum shear strength. Minor Weakness.

2

3

Excellent fit to the shape and magnitude of the experimental data.

Theory predicts 90% increase strength at SR = –14 :–1 in comparison with uniaxial strength values. Seems doubtful, but lack of experimental data in (–x, –y) quadrant to prove or disprove this possible Significant weakness under biaxial compression.

3

4

Broadly fits the experimental data. Slightly conservative predictions in the ( + x, + y) and the ( + x, –y) quadrants.

A little unconservative in the (–x, –y) quadrant (though experimental data is sparse).

4

Pt B revised predictions fit the shape and magnitude of the experimental envelope very well.

Slightly more unconservative in the (–y, –y) quadrant (though experimental data is sparse). Minor Weakness.

Broadly fits shape of the experimental data. Predictions are very conservative (ie safe) for both quadrants.

Overall the predictions are approx 50% too conservative suggesting that there is some weakness in the theory or its application.

Pt B revised theory produces a significantly different predicted envelope which is in slightly better agreement with experiment.

Pt B revision gives a little better accuracy in the ( + x, xy) quadrant but there is still Minor Weakness.

5

Minus Points

Fig No. 1

5

1200 Test Case

Plus Points

Minus Points

Fig No.

6

Unusual and distinctive envelope broadly fits some of the experimental data. Matches experimental data well in the (–x, + y) quadrant.

Theory predicts 300% increase in strength at SR = –1 : –1 in comparison with uniaxial strength values. Doubts over the validity of the experimental data in (–x, –y) quadrant mean that definitive proof or disproof of this theory is not possible. May be a Significant Weakness here.

6

Pt B revision slightly improves the fit to experiment in tension-dominated quadrants.

Pt B revision reproduces perceived weakness in the Pt A theory. Thus there remains a possible significant weakness in the theory for biaxial compressive modes of failure.

Shape of the predicted stress-strain curve in fair agreement with experiment (failure strength within 15% and failure strain within 10%).

Softer major and minor strain response than experiment.

Pt B revised prediction gives improved accuracy (failure strength within 5%).

Pt B revision does not change the oversoft strain prediction. Minor Weakness.

Shape of the predicted stress-strain curve in quite good agreement with experiment (failure strength and strain within 10%).

Major strain response is slightly softer than experiment.

Pt B revised prediction gives slightly improved accuracy.

Pt B revision does not change the oversoft strain prediction. Minor Weakness.

Envelope broadly fits the experimental data in shape (80%) and magnitude (40%). Strength predictions are conservative in all quadrants.

No leakage prediction. Theory is quite conservative in the (–x, –y) and ( + x, + y) quadrants, particularly at the –2 : –1 ‘netting analysis’ condition. Significant Weakness.

Pt B revised theory produces some improved accuracy in the (–x, –y) and ( + x, + y) quadrants. Good improvement.

Pt B revisions still conservative in places. Minor Weakness.

7

8

9

7

8

9

1201 Test Case

Plus Points

Minus Points

Fig No.

10

Prediction captures initial modulus.

Linear response, replicates approximately 5% of the actual stressstrain response. Significant Weakness in the theory or its application The Pt B revised curves produced a horizontal plateau not seen in experiment. The revised theory now replicates approximately 60% of the observed stress-strain response. Clearly, Pt B revisions have corrected some of the Significant Weaknesses observed in the Pt A theory. Minor Weakness remains.

10

Theory predicts the initial part of major stress (y) vs minor strain (y) curve well, up to approx 250 MPa. Thereafter the theoretical curve is very unrepresentative of the experiment (too stiff).

11

Pt B revised prediction gives improved accuracy. The magnitude of the stress at ‘final failure’ equates well (within 5%) to the leakage stress observed experimentally.

11

Major stress (y) vs major strain (y) curve fits shape of experiment reasonably well for most of the curve. Pt B revision provides an improvement in the strength prediction.

Pt B revision does not improve the (x) strain curve. There is still a Significant Weakness. 12

Only two stages (modes) of failure predicted. Final strength reached only two third of the measured value. Significant Weakness.

12

Conservative prediction of final strength.

Theory predicts both strains to be identical.

13

Pt B revision provided an improvement in strength prediction.

Minor Weakness in terms of capturing final failure and leakage.

Predicted curves resemble the observed ones. Pt B revision was able to capture the three observed modes of failure. Improvement in strength prediction (within 5% of measured strength). Very good revised prediction.

13

14

Pt B revision offered.

Theory unable to predict non-linear behaviour and capture the large final measured strains. Fundamental Weakness in of the theory for large deformation?

14

Pt B revision produced a horizontal plateau dissimilar to the observed trend. Fundamental Weakness remains.

14

1202 Table B-17 Remarks on the correlation between Wolfe’s theory and experimental data for the 14 Test Cases. Theory No: 17 Wolfe Test Case

Plus Points

Minus Points

Fig No.

1

Fits shape reasonably well in the ( + y, xy) quadrant.

Predictions slightly unconservative in parts. Max shear strength not reproduced.

1

Part B revisions fitted revised compression and shear strengths.

Pt B predictions did not greatly improve the fit. Minor Weakness.

2

Good general fit of the shape in both the + x and -x quadrants.

Max shear strength not predicted. Minor Weakness.

2

3

Conservative prediction.

Theory produced discontinuity in the envelope near uniaxial longitudinal loading. Significant Weakness.

3

4

5

Pt B revisions provide a better fit with tests data Also remove the discontinuity in the envelope.

Revised Pt B predictions introduced a radically different shape of failure envelope with an unlikely bulge in the ( + x, + y) quadrant. Lack of experimental data in the( + x, + y) quadrant to prove or disprove this – Significant Weakness.

Conservative predictions in the ( + x, + y) , (–x, + y) & (–x, + y) quadrants.

Overly conservative (reaching a factor of 5 under SR = 1/1) but generally within 50% of the measured data. Significant Weakness in post initial failure theory or its application.

Pt B revised theory gives a good fit to experiments in three quadrants.

Un-conservative in the (–x, –y) quadrant (though experimental data is sparse). Significant Weakness.

Very conservative predictions for both quadrants.

Extremely small envelope that does not match the experimental data (by a factor reaching 5 at SR = 1/1 but generally more than 100% deviation in many places). Overly conservative predictions seem to stem from Significant Weakness in post initial failure theory.

Pt B revised theory produces a radically different predicted envelope which is in better agreement with experiment.

With Pt B revision only Minor Weaknesses remains.

3

4

5

1203 Test Case

Plus Points

Minus Points

Fig No.

6

Predicted envelope fits the experimental data quite well in parts.

In-accurate prediction of fibre failure, Unconservative in the (–x, –y) quadrant (though experimental data is possibly unreliable, due to premature structural buckling rather than material failure).

6

Pt B revision slightly improves the fit to experiment in the (–x, + y) quadrant.

After Part B revision Minor Weakness still remains.

Shape of the predicted stress-strain curve in good agreement with experiment up to approx 450 MPa (60% of failure point).

Steps in curves and premature final failure. Significant Weakness.

Pt B revised theory produces a radical improvement – now in quite good agreement with experiment.

Pt B revision has greatly improved the initial to final failure prediction technique for this case.

Shape of the predicted stress-strain curve in fair agreement with experiment (failure strength and strains within 15%).

Steps in stress-strain curves, small changes in slope before final failure. Minor Weakness in the theory or its application?

Pt B revised prediction gives improved accuracy (failure strength and strain within 10%).

The Pt B revisions moderate the Minor Weakness observed in the Pt A theory.

Moderate fit to experimental envelope overall. Best fit in the (–x, –y) quadrant.

Very conservative by a factor of 6.5 at SR = 2/1 in the ( + x, + y) quadrant, possibly due to inadequate treatment of post failure process and lack of discrimination between initial and final failure when there is high combined shear and transverse tensile stress acting at the lamina level.

Pt B revised theory produces some improved accuracy in the (–x, –y) quadrant and very minor improvement in the ( + x, + y) quadrant.

Pt B revisions do not overcome the Pt A shortfall. Significant Weakness remains.

Theory captures part of the nonlinear behaviour, predicts Initial modulus and initial part of the nonlinear curves very well.

The curves terminate at approx 1.5% strain (one fifth of the measured hoop strain and one seventh of the measured axial strain). Significant Weakness.

Pt B revised theory predicts a very small increase in strength.

Pt B revisions did not correct the Pt A shortfall. Significant Weakness remains.

7

8

9

10

7

8

9

10

1204 Test Case

Plus Points

Minus Points

Fig No.

11

The theory replicates the early part of the curve well.

The curves terminate at very low strain. Theory appears to have difficulty in predicting beyond onset of initial failure. Fundamental Weakness in the theory or its application.

11

Pt B revision predicts a very small increase in strength.

Pt B revisions do not address the Pt A shortfall. Fundamental Weakness remains.

Captures the initial behaviour of the experimental curves.

The theory did not predict the intermediate mode of failure. Final strength less than 50% of the measured value. Significant Weakness in the theory or its application?

Pt revision provides a better strength prediction and reaches 85% of the measured value.

Pt B revision is still unable to predict the second mode of failure. Significant Weakness.

Initial part of curve predicted.

Curves truncated at very low strains. Does not capture the softening behaviour of stress strain curves. Final strength prediction is more than 7 times lower than measured strength. Fundamental Weakness.

Pt B revision resulted in a very small improvement in the strength.

Pt B revision still under-predicts the strength (by a factor of approx 5). There is still a Fundamental Weakness in the Pt B prediction.

Predicted curves are in agreement with measured data up to 2% strain.

Theory unable to capture the large final measured strains by a factor of 5 to 6.

12

13

14

Fundamental Weakness in the theory for large deformation?

12

13

14

1205 Table B-18 Remarks on the correlation between Zinoviev’s theory and experimental data for the 14 Test Cases. Theory No: 18 Zinoviev Test Case

Plus Points

Minus Points

Fig No.

1

Non-interactive failure theory does not fit the general shape of the experiments & is a little unconservative at some stress ratios. Minor Weakness?

1

2

Does not fit general shape. A little conservative in one quadrant, and unconservative in the other. Minor Weakness.

2

3

Non-interactive failure theory is a little unconservative (particularly in the ( + x, –y) quadrant). Minor Weakness.

3

4

Broadly fits shape and magnitude of the experimental data.

A little unconservative in (x, –y) and (–x, –y) quadrants (though experimental data is sparse in the latter). Minor Weakness.

4

5

Best of theories overall (with Puck) in general shape & fit to experiment. Absence of zones where predictions are un-conservative makes this the most appropriate theory for Test Case 5.

A little too conservative (30%) in the (–x, xy) quadrant?

5

Minor Weakness in theory. Un-conservative in the (–x, –y) quadrant (though experimental data is possibly unreliable, due to premature structural buckling rather than material failure). Minor Weakness in theory.

6

6

Shape and magnitude of the predictions are near identical to those of Edge, Puck, Hart-Smith (2). Broad agreement with the experimental envelope in shape and magnitude for the (x, y) & (–x, y) quadrants.

7

Shape of the predicted stress-strain curve in good agreement with experiment (failure strength within 5% and strain within 10%).

7

8

Almost exact fit with experiment (failure strength and strain well within 5%). The best fit (together with Puck) of all Pt A theories for this Test Case.

8

9

Envelope broadly fits the experimental data. Strength predictions are conservative in all quadrants.

Upper line in the ( + x, + y) quadrant is not real as the envelope is open in this area. Significant Weakness remains here.

9

1206 Test Case

Plus Points

Minus Points

Fig No.

10

The magnitude of the stress at ‘final failure’ equates well (within 5%) to the leakage stress observed experimentally.

Prediction is a bi-modular, linear approximation to the non-linear experimental curve – a reasonable fit with experiment up to approx 380 MPa. Curves terminate at this point, as final failure is detected. Thus the theory only replicates approximately 55% of the actual stress-strain response. Significant weakness in the theory or its application?

10

The theory predict a bilinear behaviour of major stress (y) vs minor strain (x) curve. Both branches of the predicted curves are stiffer than the experimental curves. Minor Weakness.

11

11

12

12

All three modes of failure were captured. Final Strength prediction is 8% higher than measured value. Very good agreement.

13

The predicted and measured curves are broadly similar.

Predicted curves stiffer than the measured curves and predicted final strength is 20% higher than average measured strength. Minor Weakness.

13

14

Theory gave a good prediction of the stress strain curve up to final failure. Good for large deformation.

Theory could be improved by taking the nonlinear behaviour into account. Minor Weakness.

14

1207 Table B-19 A summary showing the places where Weaknesses are observed in each of the theories used. Theory designation

Test Cases marked by degree of Weakness as N: No Weakness, M: Minor, S: Significant, F: Fundamental, *: Not attempted 1

2

3

4

5

6

7

8

9

10

11

12

13

14

Bogetti

M

M

3S

M

M

M

M

M

S

S

M

S

S

F

Chamis

M

M

M

S

S

F

S

N

S

F

S

S

S

F

Cuntze

M

M

M

M

M

M

S

M

M

S

S

M

S

F

Cuntze-B

M

M

M

M

M

M

M

M

M

S

S

M

M

M

Eckold

S

*

S

S

*

*

*

*

S

F

S

F

S

*

Edge

M

M

M

M

M

M

N

N

M

F

F

M

M

F

Hart-Smith(1)

M

M

M

M

M

M

*

*

F

*

*

*

*

*

Hart-Smith(2)

M

M

S

M

M

M

*

*

F

*

*

*

*

*

Hart-Smith(3)

S

M

S

M

M

M

M

M

F

M

M

S

M

F

Huang

S

M

S

S

M

S

S

F

F

S

F

S

F

S

Mayes

M

M

M

M

M

M

S

M

F

S

F

M

F

F

Mayes-B

M

M

M

M

M

M

S

M

S

S

F

M

S

F

McCartney-B

*

*

*

*

*

*

F

S

*

S

S

S

S

*

Puck

N

M

M

M

M

M

N

N

S

S

S

M

M

F

Rotem

M

M

M

S

S

S

S

S

F

S

F

S

F

S

Sun (L)

M

M

M

M

M

M

M

M

M

S

S

M

S

F

Sun (NL)

*

*

*

*

*

*

M

N

F

S

S

S

S

F

Tsai A

N

M

S

M

M

S

M

M

S

S

S

S

M

F

Tsai B

N

M

S

M

M

S

M

M

M

M

S

N

M

F

Wolfe A

M

M

S

S

S

M

S

M

S

S

F

S

F

F

Wolfe-B

M

M

S

S

M

M

N

N

S

S

F

S

F

F

Zinoviev

M

M

M

M

M

M

N

N

S

S

M

N

M

M

1208 Table B-20 A summary showing the total number of Weaknesses scored by each theory. Theory Descriptor

Type and number of Weaknesses observed Fundamental

Significant

Minor

No Weakness

Group (1) Zinoviev

0

2

9

3

Cuntze-B

0

2

12

0

Puck

1

3

7

3

Cuntze

1

4

9

0

Sun (L)

1

3

10

0

Tsai-B

1

3

8

2

Tsai A

1

6

6

1

Bogetti

1

6

9

0

Group (2) Hart-Smith(3)

2

3

10

0

Mayes-B

2

6

6

0

Chamis

3

7

3

1

Edge

3

0

9

2

Rotem

3

8

3

0

Wolfe

3

7

4

0

Wolfe-B

3

5

4

2

Mayes

4

3

7

0

Huang

4

8

2

0

Group (3) Eckold

8

6

0

0

Hart-Smith(1)

8

0

7

0

Hart-Smith(2)

8

1

6

0

Sun (NL)

8

4

1

1

McCartney-B

9

5

0

0

1209 10. Appendix C: Details of procedures used for the Quantitative assessment of the correlation between the theoretical predictions and the test results 10.1. Introduction The ranking procedure described here is aimed at exposing the shortfalls and highlighting the successes of the theories in matching many different aspects of the experimental data. The procedure is described in some detail and tables of results are presented to allow the derivation of the final ranking to be traced. An attempt has been made to make the procedure as objective and reproducible as possible so that, if required, the same procedure could be applied to assess the relative performance of other theories. The overall ranking and results for the five major areas are presented in graphical form to aid in visualising the trends.

10.2. The ranking procedure The capabilities of the theories were grouped according to the following ranking categories: (1) First ranking category: Ability to predict the biaxial failure stresses of a unidirectional lamina. (2) Second ranking category: Ability to predict the initial failure properties of multidirectional laminates. (3) Third ranking category: Ability to predict the final strength of laminates subjected to biaxial loads. (4) Fourth ranking category: Ability to predict the deformations of multi-directional laminates. (5) Fifth ranking category: Ability to predict the general trends observed in the test data (shape of stress strain curves, shapes of the biaxial failure envelopes, modes of failure, leakage etc.). Various ranking cases were selected for each of the above categories. More than one Ranking Case could be selected to demonstrate various features in any given Test Case. For instance, the following five Ranking Cases may be selected from a single Test Case representing a stress-strain curve prediction: (1) final failure strain in hoop direction; (2) final failure strain in axial direction; (3) failure strength; (4) identification of modes of failure; and (5) initial failure stresses and strains. Ranking cases were selected where experimental results were available and a total of 125 ranking cases were selected. Whenever possible, the ratio of the predicted to measured values, referred to here as a Property Correlation Ratio (PCR), were computed for each theory for each of the selected ranking cases.. Values of PCR greater than unity indicate that the theory over-predicts the measured data (ie predicted stresses are un-conservative) and if PCR values are less than unity, the theory under-predicts the test data (i.e predicted stresses are conservative). All the ranking cases and the values of PCRs obtained for each theory are listed in Tables (C1 to C5). For presentation purposes, grades were given to the computed ratios (PCR) to represent the degree of correlation between test data and theoretical predictions, as follows:

1210 – Grade A if the ratios lie within 0.9 and 1.1 (i.e. the prediction is within ± 10% of the measured value). – Grade B if the PCR ratios lie between 0.5–0.9 or 1.1–1.5 (between ± 10% and ± 50% of the measured value). – Grade C if the ratios lie below 0.5 or above 1.5 (i.e. below 50% or above 150% of the measured value). – Grade NA to those theories that did not attempt to provide a solution for the selected ranking case. The ranking cases in ranking categories 1–4 are easily specified as the predicted values were normalised with respect to the measured experimental values in a quantitative manner. The values of experimental results employed are given in the last column of the table so that the PCR ratios can be reproduced. The ranking cases in the fifth category are often descriptive or qualitative because this category was aimed at demonstrating experimental features not captured directly in numerical form by the rest of the categories. The wording of features has been chosen carefully in an attempt to make this category as objective as possible. 10.3. Presentation of the results 10.3.1. Ranking according to the first category (unidirectional lamina strength) Eighteen ranking cases were selected to assess the correlation between the theories and measured data for the three unidirectional laminae used, see Table C-1. Fig. C-1 shows the number of grades scored by each theory. In this case the theories are ranked according to the number of Grade A’s achieved. With the highest ranked theory on the left. 10.3.2. Ranking according to the second category (initial failure) Fifteen ranking cases are used to address the capability of the theories to predict the initial failure properties (strains, stress and Poisson’s ratio), see Table C-2. The grades scored by each theory are shown in Fig. C-2. The percentage of Grade A scores was small in this case so in Fig. C-2 the theories are arranged according to their total Grade A plus Grade B scores (i.e. the total number of predictions that are within ± 50% of the experimental measurements, divided by the total number of ranking cases and expressed as a percentage). 10.3.3. Ranking according to the third category (Laminate final failure strength) Twenty eight ranking cases were selected in this category, see Table C-3. The results are displayed in Fig. C-3. The percentage of Grade A’s awarded for this category is generally lower than for the first category so in Fig. C-3 the theories are again ranked according to the total number of Grade A plus Grade B scores. 10.3.4. Ranking according to the fourth category (deformation of laminates) In some applications, the magnitude and nature of deformation in a structure is an important design parameter. In order to capture the correlation between the predicted and measured

1211 deformation observed in the Test Cases, twenty three ranking cases were chosen which cover the ultimate failure strains from the stress strain curves, see Table C-4. The results of this ranking are shown graphically in Fig. C-4 where the theories are ordered according to the total number of A plus B Grades awarded. 10.3.5. Ranking according to the fifth category (general trends) In this case the theories were ranked according to their abilities to predict specific qualitative features that were observed in the experiments for each Test Case and are listed in Table 4. For this purpose a total of 41 features were chosen (see Table C-5) from those listed in Table 4. The only features from Table 4 that were not selected for ranking purposes were those already assessed by one of the other ranking categories and those which might possibly arise from imperfections in the experimental results. The features were assessed in qualitative but objective, go/no-go manner. For example, in assessing predictions of feature TC-1.1 in the right hand quadrant of Fig. 1, the question was asked: Is the predicted shear failure stress decreased due to the application of transverse tensile stress? If the answer was yes (as for Tsai’s theory) then a high grade (Grade A) was assigned. However, if the answer was No (as for Zinoviev or one of the other non-interactive theories) then a low grade (Grade C) was assigned. The results of this ranking are shown in Fig. C-5.

10.4. The overall results and general remarks on the ranking procedures Tables 6 and C-6 summarise the number of A, B, C and NA grades scored by each theory. The data in Table 6 are arranged in such a way as to show the best theories at the top and the less favourite theories at the bottom. The theories are ranked according to their total Grade A plus Grade B scores. The data listed in Table 6 are shown graphically in Fig. 15 in the body of this Chapter 6.1 (page 1158) paper. In Fig. 15 the grades scored in each of the individual ranking categories are added up and divided by the total number of cases (125), to give an overall percentage score for each theory. The theories achieving the highest Grade A plus Grade B scores are arranged on the left.

1212

Table C-1 Ratios of predicted to measured values of the lamina strength: Results of ranking the theories according to the first ranking category.

* These are the unidirectional data provided. + Data taken were based on ‘ultimate’ failure envelope. # These data were obtained from parameters that gave the best-fitted curve.

Table C-2 Ratios of predicted to measured values of the initial failure properties: Results of ranking the theories according to the second ranking category (Experimental data are in MPa for stress and in % for strain).

* At initial failure.

1213

* Based on ‘ultimate’ failure envelope provided.

1214

Table C-3 Ratios of predicted to measured values of the final strength of multi-directional laminates: Results of ranking the theories according to the third ranking category.

Table C-4 Ratios of predicted to measured values of multi-directional laminates: Results of ranking the theories according to the fourth ranking category.

1215

* Based on the 0.5% extended strain. + These are based on an extrapolation process. (a) These taken at final predicted values.

1216

Table C-5 Predicting the trend of the general features exhibited in the experimental results. Results of ranking the theories according to the fifth ranking category.

Table C-5 Continued.

1217

1218

Table C-6 A summary of the number of grades scored by each theory for each of the five ranking categories.

1219

Fig. C-1. Ranking the failure theories according to their ability to predict the biaxial failure stresses of unidirectional lamina.

Fig. C-2. Ranking the failure theories according to their ability to predict the initial failure characteristics in the multi-directional laminates.

1220

Fig. C-3. Ranking the failure theories according to their ability to predict the biaxial final failure stresses of multi-directional laminates.

Fig. C-4. Ranking the failure theories according to their ability to predict the deformation in the multidirectional laminates.

1221

Fig. C-5. Ranking the failure theories according to their ability to predict the general features exhibited in the test results.

Section 7 Recommendations for Designers and Researchers Resulting from the World-Wide Failure Exercise

Failure Criteria in Fibre Reinforced Polymer Composites © 2004 QinetiQ Ltd. Published by Elsevier Ltd. All rights reserved

1223

CHAPTER 7.1

Recommendations for designers and researchers resulting from the world-wide failure exercise P. D. Sodena, A. S. Kaddourb and M. J. Hintonc b

a UMIST, Manchester, UK QinetiQ, Farnborough, Hampshire, UK c QinetiQ, Fort Halstead, Kent, UK

Abstract The World-Wide Failure Exercise (WWFE) contained a detailed assessment of nineteen theoretical approaches for predicting the deformation and failure response of polymer composite laminates when subjected to complex states of stress. The leading five theories are explored in greater detail to demonstrate their strengths and weaknesses in predicting various types of structural failure. Recommendations are then derived, as to how the theories can be best utilised to provide safe and economic predictions in a wide range of engineering design applications. Further guidance is provided for designers on the level of confidence and bounds of applicability of the current theories. The need for careful interpretation of initial failure predictions is emphasised, as is the need to allow for multiple sources of non-linearity (including progressive damage) where accuracy is sought for certain classes of large deformation and final failure strength predictions. Aspects requiring further experimental and theoretical investigation are identified. Direction is also provided to the research community by highlighting specific, tightly focussed, experimental and theoretical studies that, if carried out in the very near future, would pay great dividends from the designer’s perspective, by increasing their confidence in the theoretical foundations.

1. Introduction The results of Parts A, B and C of an international exercise (the ‘World-Wide Failure Exercise’ (WWFE)) to assess the accuracy of current theoretical methods of failure prediction in composite laminates have been published in three special issues of the journal Composites Science and Technology, Refs [1–3]. The background and approach adopted in Parts A, B and C have been described in Refs [4,19,34]. Nineteen different methodologies, representing a wide range of theoretical approaches to predicting failure in composite laminates were considered. The methods have been described in detail in Refs [6–17,21,35–38]. The contributors each applied their own theories to predict the strength and deformation of fibrereinforced composite laminates in 14 test cases set by the organisers (the authors of this

1224 paper), Ref. [5]. The theoretical predictions were compared with one another in Refs [18,43] and with the experimental test data in Refs [33,44]. In their individual papers, Refs [6–17,21–32,35–42], the theoreticians described their approaches in detail, indicated some of the limitations of their methods and specified steps to improve the predictive capability of their theories. The majority of the participants attempted to provide a view and recommendations on how their methodology could be used in design applications. Each participating group was requested to comment only on their own theory. Nevertheless, their recommendations are very useful to those wishing to adopt one of the methodologies. Having now completed the WWFE, the objective of this paper is to distil the key lessons that have been learned and the actions that are required, into a readily accessible form, so that both the design and research communities can advance. The paper considers how the overall results of the exercise can be applied in design applications. Specific recommendations are then given as to how the theories can be best utilised to provide safe and economic predictions in a wide range of engineering design applications. Shortfalls and problems that have been revealed in the theoretical predictions and experimental results will be touched upon where they have a direct bearing on the current discussion. Finally, the paper highlights specific experimental and theoretical studies that should be carried out in the very near future, to advance the subject and thereby increase confidence in the theoretical foundations.

2. Fitting the lessons learned from the WWFE into the design environment 2.1. Basic design objectives Before attempting to make suggestions or recommendations about suitable methods for use in engineering design we will first consider, briefly, some of the diverse factors that influence the choice of methods of failure analysis for the design of composite laminates in load bearing applications. The basic task facing the designer of a composite laminate is to determine the number, type and arrangement of layers in the laminate and the amount of reinforcement required in each layer, to ensure that the component performs safely throughout the lifetime of the component. Theories that give detailed descriptions of mechanisms of failure and embody the physical principles and phenomena involved may not be absolutely essential to achieve the basic design task outlined above. However they are highly desirable in order to provide designers with insight and foresight into possible faults, and possible improvements in laminate design. In designing a load bearing laminate, or any other structural component, the designer has to assess the types of load that the structure must support. These will obviously depend upon the application but will include a variety of types and magnitudes of loads expected under normal service conditions. In most practical applications these will involve two-dimensional or three-dimensional loading. Repeated and long-term loads and environmental effects may

1225 also be encountered. Loads imposed during normal handling, transport and erection must be considered, and also occasional or ‘upset’ loads (for example those imposed by accidental impacts, collisions, storms or earthquakes). The designer must also consider the different possible types of structural failure that are relevant to the design application, which may include some of those observed in the experiments referred to in this exercise: • • • • •

unacceptable damage (discussed in Section 5.2); rupture; excessive deformation; buckling; leakage.

The designer usually has to consider other requirements in addition to failure strength. For example, material cost (and sometimes weight) may need to be minimised and assurance is required that the resulting laminate can be made easily and economically with the available manufacturing processes.

2.2. Dealing with variability and doubt Variability and doubt are unavoidable features of the design process. Sources of doubt include: • • • •

uncertainty in the possible combinations of loading that might occur; variability in some loads (wind loading for example); variability in material properties; variations in the manufactured laminate thickness and fibre orientation.

Probabilistic methods have been proposed and are sometimes used to deal with these uncertainties, Refs [49–50], but designers often apply more pragmatic approaches, such as the judicious use of safety factors. Different safety factors may be applied to allow for specific circumstances (e.g. regular service versus upset loads). Furthermore, the possible consequences of various modes of failure may influence the choice of safety factors and the level of allowable stresses and strains. The concept of classifying components into primary (i.e. man critical), secondary (i.e. mission critical) and tertiary (i.e. mission non-critical) is common in the military and aerospace environments, whereby differing safety factors are applied to each. Safety factors not only have to make allowance for uncertainties in understanding the service conditions and in the build standard of the components, but they must also allow for inadequacies in the design tools themselves. Where doubt exists and the designer has to choose between two methods of calculation, one of which is non-conservative (i.e. unsafe) and the other being conservative, it is normal practice to err on the safe side by favouring the conservative method, even though that will result in a thicker, heavier and more expensive laminate.

1226 2.3. Design for fibre dominance or to utilise the matrix strength? Some contributors to the exercise (particularly those with an aerospace background) expressed strong preferences for ‘well-designed’ laminates with fibres typically arranged in three or more directions to provide increased robustness against primary and secondary loading conditions for the application in hand. This philosophy stems from the early usage (1970s) of carbon fibre materials to replace aluminium alloys in aircraft components. The initial designs were naturally conservative, and were based on simple 0°/ ± 45°/90° configurations that gave fairly good omni-directional strength properties whilst also broadly matching the in-plane Young’s modulus of aluminium (useful when retrofitting a composite part for an aluminium item). The reader should note that Test Cases 6, 7 and 8 in the WWFE were chosen, Ref. [20], to explore the ‘black aluminium’ philosophy and determine how accurately the theories could predict the strength of this type of laminate. The adoption of so called ‘black aluminium’ did not exploit the benefits of the CFRP material to the full, but did simplify the design rules by removing the need to worry about matrix induced failure modes. Some theoreticians with an aerospace background started from the position that designers should not use other types of laminate and that therefore it was not important for analytical methods to address any other situations (e.g. matrix induced failures). In contrast, although multi-directional laminates can be advantageous under multiple loading conditions, many thousands of tons of simple ±  angle-ply E glass-fibre/epoxy composite tube has been successfully employed as pipes in chemical and process plant where complex loadings are encountered regularly. The ±  configuration is used because this simpler fibre arrangement can be readily and economically manufactured (by automated filament winding machines). Such components will have fibre orientations designed to resist the primary loading (internal pressure) and then strength will be checked for other occasional, less dominant or less likely loads (e.g. torsion, bending and upset loads). Experiments confirm that simple fibre arrangements of this type are often capable of supporting much higher complex and ‘off design’ loads than might be expected. It is therefore important that theories for use in design should be able to deal with all of these common and simple types of laminates, and loading conditions as well as ‘well designed’ laminates and loading conditions. The reader should note that Test Cases 9, 10 and 11 in the WWFE were chosen, Ref. [20], to challenge the applicability of the theories to simple angle-ply laminates.

2.4. Micro or meso analysis? In approaching the design of composite laminates it is possible to start at different levels. Most design methods take the laminate layers as the basic building blocks (meso modelling) and attempt to utilise data from simple tests on isolated laminae to predict the behaviour of the whole laminate. The individual layers are commonly reinforced with continuous unidirectional fibres, woven or non-woven fabrics. The WWFE concentrated on continuous unidirectional materials, these being the most fundamental class. An alternative is to start at the constituent level (micro modelling) and use experimental data on the properties of the fibres and matrix materials to predict the properties of the individual laminae, then the laminate and finally the structure. If successful this approach has

1227 the potential of allowing the designer to predict effects of material changes before lamina test data is available. However, it does introduce an additional set of calculations and the associated possibility of introducing inaccuracies into predicted lamina properties. Three of the contributors Chamis1 [6], Mayes [37] and Huang [38] entered at this level and used micromechanics to evaluate lamina properties. The accuracy of their laminate strength predictions was judged to be moderate compared with that of the other theories in Ref. [44]. Several of the other contributors employed some micro-mechanics to a lesser extent in their theories, Refs [11–13,15]. 2.5. Complexity of calculations The simplicity, speed and availability of methods for analysis and calculation relative to the time available to carry out the design may also influence the choice of analysis method. Reasonable accuracy of prediction of laminate behaviour under the complex loading frequently encountered in structural design, normally requires analytical methods of complexity comparable with most of the methods presented in the WWFE. Such methods must be embodied in reliable, well-validated and user-friendly laminate analysis software, which must be made readily available to designers if the methods are to be used in practice. We will return to this issue in a later section of the paper (see section 7.3). 2.6. Influences of structural form and stress concentrations Load bearing structures rarely consist entirely of laminated flat plates or gently curved shells subjected to uniform stress fields. The designer usually has to consider complicated structural shapes and the effects of stress concentrations due to abrupt changes in geometry (e.g. at an opening or protrusion) or local loads (e.g. from a support or impact indentation). Luckily a variety of computational aids and design methods are available to assist the modern designer with these types of problems. However their solution usually involves lengthy finite element analysis or similar computational methods. 2.7. Preliminary sizing calculations Before using such complex methods of analysis and software the designer will normally have to arrive at a preliminary design for the laminate, and to do so will employ judgement, experience, and quick and approximate design methods. For the design of laminates, such methods include netting analysis2, which was not represented in the exercise and the ‘10% rule’ a development of which was presented by Hart-Smith, Ref. [21]. Considering the simplicity of the latter approach it performed remarkably well in predicting the final strength of laminates in a wide variety of cases, in fact better than some of the more complex 1

Each of the theories presented in the exercise has been identified using the name of a single key participant, see Sections 4 and 6. 2 Netting analysis assumes that the stresses induced in the structure are carried entirely by the filaments, and the strength of the resin is neglected. It is also assumed that the filaments possess no bending or shearing stiffness, and carry only the axial tensile loads.

1228 approaches, Ref. [33]. Stress concentrations may be accounted for by employing design data from notched rather than plain test specimens, Ref. [46]. Relatively simple and reliable laminate design methods have been evolved and become standard practice for specific applications. They are most useful when they are made available as national (e.g. BS or ASTM) or international (e.g. ISO, API) standard design codes. Methods of this type often employ apparently simple design rules such as limiting the maximum strains to specified values based on service experience. Eckold’s contributions to the WWFE, Refs [7,23], are an example of this type of design method, which is related to procedures used in a British Standard (BS) code for designing glass-fibre reinforced pressure vessels. Hart-Smith, Ref. [10], presented the Truncated Maximum Strain method that is employed in the aerospace industry for designing carbon fibre reinforced laminates. Such methods are undoubtedly very useful but are only well tuned and certified for application to a very limited range of design problems. It is very important to establish their limitations and scope. This is demonstrated by Eckold’s brave and cooperative attempts in the WWFE to apply the BS design philosophy to as many of the test cases as possible. The method proved extremely limiting (Ref. [33]) and, indeed, led to non-conservative predictions in certain situations when extrapolated beyond the normal usage. A further perspective is given through two theories presented by Hart-Smith, Refs [10,11], which were developed through experience with aircraft structures where behaviour is dominated by fibre stiffness and fibre failure, and where the designer’s main concerns are minimising weight and avoiding buckling and fracture. Those theories do not address matrix cracking or other matrix-dominated behaviour and were deliberately not used by Hart-Smith to solve problems requiring prediction of initial failure or cases such as a cross-ply laminate loaded in shear (Test Case No 14). Sometimes new or innovative design applications or new situations arise in completely different fields where there is no previous experimental work available and no prescribed form for the laminates. Great care is needed in applying the specialist methods in such situations. 2.8. The design needs from the WWFE What we seek then is a general approach or a set of methods that is capable of analysing a wide variety of laminates and loading conditions and predicting as many aspects of laminate behaviour as possible with reasonably good accuracy. The identification of such methods from those assessed within the WWFE is addressed in the following sections. 3. Procedure used for evaluating the overall performance of the theories The WWFE included a comprehensive assessment of the overall performance of each theory in all test cases, using qualitative and quantitative methods. The qualitative approach concentrated on identifying apparent weaknesses in the theories whereas the quantitative method considered both weak and strong features. In the quantitative method the theories were ranked according to the accuracy with which they could predict failure stresses and deformations that had previously been measured experimentally on a range of laminates, lay-up configurations and loading conditions. A total of 125 ranking cases were considered in five ranking categories.

1229 Full details of the methods of assessment, the cases considered and the results are given in Ref. [33,44]. Fifteen theories were evaluated in Ref. [33] and a further four in Ref. [44], using exactly the same method of assessment. These have been combined to provide a comprehensive evaluation of all nineteen theories in Section 6 of this book. The assessment and ranking of the theories took into account two sets of predictions. The primary set (i.e. ‘blind’) was produced by the contributors before they had sight of the experimental results, see Refs [6–17,21,35–38]. The secondary set was produced by certain contributors, after examining the experimental results and where, for instance, significant shortfalls had been exposed. Tsai, for example, made significant improvements to the accuracy of his predictions by making detailed and fundamental changes to the method employed to calculate the post initial failure response. In situations where the original and revised predictions were provided, care was taken to discriminate between each (by means of a key in each reference (A = original, B = revision)). Clearly, in the purest sense of the WWFE, the primary set of predictions is considered to be most representative of the situation faced by a designer when previous experience of the proposed laminate is limited and few, if any, experimental results are available. There are always suspicions about ‘after the fact’ predictions and it is left to the reader to discriminate (with the help of Refs [33,44]), between the genuine improvements introduced in the revisions and ‘fine tuning’ to suit the particular test cases used in the WWFE.

4. Selecting the theories which have utility for designers The performance assessment of the theories carried out within the WWFE, only has utility if the results can be translated into improved design procedures. With this in mind, five leading theories were selected and used by the organisers to create recommendations and guidelines for designers. Whilst the decision to choose the leading 5 theories (rather than (say) 6 or 10) for the purpose, was somewhat arbitrary, as will be demonstrated later, this number and mix of theories provided a comprehensive framework for a designer to work with. The five approaches ranked most highly by the quantitative assessment procedure described earlier are the theories of Zinoviev, Bogetti, Puck, Cuntze and Tsai. These five leading theories are representative of different approaches and their general capabilities are listed in Table 1 together with those for all of the theories employed in the WWFE. Zinoviev [17,32] used the Maximum Stress failure criterion with a carefully developed post failure analysis. He assumed linear elastic stress–strain behaviour up to initial failure but included a continuous correction for the effects of change of fibre orientation throughout loading. Bogetti [35,39] used a three-dimensional form of the Maximum Strain failure criterion, with allowance for non-linear lamina shear stress–strain behaviour and a simple progressive failure analysis. Tsai [15,30] employed the well-known Tsai-Wu interactive failure criterion that does not explicitly identify failure mechanisms, assumed linear elastic material properties and reduced matrix stiffness after initial failure. The theories used by Puck[12,27] and Cuntze[36,40] considered three-dimensional failure mechanisms in some detail and applied non-linear analysis to predict progressive failure. Cuntze’s approach is similar to Puck’s in some respects but assumes interaction between

1230 Table 1 A list of characteristics exhibited by the theories. (✓ means the theory attempted to consider the characteristics and ✔ means the theory performed well in that respect).

* See Table (4) in Section 6 for a definition of features.

failure modes due to probabilistic effects. These two theories produced the highest number of accurate predictions (i.e. within ± 10% of the measured values) and captured more general features of the experimental results and laminate behaviour in the ranking study than the other theories. 5. Further consideration of the detailed characteristics of the selected theories This section focuses on how well the five selected theories are expected to perform in assisting a designer to predict and guard against the particular types of failure mentioned in Section 2.

1231 Where possible an attempt is also made to recommend what we (the organisers) regard as the most appropriate approach to adopt in any given situation based on the results of the exercise. In assessing the alternative theories one consideration will be whether the predictions are likely to be unsafe (non-conservative) or uneconomic (overly conservative) by comparison with the experimental results. This further examination will be illustrated by reference to the correlation between the predictions of the five theories with the experimental results shown in Figs 1–8. Firstly, a brief re-examination of the ability of the theories to predict the strength of an isolated unidirectional (UD) lamina under combined loading is necessary. 5.1. Predicting the response of a lamina Figures 1 and 2 show failure envelopes predicted by the five selected theories for unidirectional (UD) fibre reinforced glass/epoxy material subjected to combined loading. The available experimental results are superimposed upon the theoretical predictions. The experiments were mainly on UD fibre reinforced tubes and all of the failures were by fracture. Structures will not usually be designed with all of the fibres arranged in one direction if the structure is likely to be subjected to any stresses acting perpendicular to the fibres. However, UD fibre reinforced laminae are the basic building blocks in many laminates and within the laminates they will usually be exposed to simultaneous direct and shear stresses in directions parallel and perpendicular to the fibres. Therefore it is important for the designer to appreciate the basic characteristics of any failure criteria he intends to use. If a theory is incapable of

Fig. 1. Comparison between the predicted and measured failure stresses for a unidirectional glass fibrereinforced lamina subjected to combined shear and normal stresses perpendicular to the fibres. Material: E-Glass/LY556/HT907/DY063, (Test Case No 1).

1232

Fig. 2. Comparison between the predicted and measured biaxial failure envelope for a unidirectional glass fibre-reinforced lamina under combined normal stresses in directions parallel (x) and perpendicular (y) to the fibres. Material: E-glass/MY750 epoxy (Test Case No 3).

describing the response of the UD lamina building block then it is unlikely to be able to predict the response of more complex laminates. Tsai, followed by Cuntze and Puck, were among the theories that were highly ranked for predicting the strength of UD laminae under combined loads in Ref. [44]. Puck did well in predicting the shear stress versus transverse normal stress envelope ((12-2), see Fig. 1) but his predictions were a little unconservative in the tension-compression quadrant of the transverse versus longitudinal normal stress envelope ((2-1), see Fig. 2). Cuntze’s predictions were not quite as good as Puck’s for 12-2 and slightly better for 2-1. Tsai’s theory fitted all of the available experimental results well but doubt was expressed about the potentially unconservative predictions in the compression-compression quadrant of the 2-1 envelope where there is a shortage of critical experimental data. The theories of Zinoviev and Bogetti were un-conservative in some regions of both of the failure envelopes shown in Figs 1 and 2. Bogetti’s version of the Maximum Strain criterion did not reproduce the measured uniaxial transverse compressive strength in Fig. 2 and many parts of the predicted 2-1 envelope were also much less conservative than those predicted by other theories. Also in the biaxial compression quadrant of Fig. 2, Bogetti’s theory produced an initial failure envelope due to the through-thickness Poisson’s strain exceeding the failure strain, although no stress was assumed to act in that direction. 5.1.1. Recommendations Bearing these limitations and reservations in mind, it is recommended that the theories of Puck (or Cuntze) and Tsai are used together for predicting the strength of unidirectional laminae under combined loading. The innermost portions of the superimposed failure envelopes for these theories should be used for design purposes. Although Tsai gives the best fit to the available data, to use his less conservative predictions in the biaxial compression portion of the 2-1 envelope where data is not available would act against the normal design practice, referred to in Section 2. Taken in conjunction these theories are capable of ensuring

1233 conservative and reasonably accurate predictions of all three lamina failure envelopes considered in the exercise. If Puck and Cuntze’s theories have not been implemented then for design purposes the simpler Maximum Stress criterion (Zinoviev) could be used in conjunction with Tsai’s theory.

5.2. Predicting initial strength of multidirectional laminates (i.e. the onset of first failure) Most of the theories were able to predict initial failure in multi-directional laminates (see Table 1) and it was notable that nearly all of the theories, predicted initial matrix failures at low loads in the ‘black aluminium’ laminate test cases (Test Cases 6–8). Interestingly, this result suggests that any first generation aircraft parts that have been designed on the ‘fibre dominates’ principle outlined earlier, are likely to contain matrix cracks. The level of agreement between the theoretical predictions and the available experimental results, provided in the WWFE, was generally rather poor for initial failure, Ref. [44]. Doubts arise from: • the shortage of reliable experimental data on first ply failure; • whether the event recorded as the first significant failure in some of the experiments corresponds with the definition of initial failure in the theoretical predictions; • the rather unsatisfactory variation in the theoretical treatment of residual thermal stresses, Refs [33,45]. Of the five theories under consideration Bogetti and Zinoviev were recorded as performing best in the ranking for predicting initial failure in multidirectional laminates in Ref. [44]. However they both ignored residual thermal stresses completely and their initial failure predictions were a little un-conservative in Test Case 13 shown in Fig. 5, which is one of only two cases that provided visual experimental evidence of initial failure in the form of resin cracking. Those theories that made some allowance for thermal stresses (eg. Puck) were generally more conservative. Taking the thermal stresses into consideration would be a safe option. Puck’s approach, which made partial allowance for thermal stresses and performed well on a lamina level, gave conservative predictions of initial failure stress for all of the test cases considered in the exercise and was favoured by us in an earlier publication, Ref. [33]. However, the treatment of micro and macro residual stresses clearly requires deeper attention than has been possible in this exercise and those considerations should include due allowance for the effects of curing and also the expected moisture content of the laminate where such information is available. Careful consideration must also be given to the nature of the event that is predicted as first failure, in terms of the effect that such an event will have upon the functional performance of the laminate. Through such consideration, the designer should make decisions on the level of minor damage that might be acceptable in service. For example, assume that the analysis predicts or implies initial failure to be the onset of a few through-thickness cracks in one lamina. In an isolated lamina the formation of even one through-thickness crack would result in fracture into separate parts but in an embedded lamina the cracks could be arrested by adjacent layers and may have no detectable effect on laminate stiffness or performance. The

1234 designer of an aircraft wing may regard such an event as insignificant while the designer of a pressure vessel for storing aggressive substances may fear that it will signal the onset of accelerated diffusion through, and hence degradation of, the vessel wall and therefore regard this as unacceptable damage. 5.2.1. Recommendation Unfortunately the doubts discussed above make it difficult to arrive at straightforward conclusions regarding the best theory to use for predicting initial failure. We shall refrain from making firm recommendations for designers in this case – there simply isn’t sufficient, definitive, information from the WWFE on this matter.

5.3. Predicting final strength of multidirectional laminates None of the theoretical approaches considered in the exercise could claim very great accuracy in predicting final failure in multidirectional laminates – None of them could predict to within ± 10% of the measured strengths in more than 40% of the test cases. However, if the requirement for accuracy is relaxed somewhat then all of the selected theories can be applied for design purposes. All five were able to predict to within ± 50% of the measured final failure strengths in more than 75% of the ranking test cases and Puck and Cuntze predicted within that accuracy in more than 85% of the cases. In the remaining test cases some of the predictions were excessively conservative. The few predictions that overestimated the measured strength by a large margin were for experiments on multi-directional laminates loaded in uniaxial and biaxial compression, where there was a possibility that experimental values may have been reduced by the influence of buckling (see for example the biaxial compression quadrant of Fig. 3 which describes the theoretical and experimental failure envelopes for Test Case 6 ). Therefore, the theoretical predictions for those cases may not be as unsafe as they appear to be. However, Tsai’s theory over- predicted the experimental biaxial compression result in Test Case 6 (Fig. 3) by a larger factor than any of the others and that reflects the large and unvalidated biaxial compressive strength predicted by his theory for a UD lamina (Fig. 2). Figure 4 compares the theoretical predictions of the 5 theories superimposed on the experimental measurements of rupture strength for Test Case 9 ( ± 55° filament wound glassfibre reinforced epoxy tubes). All of the five theories appear to predict the experimental results reasonably well and buckling did not influence the biaxial compression results in these tests. Some approaches, including Puck, Cuntze and Zinoviev, encountered difficulties in predicting the response for angle-ply laminates and loading regimes of this type. Puck and Zinoviev reported numerical convergence problems and predicted open failure envelopes in the biaxial tension quadrant. Convergence problems were also encountered in Cuntze’s predictions but some of these problems were reported to have been reduced in his Part B revised results, Ref. [40]. Tsai’s predicted final failure stresses for this laminate were all reasonably conservative and the improvements to his theory presented in Part B of the exercise improved the fit to the experimental results. There were no convergence problems with his method.

1235

Fig. 3. Comparison between the predicted and measured final failure stresses for (0°/ ± 45°/90°) AS4/ 3501-6 carbon-fibre/epoxy laminates subjected to biaxial loads (Test Case No 6).

5.3.1. Recommendations On balance, and bearing in mind the reservations expressed above, we recommend Puck and Cuntze’s theories for predicting final failure strength of composite laminates, although Zinoviev had much to offer. Tsai’s method gave many sensibly conservative results and could be used in conjunction with the other methods (see Section 5.1.1). Finally, a word of caution: There is a multiplicity of software available, commercially, that purports to perform complex stress analysis of a structure, determine stresses in a lamina and offer the user a choice of well known failure criteria, allowing failure loads to be determined without employing any post failure analysis. The reader should note that such software will usually only predict initial failures but that may not be clear to an inexperienced user. Also, several theories that were used in the WWFE employed limited and poorly validated post failure analysis methodologies that resulted in falsely predicting final failures at loads that

1236

Fig. 4(a). Comparison between the predicted and measured final failure stresses for ( ± 55°) E-glass/ MY750 laminates subjected to biaxial loads (Test Case No 9).

were not much higher than the initial failure loads for some loading cases, Refs [33,43,44]. These theoretical approaches often greatly underestimated the actual measured final failure strength of multi-directional laminates (e.g. by a factor of up to 50 for Test Case 6). Such approaches could lead to hopelessly uneconomic or heavy designs and should not be adopted if the designer’s main concern is to determine the maximum load the structure can support. 5.4. Predicting the deformation of laminates In Figs 5–8 the stress–strain curves predicted by the selected theories are compared with the experimental stress–strain curves for four Test Cases (13, 12, 14 and 10) considered in the WWFE. All of the theories were capable of accurately predicting the linear elastic deformations of laminates at very low strains. The five selected theories all predicted the shape of the entire stress–strain curves quite well for Test Cases 12 and 13 (Figs 5 and 6) and in other fibre dominated cases (e.g. for the quasi-isotropic laminates in Test Cases 7 and 8, Ref. [44]).Three of the five theories (Puck, Cuntze and Bogetti) could predict the nonlinear behaviour that was observed in Test Cases 10 and 14 (Figs 7 and 8), at moderate strains before initial failure. The other two approaches (Tsai and Zinoviev) assumed linear elastic behaviour up to initial failure and this sometimes greatly under-estimated the measured strains and over-

1237

Fig. 4(b). Comparison between the predicted and measured ‘initial’ failure stresses for ( ± 55°) E-glass/ MY750 laminates subjected to biaxial loads (Test Case No 9).

estimated the stiffness, usually an unconservative feature and therefore undesirable from a design point of view. The application of linear elastic theories with design rules that simply limit the maximum strain are likely to be less safe than anticipated in cases (eg Test Cases 10, 14) where the stress–strain curves are actually non-linear (e.g. Figs 7 and 8). Beyond initial failure all of the theories were generally good at predicting deformation in cases where fibre stiffness was dominant, such as uniaxial and equal-biaxial tension loading of cross ply laminates (Figs 5 and 6). Even in these cases the abrupt decrease in lamina stiffness assumed by Bogetti produced steps in the predicted curves that were not observed in the experiments. The theories tended to under-perform in situations where shear stresses and matrix behaviour play a significant role in the failure process, and where large deformations were present before final failure occurred in the experiments (e.g. Figs 7 and 8). In Refs [33,44], Zinoviev was ranked best at predicting final failure strains and deformations of laminates. His theory, Ref. [17], allowed for the geometrical non-linearity associated with fibre reorientation, he assumed constant transverse tensile and shear stress characteristic behaviour in embedded laminae after initial failure, and gave some consideration to lamina unloading. Cuntze also predicted large strains at final failure in some cases (e.g. Fig. 8) but only after examining the experimental results and revising the assumed shear stress–strain curves for the embedded lamina to fit the results. In designs where the magnitude of deformations or strains are a major concern the accuracy of the predictions of

1238

Fig. 5. Comparison between the predicted and measured stress–strain curves for ± 45° E-glass/MY750 laminate under equal biaxial tension (y/x = 1/1). (Test Case No 13). The ranges of initial and final failure stresses shown are those predicted by all the theories presented in Refs.[1–3]. The stresses at which resin cracking, leakage and final failure were observed in the experiments are also indicated.

strain at the design stress or design strain level may be of even greater importance than at rupture. Then, if the design stress or strain levels are likely to produce non-linear behaviour (see for example Figs 7 and 8) that would favour the use of non-linear theories such as those employed by Puck, Cuntze and Bogetti. 5.4.1. Recommendations We recommend Zinoviev’s approach for predicting final deformations but it should be used in conjunction with one of the non-linear approaches (e.g. Puck) when a better description is required of non-linear behaviour at moderate strains. Special consideration should be given to cases where large deformations may be involved (e.g. in certain very restricted cases with angle-ply laminates) because none of the theories coped very well in this situation. The designer should assess whether large or non-linear deformations are likely to be significant in the application in hand and if so should avoid using theories that only employ simple linear assumptions (Table 1). 5.5. Predicting buckling failures The exercise did not address the prediction of buckling modes of failure, as they were deemed to be structural rather than a laminate problem for the purposes of the WWFE. The high

1239

Fig. 6. Comparison between the predicted and measured stress–strain curves for a 0°/90° E-glass/ MY750 laminate under uniaxial tension (y = 0) (Test Case No 12).

strength of composite laminates often allows the design of slender and thin walled structures that are prone to buckling, particularly if the effective laminate stiffnesses are low. Buckling can occur at various levels of strain. If it occurs at low strains then any of the selected laminate theories could be used with confidence to predict the effective laminate stiffnesses for use in conventional elastic buckling equations. In cases where the laminate can sustain higher strains, then the nonlinear theories identified in the previous section could be employed to generate reduced effective tangent or secant stiffness data for the laminate, which could then be used subsequently in buckling calculations. A deeper consideration of buckling is beyond the scope of this paper and of the WWFE.

5.6. Predicting and designing against leakage failure It was commonly assumed that the first obvious signs of failure, which took the form of leakage through the tube walls in experiments on filament wound tubes, should agree with the theoretical initial failure predictions. In the WWFE it was shown that nearly all of the theories predicted initial failure stresses that were lower than the stresses that produced leakage in the

1240

Fig. 7. Comparison between the predicted and measured stress–strain curves for ± 55° E-glass/MY750 laminate under uniaxial tension (y/x = 1/0) (Test Case No 10).

experiments, see Fig. 4b. It now seems probable that a considerable amount of damage in the form of resin cracking and delamination, has to accumulate before there are sufficient leakage paths present to allow detectable leakage of fluid through the tube wall. It therefore follows that significantly higher stresses are required to propagate the required amount of damage than is needed to initiate the first cracks. This is confirmed by the results from Test Case 13 (Fig. 5). In those tests the stress at which cracks were first observed were much lower than those at which leakage was observed. 5.6.1. Recommendations Further theoretical work is required to allow leakage to be predicted with greater accuracy (see Section 7.1). However, as an interim measure, any of the chosen five theories can be used to predict initial failure strengths for typical pressure vessel configurations. As indicated above, the initial failure strength can be used as a lower bound (hence conservative) value. In practice laminated composite pressure vessels are often designed with impermeable metal or plastic liners to prevent leakage. This may allow very high stresses to be carried by the composite layers without leakage or rupture failure occurring. Several standard codes exist for this type of design and to minimise the weight of such vessels safety factors against rupture as low as 1.5–2.0 are applied in some cases, Refs [51–52]. The better non-linear approaches to failure prediction identified in section 5.4.1 should be applied if angle-ply or similar laminates are considered for this type of design.

1241

Fig. 8. Comparison between the predicted and measured stress–strain curves for ± 45° E-glass/MY750 filament wound tube subjected to circumferential tension and equal axial compression stresses, y/ x = 1/–1 (Test Case No 14).

6. Shortfalls in experimental data and identifying the gaps The method employed in this study for assessing the accuracy of the theoretical predictions obviously relies greatly upon the pedigree of the experimental data, which has been utilised as the benchmark. The variety and accuracy of the experimental results employed in the WWFE has been more than adequate to reveal undeniable weaknesses in many of the theories. However, some gaps or weaknesses in the experimental result were also to be expected and an important achievement of the exercise has been to identify and publicise just where some of those weaknesses and shortfalls lie. This information forms a useful launch pad for followon projects which could be aimed at further consolidation of the theoretical methods highlighted in the WWFE. An initial view from the organisers on the shortfalls in the experimental data was presented in Table 3 of Ref. [33]. The consequences on the outcome of the exercise were discussed in Ref. [33]. A more specific list of the experiments required to build on the WWFE and the rationale for them is provided in Table 2, together with a brief commentary, below.

6.1. Failure stresses for isolated laminae under biaxial loads There is a need for reliable and more complete experimental failure envelopes for unidirectional fibre-reinforced laminae under combined axial and transverse loading. Missing data at particular ratios of biaxial compression and biaxial tension for an Eglass-fibre/epoxy

1242 Table 2 Further key experimental tests required for validating the theories. Test Description of the required loading cases Case 2

Tests of carbon/epoxy unidirectional lamina under combined 1 and 12 at stress ratio SR = 1/12 = 18.7/1, 8/1, –11.4/1,–1/0, 1/0, 0/1. Reasons: No test data available to distinguish between the extreme theories and also to confirm the peak in the reported shear strength. Methods of testing could include axially wound tubes under combined axial load and torsion. Data measured include failure strains and loads and the shape of the stress strain curves.

3

Tests of glass/epoxy unidirectional lamina under combined 1 and 2 at stress ratio SR = 1/2 = 7.8/1, –3.75/–1 and -14/–1, –20/1. Reasons: (a) No test data available to distinguish between the extreme theories, particularly Tsai and Maximum Strain criteria (Bogetti and Hart-Smith(2)) and (b) Check for initial failure (Bogetti). Methods of testing may include tubes under internal and external pressure combined with axial tension/compression. Data measured include failure modes, failure strains and loads/pressure and the shape of the stress strain curves.

6

Tests on (0°/ ± 45°/90°) quasi-isotropic carbon/epoxy laminates covering the biaxial compression-compression quadrant and in particular under y /x = –1/–1. Reasons: (a) The test data provided were compromised by buckling and (b) crucial, true compressive tests are needed to check the accuracy of Tsai’s and Bogetti’s predictions. Methods of testing include thick tubes or cruciforms under biaxial loads. Data measured include failure stresses and strains and failure modes.

6

Tests on (0°/ ± 45°/90°) quasi-isotropic carbon/epoxy laminates under equal biaxial tension y /x = 1/1. Reasons: (a) To test maximum strength of laminate and maximise the ability to discriminate between predictions in biaxial tension (Huang, Cuntze-B, Wolfe, Tsai, Rotem). Methods of testing include tubes or cruciforms under biaxial loads. Data measured include failure stresses and strains, failure modes and damage.

7

Tests on (0°/ ± 45°/90°) quasi-isotropic carbon/epoxy laminates under y /x = 1/0. Reasons: To determine initial and intermediate failure stresses and modes of failure. Methods of testing by using either tubes or coupons. Data measured: Crack developments at various loads.

10

Tests on ± 55° glass/epoxy laminate under y /x = 1/0. Reasons: To determine the initial failure stresses, observe damage, damage progression, initial and leakage and final failure modes. Methods of testing include tubes under internal pressure or cruciforms. Data measured include crack measurement.

1243 Table 2 Continued. Test Description of the required loading cases Case 11

Tests on ± 55° glass/epoxy laminate under y /x = 2/1. Reasons: to determine initial failure stresses, crack density, damage progression, damage state at leakage. Methods of testing include tubes under internal pressure or cruciforms. Data measured include crack density measurement.

13

Tests on ± 45° glass/epoxy laminate (long tubes) under y /x = 1/1. Reasons: to determine accurate stress strain curves and final strength. Methods of testing include tubes under internal pressure and axial tension or cruciforms. Data measured include load or pressure and strains during tests.

14

Repeat Tests on ± 45° glass/epoxy laminate under y /x = –1/1. Reasons: to determine the initial and final failure stresses, observe failure modes. Methods of testing include tubes under internal pressure or cruciforms. Data measured include microscopic damage and crack measurement at various levels of load/ pressure.

composite used in the exercise was critical in preventing full evaluation of the theoretical predictions including those by Tsai[15,30] and the Maximum Strain theory (HartSmith[10,25] and Bogetti[35,39]). Also, further experiments are required to confirm or deny the existence of the measured peak shear stress that was not present in any of the theoretical predictions of lamina failure under combined longitudinal direct and shear loading, Ref. [20]. 6.2. Structural failure (buckling) masking true material failure at higher loads There were indications that buckling under uniaxial and biaxial compression loading had resulted in premature failure in the available test results for Test Cases 4 and 6 (the 0°/ ± 45°/90° carbon fibre /epoxy and 90°/ ± 30° glass fibre/epoxy laminates). In the authors’ opinion true material failure would have occurred at much higher loads, somewhat closer to those predicted by the majority of the theories. Further tests in this loading region are suggested, particularly for the quasi-isotropic carbon-fibre reinforced laminate (see Table 2). In fact, given the heavy usage of this type of laminate configuration in aerospace, it is quite possible that such data already exists in the database of one or more of the industry primes. If so, then we would be pleased to hear from any reader with knowledge and access. 6.3. Lack of information on initial failure loads A common shortfall in the data set was a lack of emphasis on gathering information associated with the initial failure event, during testing. Thus, data on the magnitude of the

1244 initial failure stress (or load), the mode of failure that occurred and the practical significance of the mode of initial failure on the post-initial laminate performance is sparser than ideal. The exceptions were from Test Case 12 (0°/90° laminates tested under uniaxial tension (Fig. 6)) and Test Case 13 ( ± 45° filament wound tubes loaded under equal biaxial tension (Fig. 5)). In those cases data was available on the magnitude of the loads at onset of lamina cracking (due to tension perpendicular to the fibres) and on the relationship between crack multiplication and change in stiffness as loading progressed beyond initial failure. However, more experimental data on initial failure loads and knowledge of the real effects in a wider range of laminates and the significance of initial failure on subsequent laminate performance is important in many design situations. To facilitate comparison with theoretical predictions the test temperature, curing cycle and moisture content should be recorded.

6.4. Recommendations We recommend that the community undertake the work necessary to obtain the additional experimental data, outlined above and listed in Table 2. However, this is not routine testing and the task cannot be undertaken lightly. There is a very good reason why such information is rather sparse in the literature – these are hard experiments to conduct! It is important to realise the dedication, skill, care, effort and resources required to undertake and complete such work and to obtain the necessary accurate, reliable and meaningful results. High levels of engineering, experimental and analytical skills, ingenuity and attention to detail will be required in designing and manufacturing specimens and test rigs, conducting tests and interpreting the results. Ref. [53], for example, gives an indication of just some of the problems to be overcome in testing tubes under external pressure. Even converting a tentative conceptual list like Table 2 into a proper, thoroughly considered programme of work will require considerable effort.

7. Shortfalls in the theories and developments required Table 3 contains a list of major issues addressed or touched upon by the WWFE and indicates whether the confidence generated in the theoretical predictions of those aspects is high or low. In some cases where confidence is low further experimental work would clearly help to improve the situation. Of course, there is a huge body of theoretical and experimental activity in the composites community beyond that reported in the WWFE. Thus many of the issues listed in Table 3 may be/are being addressed elsewhere. It behoves the reader to keep studying the literature. However, a number of the key deficiencies are considered below.

7.1. Prediction of leakage As noted in Section 5.6 none of the theories was able to predict leakage. A deeper theoretical study is required to explore the nature of damage accumulation after initial failure and hence predict leakage failures. Development of crack multiplication, damage and delamination

1245 Table 3 A list of the major issues addressed in the exercise.

No

Test Case

Issues to be addressed

1 2 3 4 5 6 7 9 10 11 12 13 14 15 16

1–14 1–3 4–14 4–7 1–14 4–14 4–14 4–6 4–14 4–14 6,9,14 4–14 4–14 4–7 6–8,15

17

9,10,14

Use of micro-mechanics for prediction properties/failure Prediction of the biaxial failure of a lamina in isolation Prediction of various modes of failure Prediction of the whole laminates failure envelopes Thermal residual curing stress consideration In situ strength of an embedded lamina Matrix failure in tension, shear and compression Leakage of pressurised pipes Single material non-linearity Multiple material and structural non-linearities Thin and thick laminae Effects of lay up sequence Post failure modelling or degradation of composite Prediction of fibre failure Integration of laminate and structural failure analysis for specimen design Prediction of large strains and deformations

Degree of confidence in predictions* High High High High Low Low High Low High Low Low Low High High Low Low

* High: This implies that participants have dealt with the issue relatively well. Low: This implies that the participants either did not take the opportunity to tackle the issue or they took it but the overall results lacked unanimity or accuracy.

theory may provide the solution to this problem. Refs [14,29,9,26,47,48]. The reader is reminded that ‘Damage Mechanics’ was in its infancy at the start of the WWFE and McCartney and Sun, Refs [9,14,26,29], were the only ones willing to represent that approach in the WWFE. McCartney’s theory did not feature well in the overall performance assessment, primarily because it was not sufficiently well developed or robust enough. However ‘Damage Mechanics’ has made significant progress in recent years and remains a rapidly moving field within the composites research community today. The authors anticipate that any follow-on studies from the WWFE that feature leakage will stimulate likely solutions from the ‘Damage Mechanics’ experts. 7.2. Prediction of laminate behaviour at high strains None of the theories was able to predict the nonlinear shape of the stress–strain curves at high strains where behaviour was matrix dominated (e.g. Test Cases 10, 14 (Figs 7 and 8)), nor predict the large final failure strains accurately in those cases. Developments in this area should incorporate the simultaneous effects of progressive damage, shear non-linearity, change in fibre orientation and differences between the behaviour of embedded and isolated

1246 laminae. Improved modelling of progressive damage and work of the type presented in this exercise by McCartney [9,26] on crack multiplication may help improve the prediction of stiffness degradation after initial failure. The work described in Refs [17,12,47] may also be of assistance in this task.

7.3. Availability of computer codes for the most promising laminate analysis theories The laminate theories discussed in this paper are believed to be the best available at present and, as indicated in Table 1, they are already embodied in computer codes of one form or another. The accuracy of the predictions produced by them depends upon details of the implementation and the reliability of the numerical methods employed. Generating a computer model that embodies a given theory is clearly a task best carried out by the originator of that theory, rather than leaving it up to individual designers to attempt to replicate their own versions that have high likelihood of containing transcription errors, and is certain to be more costly in time and effort. We believe it is incumbent on the originators of the most promising theories to make such user-friendly computer codes readily available to practicing designers. One certainty is that the best theory in the world is useless if it is not made available in an accurate and user friendly form. Professor Tsai has set a good example by making his codes widely available at very low cost and there is further scope for all of the participants to make software available via other routes such as the world wide web.

7.4. Integration into finite element structural analysis packages Predictions of the behaviour of realistically complex, laminated load bearing composite structures requires the use of finite element (FE) or other structural analysis methods capable of handling large displacements and multiple non-linearities, together with properly validated laminate analysis methods. As noted in Table 1 some of the contributors, for example Chamis et al. [6] and Mayes [37], have already incorporated their methods into FE codes and it is important that in due course, all of the leading theories are made available in this form, preferably in conjunction with well known and validated FE packages. This puts an increased burden on the theoreticians and on the code implementers, to provide better in-built controls to keep the user within the bounds of applicability of the theory. The organisers feel that this is a duty of care issue for future users.

8. Issues not addressed by the exercise There are a number of topics relating to the prediction of failure in composite laminates that have not been addressed by the current failure exercise including the following: 3-D failure criteria and stress systems Delamination initiation and propagation Buckling in thin walled and slender structures Effects of other matrix and reinforcement materials Other forms of fibre reinforcement, woven and non- woven cloths

1247 Effects of ply thickness and stacking sequence Long-term behaviour, creep, fatigue and degradation Environmental effects, including the presence of moisture and aggressive substances Effects of different temperatures, including high and low and temperatures Sequential loading, e.g. compression after impact, leak after impact Integration into finite element and other forms of structural analysis Many of these issues have, of course, been addressed in other studies elsewhere. Those that haven’t are natural contenders for further, comprehensive assessment studies, along the lines of the WWFE. 9. Possible Use of the WWFE framework for assessing new theories in future Within the WWFE, careful consideration has been given to making the quantitative method of assessment and ranking, used here and described in detail in Appendix C of Ref. [33], as objective and repeatable as possible, so that the same ranking criteria and experimental results could be used for assessing new theories. Whilst the organisers recognise that the WWFE has been too protracted, the time that has elapsed has allowed existing theories to be refined, Ref. [2] and new theories to be developed and included in the exercise, Ref. [3]. The production of Chapter 6 of this book which includes the assessment of additional theories has also clearly demonstrated that the assessment methods are repeatable. Should assessment of any new theories be attempted in the future by using the same Test Cases, one critical factor will be lost (i.e. the predictions will not be ‘blind’, given that the three special journal editions covering the WWFE are now in the public domain). However usage of the methodology in an open manner would be similar to that employed to assess the revised predictions in Part B. Using this method would again force a wide range of problems to be considered and would allow comparison with the results presented here. Clearly, any new theory that performed less well against the ‘open’ WWFE test assessment is unlikely to offer any improvements against those theories already assessed to date. An open WWFE test assessment could be used as a ‘necessary but insufficient’ first benchmark of any new theory. More discerning or critical test cases could be added (or substituted for less critical ones) as important missing experimental data (see Section 6 and Table 2) becomes available. This would require an extension of the quantitative ranking process from the WWFE and would enable a continuous improvement process to be implemented for the leading theories. Refinements and improvements could be made to the method, selected subsets of the method could be employed and, if required, similar approaches could be applied in some of the areas where confidence is low (Table 3) and to related topics not covered by the existing exercise (see Section 8 above). The authors would be happy to pass on the baton to the next team of organisers! 10. Concluding remarks 1. In this chapter the results of the WWFE have been used to select five promising methods (Puck, Zinoviev, Tsai, Cuntze and Bogetti) for predicting and designing against failure in

1248

2.

3.

4.

5.

6.

7.

8.

a variety composite laminates subjected to a range of loading conditions. The accuracy with which these theories performed in predicting different types of failure of importance to designers has been compared and we have indicated the approaches that we consider most appropriate for use by designers in particular situations. These theories provide a powerful set of design tools that can be used to tackle most design problems. Many other theories in the WWFE exhibit particularly useful features and attention has been drawn to some of them. At a lamina level many theories described the available experimental strengths for unidirectional fibre reinforced composites subjected to biaxial loading and combined direct and shear stresses reasonably well. More experimental data is required to verify potentially non-conservative predictions by some theories (e.g. Tsai and Bogetti). For design purposes it was recommended that Tsai’s theory be used in conjunction with Puck’s because together they gave reasonably accurate and conservative predictions for all the failure envelopes considered. Difficulties were encountered in assessing the accuracy of the initial failure results due to a number of factors (e.g. different approaches adopted for calculating residual stresses, absence of information on moisture content and shortage of good experimental data on initial failure). Attention was drawn to the need for careful consideration of the nature and practical significance of the initial failures predicted. Nearly all of the theories gave accurate predictions of laminate properties at very low strains before initial failure but none of them dealt very well with the large non-linear deformations observed in some tests and this area requires further attention. The theories performed reasonably well in predicting the final fracture strength of multi directional laminates for design purposes, the most accurate (Puck and Cuntze) predicting the experimental results to within ± 50% in more than 85% of the test cases. It is important that the best theories be made readily available in accessible user-friendly codes and that they be integrated into accessible and reliable ‘state of the art’ finite element packages with the capabilities of handling multiple non-linearities. The organisers believe that all of the objectives of the exercise have been achieved. Many lessons have been learned by addressing a wider range of practical laminates and loading cases than usual and by comparing theoretical predictions with experimental results. The strengths and weaknesses of the current theories have been exposed and areas where further experimental and theoretical work is needed have been identified. A number of the participants have already made improvements to their theories to address some weaknesses and the level of confidence in the use of design tools could be further improved if future collaborative activities focus on the outstanding experimental and theoretical issues identified as a result of this exercise.

References 1 M J Hinton, P D Soden and A S Kaddour, ‘Failure Criteria In Fibre Reinforced Polymer Composites’, a special issue of Compos Sci Techno, V58: No 7, published by Elsevier Science Ltd, Oxford, UK, 1998. 2 M J Hinton, P D Soden and A S Kaddour, ‘Failure Criteria In Fibre Reinforced Polymer Composites: Comparison with experiments’, a special issue of Compos Sci Techno, V62: No 11/12, published by Elsevier Science Ltd, Oxford, UK, 2002.

1249 3 P D Soden, M J Hinton and A S Kaddour, ‘Failure Criteria In Fibre Reinforced Polymer Composites: Additional Theories, Comparison and Recommendations’, a special issue of Compos Sci Techno, V64: No 3–4, 2004, Published by Elsevier Science Ltd, Oxford, UK, 2004 4 M J Hinton and Soden P D, ‘Failure criteria for composite laminates’, Compos Sci Techno, V58, pp 1001–1010, 1998, see also Chapter 1 in this book. 5 P D Soden, M J Hinton and A S Kaddour, ‘Lamina properties and lay-up configurations and loading conditions of a range fibre reinforced composite laminates’, Compos Sci Techno, V58, pp 1011–1022, 1998, see also Chapter 2.1 in this book. 6 P K Gotsis, C C Chamis and L Minnetyan, ‘Prediction of composite laminate fracture: micromechanics and progressive fracture’, Compos Sci Techno, V58, pp 1137–1150, 1998, see also Chapter 3.1 in this book. 7 G C Eckold, ‘Failure criteria for use in the design environment’, Compos Sci Techno, V58, pp 1095–1106, 1998, see also Chapter 3.2 in this book. 8 E C Edge, ‘Stress based Grant-Sanders method for predicting failure of composite laminates’, Compos Sci Techno, V58, pp 1043–1044, 1998, see also Chapter 3.3 in this book. 9 L N McCartney, ‘Predicting transverse crack formation in cross-ply laminate’, Compos Sci Techno, V58, pp 1069–1082, 1998, see also Chapter 3.4 in this book. 10 L J Hart-Smith, ‘Predictions of the original and truncated maximum strain failure models for certain fibrous composite laminates’, Compos Sci Techno, V58, pp 1151–1178, 1998, see also Chapter 3.5 in this book. 11 L J Hart-Smith, ‘Predictions of a generalised maximum-shear-stress failure criterion for certain fibrous composite laminates’, Compos Sci Techno, V58, pp 1179–1208, 1998, see also Chapter 3.6 in this book. 12 A Puck and H Schürmann, ‘Failure analysis of FRP laminates by means of physically based phenomenological models’, Compos Sci Techno, V58, pp 1045–1068, 1998, see also Chapter 3.7 in this book. 13 A Rotem, ‘Prediction of laminate failure with Rotem failure criterion’, Compos Sci Techno, V58, pp 1083–1094, 1998, see also Chapter 3.8 in this book. 14 C T Sun and J X Tao, ‘Prediction of failure envelopes and stress strain behaviours of composite laminates’, Compos Sci Techno, V58, pp 1125–1136, 1998, see also Chapter 3.9 in this book. 15 K-S Liu and S W Tsai, ‘A progressive quadratic failure criterion of a laminate’, Compos Sci Techno, V58, pp 1023–1032, 1998, see also Chapter 10 in this book. 16 W E Wolfe and T S Butalia, ‘A strain energy based failure criterion for nonlinear analysis of composite laminates subjected to biaxial loading’, Compos Sci Techno, V58, pp 1107–1124, 1998, see also Chapter 10 in this book. 17 P Zinoviev, S V Grigoriev, O V Labedeva and L R Tairova, ‘Strength of multilayered composites under plane stress state’, Compos Sci Techno, V58, pp 1209–1224, 1998, see also Chapter 3.11 in this book. 18 P D Soden, M J Hinton, and A S Kaddour, ‘Comparison of the predictive capabilities of current failure theories for composite laminates’, Compos Sci Techno, V58, pp 1225–1254, 1998, see also Chapter 4 in this book. 19 M J Hinton, Soden P D and Kaddour A S, ‘Evaluation of failure prediction in composite laminates: background to Part B of the exercise’, Compos Sci Techno, V62, pp 1481–1488, 2002, see also Chapter 1 in this book. 20 P D Soden, Hinton M J and Kaddour A S, ‘Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates: Failure exercise benchmark data’, Compos Sci Techno, V62, pp 1489–1514, 2002, see Chapter 2.2 in this book. 21 L J Hart-Smith, ‘Expanding the capabilities of the ten-percent rule for predicting the strength of fibre-polymer composites’, Compos Sci Techno, V62, pp 1515–1544, 2002, see also Chapter 3.17 in this book. 22 P K Gotsis, C C Chamis and L Minnetyan, ‘Application of progressive fracture analysis for predicting failure envelopes and stress–strain behaviors of composite laminates: A comparison with

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26 27

28 29

30 31

32

33

34

35

36

37 38

39

40

experimental results’, Compos Sci Techno, V62, pp 1545–1560, 2002, see also Chapter 5.1 in this book. G C Eckold, ‘Failure criteria for use in the design environment – Part B’, Compos Sci Techno, V62, pp 1561–1570, 2002, see also Chapter 5.2 in this book. E C Edge, ‘Theory v. experiment comparison for stress based Grant-Sanders method’, Compos Sci Techno, V62, pp 1571–1590, 2002, see also Chapter 5.3 in this book. L J Hart-Smith, ‘Comparison between theories and test data concerning the strength of various fibre-polymer composites’, Compos Sci Techno, V62, pp 1591–1618, 2002, see also Chapter 5.4 in this book. L N McCartney, ‘Predicting ply crack formation and failure in laminates’, Compos Sci Techno, V62, pp 1619–1632, 2002, see also Chapter 5.5 in this book. A Puck and A Schürmann, ‘Failure analysis of FRP laminates by means of physically based phenomenological models- Part B –’, Compos Sci Techno, V62, pp 11633–1672, 2002, see also Chapter 5.6 in this book. A Rotem, ‘The Rotem failure criterion theory and practice’ Compos Sci Techno, V62, pp 1663–1672, 2002, see also Chapter 5.7 in this book. C T Sun, J. Tao and A S Kaddour, ‘Prediction of failure envelopes and stress–strain behavior of composite laminates: Comparison with experimental results’, Compos Sci Techno, V62, pp 1672–1682, 2002, see also Chapter 5.8 in this book. A Kuraishi, S W Tsai, K Liu, ‘A progressive quadratic failure criterion, Part B’, Compos Sci Techno, V62, pp 1682–1696, 2002, see also Chapter 5.9 in this book. T S Butalia and W E Wolfe, ‘Strain energy based non-linear failure criterion: comparison of numerical predictions and experimental observations for symmetric composite laminates’, Compos Sci Techno, V62, pp 1695–1710, 2002, see also Chapter 5.10 in this book. P A Zinoviev, O V Lebedeva and L P Tairova, ‘Coupled analysis of experimental and theoretical results on the deformation and failure of laminated composites under a plane state of stress’, Compos Sci Techno, V62, pp 11711–1724, 2002, see also Chapter 5.11 in this book. M J Hinton, A S Kaddour and P D Soden, ‘A comparison of the predictive capabilities of current failure theories for composite laminates, judged against experimental evidence’, Compos Sci Techno, V62, pp 1725–1797, 2002, see Chapter 5 in this book. M J Hinton, P D Soden and A S Kaddour, ‘Evaluation of failure prediction in composite laminates: background to Part C of the exercise’, Compos Sci Techno, V64: No 3–4, pp 321–327, 2004, see Chapter 1 in this book. T A Bogetti, C P R Hoppel, V M Harik, J F Newill and B P Burns, ‘Predicting the nonlinear response and progressive failure of composite laminates’, Compos Sci Techno, V64: No 3–4, pp 329–342, 2004, see also Chapter 3.13 in this book. R G Cuntze and A Freund, ‘The predictive capability of failure mode concept – based strength criteria for multidirectional laminates’, Compos Sci Techno, V64: No 3–4, pp 343–377, 2004, see also Chapter 3.14 in this book. S J Mayes and A C Hansen, ‘composite laminate failure analysis using multicontinuum theory’, Compos Sci Techno, V64: No 3–4, pp 379–394, 2004, see also Chapter 3.15 in this book. Z M Huang, ‘A bridging model prediction of the tensile strength of composite laminates subjected to biaxial loads’, Compos Sci Techno, V64: No 3–4, pp 395–448, 2004, see also Chapter 3.16 in this book. T A Bogetti, C P R Hoppel, V M Harik, J F Newill and B P Burns, ‘Predicting the nonlinear response and failure of composite laminates: correlation with experimental results’, Compos Sci Techno, V64: No 3–4, pp 477–485, 2004, see also Chapter 5.12 in this book. R G Cuntze, ‘The predictive capability of failure mode concept – based strength criteria for multidirectional laminates: Part B’, Compos Sci Techno, V64: No 3–4, pp 487–516, 2004, see also Chapter 5.13 in this book.

1251 41 S J Mayes and A C Hansen, ‘A comparison of multicontinuum theory based failure simulation with experimental results’, Compos Sci Techno, V64: No 3–4, pp 517–527, 2004, see also Chapter 5.14 in this book. 42 Z M Huang, ‘Correlation of the bridging model predictions of the biaxial failure strengths of fibrous laminates with experiments’, Compos Sci Techno, V64: No 3–4, pp 529–548, 2004, see also Chapter 5.15 in this book. 43 A S Kaddour, M J Hinton and P D Soden, ‘Further comparison of predictive capabilities of failure theories for composite laminates’, Compos Sci Techno, V64: No 3–4, pp 449–476, 2004, see also Chapter 4 in this book. 44 M J Hinton, P D Soden and A S Kaddour, ‘A further assessment of the predictive capabilities of current failure theories for composite laminates: comparison with experimental evidence’, Compos Sci Techno, V64: No 3–4, pp 549–588, 2004, see also Chapter 6 in this book. 45 M J Hinton, A S Kaddour and P D Soden, ‘Predicting failure in fibre composites: The final chapter of the world-wide failure exercise’, Proceedings of the Fourteenth International Conference on Composite Materials (ICCM-14), San Diego, USA. 46 I Eriksson and C G Aronsson, ‘Strength of tensile loaded graphite/epoxy laminates containing cracks, open and filled holes’, J Compos Mater, V24, pp 456–482, (1990). 47 S Li, S R Reid and P D Soden, ‘A continuum damage model for transverse matrix cracking in laminated fibre-reinforced composites’, Phil Trans R Soc London A, V356, pp 2379–412, 1998. 48 Z Zou, S R Reid, S Li and P D Soden, ‘Modelling interlaminar and intralaminar damage in filament-wound pipes under quasi-static indentation’, J Composite Materials, V36, pp 477–499, 2002. 49 B R Roe, ‘Probability and Statistics in Experimental Physics’, Springer-Verlag, New York, 1992. 50 K Y Lin, J Du and D Rusk, ‘Structural Design Methodology Based on Concepts of Uncertainty’, NASA/CR-2000–209847. 51 D B Johnson, N L Newhouse, D D Baldwin and K H Lo, ‘Codes And Standards For Composite Tanks And Vessels: A general Discussion’, in 3rd Int Conf on Composite materials for offshore operations, Houston, TX, October 31- November 2, 2000. 52 ASME Boiler and Pressure Vessel Code X: Fibre-reinforced plastic pressure vessels. The American Society of Mechanical Engineers, New York: see also 2002 Addenda July 1, 2002. 53 A S Kaddour, P D Soden and M J Hinton, ‘Behaviour of + 55/–55 angle ply laminates under external pressure and axial compression’, (ICCM-11), Gold Coast, Queensland, Australia, 14–18 July, 1997.

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1253

Index Bogetti, T.A., 14, 24, 50, 332, 402, 426, 647, 656, 660, 662, 663, 665, 667, 674, 677, 678, 683, 689, 690, 693, 697, 698, 701, 961, 975, 1087, 1098, 1102, 1108, 1109, 1119, 1121, 1125, 1127, 1130, 1132, 1135, 1138, 1141, 1144, 1147, 1150, 1153, 1156, 1160, 1163, 1167, 1171, 1207, 1208, 1229, 1232, 1233, 1236–1238, 1243, 1247, 1248, 1250 Burns, B.P., 24, 50, 332, 402, 426, 701, 961, 975, 1109, 1125, 1250 Butalia, T.S., 15, 26, 49, 353, 377, 425, 573, 645, 701, 922–925, 927, 928, 930–932, 936, 940, 941, 1117, 1124, 1125, 1249, 1250 Chamis, C.C., 15, 24, 49, 120, 262, 377, 378, 403, 425, 526, 527, 574, 592, 594, 644, 645, 647, 653, 656, 659, 665, 666, 670, 671, 674, 677, 678, 683, 684, 689, 690, 695, 697, 699, 700, 723–725, 1021, 1087, 1098, 1105, 1109, 1123, 1124, 1127, 1130, 1132, 1135, 1138, 1141, 1144, 1147, 1150, 1153, 1156, 1160, 1163, 1167, 1173, 1207, 1208, 1246, 1249 Cuntze, R.G., 14, 24, 50, 296, 429, 432–435, 453, 474–477, 479, 481, 483, 484, 647, 656, 663, 664, 666, 677, 678, 683, 689, 690, 695, 697, 701, 772, 805, 873, 976, 978, 1020, 1021, 1087, 1098, 1105–1108, 1110, 1121, 1125, 1127, 1130, 1132, 1135, 1138, 1141, 1144, 1147, 1150, 1153, 1156, 1160, 1163, 1167, 1174, 1207, 1208, 1229, 1232, 1234, 1236–1238, 1247, 1248, 1250 Eckold, G.C., 25, 49, 81, 83–85, 87, 92, 121, 139, 398, 425, 647, 656, 659–661, 663, 665, 670, 684, 689, 698, 700, 726, 738, 938, 942, 1092, 1105, 1108, 1110, 1111, 1123, 1124, 1127, 1130, 1132, 1135, 1138, 1141, 1144, 1147, 1150, 1153, 1157, 1160, 1164, 1168, 1176, 1207, 1249, 1250

Edge, E.C., 25, 49, 140, 144, 156, 272, 297, 425, 647, 656, 666, 670, 671, 677, 678, 684, 689, 693, 695, 696, 697, 700, 739, 768, 1087, 1105, 1111, 1123, 1124, 1127, 1130, 1132, 1135, 1138, 1141, 1143, 1144, 1147, 1151, 1154, 1157, 1161, 1164, 1168, 1178, 1205, 1207, 1208, 1249, 1250 Freund, A., 24, 50, 429, 701, 1019, 1020, 1110, 1125, 1250 Gotsis, P.K., 24, 49, 425, 645, 700, 703, 704, 724, 1109, 1123, 1124, 1249 Grigoriev, S.V., 26, 49, 379, 399, 425, 701, 960, 1117, 1124, 1249 Hansen, A.C., 14, 25, 50, 492, 497, 516, 517, 701, 1044, 1113, 1125, 1250, 1251 Harik, V.M., 24, 50, 402, 426, 701, 961, 975, 1109, 1125, 1250 Hart-Smith, L.J., 3, 14, 17, 25, 49, 50, 139, 179, 217–219, 262, 263, 297, 378, 425, 476, 597, 641, 645, 647, 653, 656, 659, 660, 663, 670, 673, 677, 678, 684, 689, 690, 693, 695, 698, 700, 701, 770, 804, 805, 873, 1087, 1092, 1098, 1105, 1108, 1111–1113, 1119, 1120, 1124, 1126–1128, 1130, 1132, 1133, 1136, 1138, 1141–1144, 1147, 1151, 1154, 1157, 1161, 1164, 1168, 1180–1182, 1205, 1207, 1208, 1227, 1228, 1243, 1249, 1250 Hinton, M.J., 4, 24, 26, 30, 49, 50, 61, 81, 86, 91–93, 98, 120, 139, 156, 175, 176, 217, 218, 262, 263, 297, 312, 333, 338, 351, 378, 399, 403, 424–426, 474, 516, 520, 521, 573, 574, 597, 598, 641, 642, 700, 701, 725, 738, 768, 769, 804, 805, 831, 833, 872, 873, 889, 902, 920, 921, 922, 933, 934, 938, 940–942, 956, 959, 960, 975, 1017, 1020, 1021, 1043, 1044, 1071, 1123–1125, 1248–1251

1254 Hoppel, C.P.R., 24, 50, 402, 426, 701, 961, 975, 1109, 1125, 1250 Huang, Z.M., 14, 25, 50, 139, 378, 518, 574, 575, 647, 653, 656, 660, 663, 664, 666, 674, 676, 678, 683, 684, 689, 690, 695, 696, 698, 699, 701, 725, 1045, 1071, 1081, 1087, 1105, 1113, 1125, 1128, 1131, 1133, 1136, 1138, 1142, 1145, 1148, 1151, 1154, 1157, 1161, 1164, 1168, 1184, 1207, 1208, 1227, 1250, 1251 Kaddour, A.S., 4, 24–26, 30, 49, 50, 61, 67, 80, 91–93, 120, 139, 156, 176, 217, 262, 297, 312, 333, 351, 378, 399, 424–426, 474, 516, 520, 521, 573, 574, 597, 598, 641, 642, 700, 701, 725, 738, 768, 769, 804, 805, 831, 833, 872, 873, 889, 902, 920–922, 935, 940–942, 956, 959, 960, 975, 1020, 1021, 1043, 1044, 1071, 1123–1125, 1248–1251 Kuraishi, A., 26, 903, 919, 921, 1116, 1125, 1250 Labedeva, O.V., 49, 425, 701, 1117, 1124, 1249 Liu, K.S., 25, 26, 49, 334, 701, 903, 920, 1116, 1124, 1125, 1249, 1250 Mayes, S. J., 14, 25, 50, 490, 492, 503, 516, 517, 647, 653, 656, 663, 664, 678, 684, 689, 690, 696, 697, 699, 701, 1044, 1087, 1105, 1113, 1125, 1128, 1131, 1133, 1136, 1139, 1142, 1145, 1148, 1151, 1154, 1157, 1161, 1164, 1168, 1186, 1207, 1208, 1227, 1246, 1250, 1251 McCartney, N.L, 25, 49, 139, 157, 175, 176, 647, 656, 659, 689, 695–697, 700, 701, 810, 830, 831, 1092, 1108, 1114, 1120, 1123, 1124, 1128, 1131, 1133, 1136, 1139, 1142, 1145, 1148, 1151, 1154, 1157, 1161, 1165, 1168, 1188, 1245, 1246, 1249, 1250 Minnetyan, L., 24, 49, 425, 645, 700, 724, 725, 1109, 1123, 1124, 1249 Newill, J.F., 24, 50, 402, 426, 701, 961, 975, 1109, 1125, 1250 Puck, A., 14, 15, 25, 49, 193, 218, 222, 262, 264, 296, 297, 425, 430, 432–435, 445, 448, 449, 464, 474–477, 479–484, 489, 647, 653, 656, 671, 673, 677, 678, 683, 689, 690, 695, 697, 698, 700, 771, 772, 793, 795, 798, 801, 804, 808, 832, 833, 872, 873, 1020, 1087, 1098,

1105–1108, 1110, 1114, 1115, 1118, 1121, 1124, 1128, 1131, 1133, 1136, 1139, 1140, 1142, 1143, 1145, 1148, 1149, 1151, 1154, 1158, 1161, 1165, 1168, 1190, 1205, 1207, 1208, 1229, 1232–1236, 1238, 1247–1250

Rotem, A., 24, 25, 49, 298, 299, 312, 332, 425, 647, 653, 656, 663, 664, 667, 671, 674, 676–678, 683, 684, 689, 690, 694–697, 699–701, 877, 886, 888, 889, 1087, 1105, 1115, 1116, 1120, 1124, 1128, 1131, 1133, 1136, 1139, 1142, 1145, 1148, 1151, 1154, 1158, 1161, 1165, 1168, 1192, 1207, 1208, 1249, 1250

Schürmann, H., 25, 49, 264, 297, 700, 1114, 1124, 1249, 1250 Soden, P.D., 4, 24, 26, 30, 49, 50, 58, 60, 61, 63, 65–67, 78–80, 86, 91–93, 120, 139, 156, 175, 176, 217, 218, 262, 263, 297, 312, 333, 351, 378, 398, 399, 403, 411, 412, 424–426, 474, 516, 520, 521, 573, 574, 597, 598, 641, 642, 700, 701, 704, 725, 738, 768, 769, 804, 805, 831, 833, 872, 873, 889, 891, 902, 920, 921–923, 935, 936, 938, 940–942, 956, 959, 960, 962, 975, 1017, 1020, 1021, 1043, 1044, 1071, 1123–1125, 1248–1251 Sun, C.T., 25, 49, 50, 88, 93, 94, 316, 325, 332, 333, 406, 425, 426, 641, 645, 647, 656, 663, 664, 673, 678, 683, 684, 689, 690, 694–697, 700, 805, 890, 891, 900, 902, 975, 1087, 1092, 1105, 1108, 1115, 1116, 1120, 1124, 1125, 1128, 1131, 1133, 1136, 1139, 1142, 1145, 1148, 1149, 1152, 1154, 1158, 1162, 1165, 1169, 1195, 1197, 1207, 1208, 1245, 1249, 1250

Tairova L.R., 26, 49, 425, 701, 960, 1117, 1124, 1125, 1249, 1250 Tao, J.X., 25, 49, 316, 333, 425, 645, 700, 890, 891, 900, 902, 1115, 1116, 1124, 1125, 1249, 1250 Tsai, S. W., 25, 26, 49, 86, 92, 138, 139, 217, 334, 351, 378, 380, 399, 403, 425, 433, 445, 447, 464, 476, 496, 497, 498, 517, 573, 574, 591, 644, 647, 653, 656, 663, 671, 674, 676, 678, 683, 689, 690, 693–695, 697, 700, 701, 768, 804, 873, 903, 919–921, 1021, 1023, 1087, 1098, 1104, 1106, 1107, 1116, 1117,

1255 1119, 1121, 1124, 1125, 1128, 1131, 1133, 1137, 1139, 1143, 1146, 1149, 1152, 1155, 1158, 1162, 1165, 1169, 1199, 1207, 1208, 1229, 1232, 1236, 1243, 1246–1250

Wolfe, W.E., 26, 49, 353, 425, 573, 645, 647, 656, 662–665, 667, 671–674, 678, 683, 684, 689, 690, 696–699, 701, 922–928, 930–932, 936, 940, 941, 1087, 1102, 1105, 1117, 1124, 1125, 1128, 1131, 1134, 1137, 1139, 1143,

1146, 1149, 1152, 1155, 1159, 1162, 1166, 1169, 1202, 1207, 1208, 1249, 1250 Zinoviev, P., 26, 49, 94, 379, 380, 398, 399, 647, 656, 660, 672, 677, 678, 683, 684, 689, 690, 694, 695, 697, 698, 701, 943, 960, 1087, 1102, 1105, 1108, 1117, 1118, 1120, 1121, 1124, 1125, 1129, 1131, 1134, 1137, 1140, 1143, 1146, 1149, 1152, 1155, 1159, 1162, 1166, 1169, 1205, 1207, 1208, 1211, 1229, 1232–1237, 1247, 1249, 1250

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